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Limit theorems for the number of summands in integer partitions Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei, 115 Taiwan November 9, 2000
1991 AMS Mathematics subject classification: Primary 11P82; secondary 60F05 60F10. Key words: Integer partitions, central and local limit theorems, large deviations, Meinardus’s scheme, Mellin transform, Lerch’s zeta function, saddle-point method.
Running head: limit theorems for partitions Mailing address: Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei, 115 Taiwan e-mail: [email protected]
Abstract Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cram´er-type large deviations and are proved by Mellin transform and the two-dimensional saddle-point method. Applications of these results include partitions into positive integers, into powers of integers, into integers [j β ], β > 1, into aj + b, etc.
1
Introduction
Let Λ = {λ1 , λ2 , . . . } be a sequence of positive integers satisfying 1 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · and λn → +∞. There is an extensive literature on the asymptotics of the number of partitions of n into parts λj (see Andrews [1] and the references therein). In contrast, considerably fewer results have appeared in the literature on the limiting distribution of the number of summands (or parts) in random restricted (no part being repeated) or unrestricted partitions (repetition allowed). This paper is concerned with this aspect of the theory of partitions. Erd˝ os and Lehner [7] were the first to give a systematic study along this line in the case Λ = Z+ . They showed that the number of summands (counted with multiplicities) in a random unrestricted partition of n follows asymptotically (as n → ∞) an extreme-value distribution, a local version being later derived by Auluck et al. in [2]. Haselgrove and Temperley [15] extended, by a powerful analytic method, the result of Auluck et al. to more general Λ-partitions, their conditions on the given sequence being further extended, in some respects, only recently by Richmond [31]. Weaker results (convergence in distribution) under different analytic settings were derived by Lee [23] by the method of moments. A detailed study on the moments can be found in Richmond [28, 30]. It should be noted that the limiting distributions in these problems are all non-Gaussian. For many other extensions of the original problems, see for example [8, 9, 16, 24] and the references therein. When each part is allowed to appear at most once, Erd˝ os and Lehner [7] derived the asymptotic normality of the number of summands in the case Λ = Z+ (see also [9, 36, 37]). No extension of this result has appeared in the literature. In this paper, we consider a general analytic scheme essentially due to Meinardus (see [25] or [1, Ch. 6]) under which central and local limit theorems will be derived, thus extending Erd˝ os and Lehner’s result. The analytic conditions under which we are developing our arguments are weaker than those of Meinardus. Our analytic method can also be applied to the problem left open in [24, p. 311] concerning the common summands in restricted partitions. It turns out that the limiting law is Gaussian for a large class of partitions. There is another way of counting the number of summands in unrestricted partitions, namely, if the multiplicity of each part is counted only once. Unlike the corresponding counting function (i.e. the ω(n) function) in the theory of primes (see [38]), this problem is rarely discussed in the theory of partitions. It was first briefly mentioned in [7] in the case Λ = Z+ . Wilf [40] introduced the study of distinct components (or sizes of components) in general combinatorial structures. Then Goh and Schmutz [14] derived a central limit theorem for the number of summands for Λ = Z+ . The
1
latter result was then extended by Schmutz [34] to multivariate cases under Meinardus’s scheme. We further improve and extend their results by establishing the corresponding local limit theorem (in univariate case) under weaker conditions. An distinctive feature of integer partitions is that the limiting distribution of the number of summands is non-Gaussian in almost all cases if the multiplicity of each summand is taken into account (see [15, 23, 31]), in contrast to the ubiquitous normal law in a large class of combinatorial structures (see [12, 18]). Intuitively, the former phenomenon may be ascribed to the predominance of small summands when the number of summands becomes large, say, larger than the mean value. However, Gaussian limiting distribution appears if the parts are counted without multiplicity, this being intuitively clear since no single part can contribute preponderantly to the corresponding counting function, in accordance with the classical law of errors. Our results show that the same phenomenon still persists if each part is allowed to occur at most once. For completeness, we add that a formal approach was introduced in Knessel and Keller [21] for characterizing the asymptotic behaviors of many quantities in partition problems satisfying suitable recurrences. Another recent reference on related problems is Fristedt [13], the methods employed there being probabilistic. We state the main results of this paper in § 2. The proof of these results is divided into two parts: central (§ 3) and local (§ 4) limit theorems. In each section, we first derive some necessary estimates and then prove the result in question. Since our assumptions are weaker than those used in [25, 34], some techniques are introduced to justify the regularity conditions (in order to apply the saddle-point method). Unrestricted partitions is treated in § 5. Finally, we discuss some examples in § 6 and conclude with some remarks for further extensions.
2
Statement of results
Throughout this paper, the symbols cj always denote absolute positive constants. The symbol ε represents always suitable small quantity whose value may vary from one occurrence to another. Given a sequence of positive integers 1 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · tending to infinity, let Π(n) = ΠΛ (n) be the set of partitions of the positive integer n into distinct parts λj (each λj occurring at most once), j = 1, 2, . . . (in the case when there are more than one λi with the same value, one can think of properly labeling these λi ’s so that each is “different” from the other). Let q(n) = |Π(n)|, the cardinality of the set Π(n). It is more convenient to work with ak , denoting the number of λj ’s such
2
that λj = k. The generating function of q(n) satisfies Q(z) = QΛ (z) = 1 +
X
q(n)z n =
n≥1
Y
j≥1
Y ak 1 + z λj = 1 + zk , k≥1
for |z| < 1. To state our results, we first introduce an analytic scheme essentially due to Meinardus [25] in which the sequence {ak } satisfies the following three conditions. P (M1) The Dirichlet series D(s) = k≥1 ak k −s converges in the half-plane <e s > α > 0, and can be analytically continued into the half-plane <e s ≥ −α0 , for some α0 > 0. In <e s ≥ −α0 , D is analytic except for a simple pole at s = α with residue A1 . (M2) There exists an absolute constant c1 such that2 D(s) |t|c1 uniformly for <e s ≥ −α0 as |t| → +∞. (M3) Define g(τ ) =
−kτ , k≥1 ak e
P
where τ = r + iy with r > 0 and −π ≤ y ≤ π. There exists 2
a positive constant c2 such that g(r) − <e g(τ ) ≥ c2 (log(1/r))2+4/α uniformly for π/2 ≤ |y| ≤ π as r→ 0+ . The assumption (M3) here is much weaker than those used in [25] and [34], the essential difference being that we did not impose a similar estimate for g(r) − <e g(r + iy) in the region r ≤ |y| ≤ π/2, which is established by other assumptions, notably by the growth properties of the sum function P k≤X ak and by (M3). Introducing a uniform probability measure on the set Π(n), we consider the random variable $n , counting the number of summands in a random partition of n. The bivariate generating function of $n satisfies Q(u, z) = 1 +
X
q(n)E(u$n )z n =
n≥1
Y
1 + uz k
ak
,
(1)
k≥1
for finite u and |z| < 1, where E(u$n ) represents the probability generating function of $n . Set κ = AΓ(α)(1 − 2−α )ζ(α + 1), (1 − 21−α )ζ(α) nα/(α+1) , α(1 − 2−α )ζ(α + 1) (1 − 22−α )ζ(α − 1) (1 − 21−α )2 ζ(α)2 1/(α+1) = (κα) − nα/(α+1) . α(1 − 2−α )ζ(α + 1) (α + 1)(1 − 2−α )2 ζ(α + 1)2
µn = (κα)1/(α+1) σn2
Here Γ is the Gamma-function, ζ is Riemann’s zeta function and the factor (1 − 2−s )ζ(s + 1) is defined to be log 2 when s = 0. Note that σn > 0 as can be checked. 1 2
In Meinardus’s original paper, the quantity α0 is assumed to satisfy 0 < α0 < 1. The Vinogradov symbol is the same as the Landau symbol O(.) and is used interchangeably as is convenient.
3
Theorem 1. Suppose that the sequence {ak } satisfies (M1), (M2), and (M3). Set $n∗ = ($n − µn )/σn . Then the random variable $n is asymptotically normally distributed with mean E($n ) ∼ µn and variance Var($n ) ∼ σn2 : Pr{$n∗
1 < x} = √ 2π
x
Z
2 /2
e−t
dt + o(1),
(2)
−∞
uniformly for all x as n → ∞. Moreover, for sufficiently large n, we have the exponential bounds: ( 2 e−x /2 1 + O (log n)−1 , if 0 ≤ x ≤ nα/(6α+6) / log n; ∗ Pr{$n ≥ x} ≤ (3) α/(6α+6) x/(2 log n) e−n 1 + O (log n)−1 , if x ≥ nα/(6α+6) /(log n), and the same inequalities for Pr{$n∗ ≤ −x}. The method of proof consists of analytic and probabilistic parts: the analytic part is based on Mellin transform and the saddle-point method; and the probabilistic part utilizes Curtiss’s theorem [4] for convergence of moment generating functions. It turns out that Lerch’s zeta function (see [6, P §1.11]) Φ(z, s, v) = n≥0 z n (n + v)−s intervenes in a natural way in our analysis. Our application of the saddle-point method differs from that in [25] and [1, Ch. 6] and yields a better error term. We complete the asymptotic normality of $n by its strong concentration property (3) using a simple technique (see [27, Ch. III]) amended from the usual Chernoff bound. We can also derive a local limit theorem in the form of Cram´er-type large deviations (see [20] [27]). It suffices to replace condition (M3) by the following stronger one. (M3’) There exists a fixed constant c3 > 0 such that g(r) − <e eiθ g(r + iy) ≥ c3 (log(1/r))2+4/α
2
uniformly for π/2 ≤ |y| ≤ π and −π ≤ θ ≤ π, as r→ 0+ . Let Y (u, s) be the Mellin transform of the function log(1 + ue−x ): Z ∞ Y (u, s) = xs−1 log(1 + ue−x ) dx for <e s > 0.
(4)
0
As we will see, Y is essentially Lerch’s zeta function. Theorem 2. Assume that the sequence {ak } satisfies (M1), (M2) and (M3’). If m = µn +xσn ∈ Z+ , where x = o nα/(2α+2) , then Pr{$n = m} =
e−x
2 /2+ξ(x/σ
√
α/(α+1) n )n
2π σn
1+O
uniformly in x, where α2 = min{1, α0 , α} and ξ(w) =
|x| nα/(2α+2)
P
j≥3 ξj w
origin whose Taylor coefficients satisfy 0 −1 k−2 00 U (w) − U 0 (0) −k ξk = [w ]U (w) k U 00 (0)w 4
j
+n
−α2 /(α+1)
,
(5)
is analytic in a neighborhood of the
for k = 3, 4, 5, . . .
(6)
with U (w) = (α + 1)α−α/(α+1) A1/(α+1) Y (ew , α)1/(α+1) − Y (1, α)1/(α+1) . Here the symbol [z n ]f (z) denotes the coefficient of z n in the Taylor expansion of f (z). Note that U is convex due to the same property of Y and that µn = U 0 (0)nα/(α+1) and σn2 = U 00 (0)nα/(α+1) . The first two terms of ξk are given by (see [20]) U 000 (0)2 (4) 1 000 1 ξ3 = 6 U (0) and ξ4 = 24 U (0) − 00 . U (0) As an interesting consequence, we state the following Corollary 1. If m = µn + xσn ∈ Z+ , where x = o nα/(6α+6) , then 2
e−x /2 Pr{$n = m} = √ 2π σn
1+O
|x| + |x|3 + n−α2 /(α+1) nα/(2α+2)
,
(7)
uniformly in x. The proof of this theorem utilizes essentially the two-dimensional saddle-point method and is technically more involved. As is usual in the application of the saddle-point method, it is the verification of the regularity conditions to which much of our analysis is devoted. Actually, we prove more (see Proposition 2 below) but content ourselves with the statement of the theorem. Our methods can also be applied to the number of distinct parts in unrestricted partitions (repetition allowed) under the same assumptions (M1)–(M3) as in Theorem 1. e Let Π(n) represent the set of unrestricted partitions of n and let p(n) be its cardinality. Let
ωn be the number of distinct parts (i.e., counted without multiplicities) in a random partition of n, where all p(n) partitions of n are equally likely. The bivariate generating function of ωn satisfies P (u, z) = 1 +
X
ωn
n
p(n)E(u )z =
n≥1
Y
k≥1
uz k 1+ 1 − zk
a k
,
for |z| < 1. Set κ1 = AΓ(α)ζ(α + 1), (κ1 α)1/(α+1) α/(α+1) µ en = AΓ(α)(κ1 α)−α/(α+1) nα/(α+1) = n , αζ(α + 1) α (κ1 α)1/(α+1) 2 −α σ en = 1−2 − nα/(α+1) . αζ(α + 1) (α + 1)ζ(α + 1) 5
(8)
Define Z(u, s) =
Z
∞
0
u log 1 + x e −1
xs−1 dx for <e s > 0 and | arg u| < π.
It is obvious, by (4), that Z(u, s) = Γ(s)ζ(s + 1) + Y (u − 1, s). Theorem 3. Under the assumptions (M1), (M2), and (M3), the random variable ωn satisfies asymptotically E(ωn ) ∼ µ en , Var(ωn ) ∼ σ en2 , and Pr{ωn = µ en + xe σn } =
e−x
2 /2+η(x/e µn )nα/(α+1)
√
2π σ en
1+O
|x| nα/(2α+2)
+n
−α2 /(α+1)
,
uniformly for all x = o nα/(2α+2) such that µ en + xe σn ∈ Z+ . Here α2 is as in Theorem 2 and P η(w) = j≥3 ηj wj is analytic at the origin with coefficients given by −1 k−2 00 ηk = [w ]V (w) k
V 0 (w) − V 0 (0) V 00 (0)w
−k
for k = 3, 4, 5, . . . ,
where V (w) = (α + 1)α−α/(α+1) A1/(α+1) Z(ew , α)1/(α+1) − Z(1, α)1/(α+1) . As the proof of this theorem parallels that of Theorems 1 and 2, only the necessary regularity conditions is worked out in § 5. That the assumptions needed for the local limit theorem of ωn are weaker than those for $n is seen by the following example. Take λj = 2j − 1. Then it is obvious that the span of the random variable $n is 2 whereas that of ωn is 1. More precisely, E(u$n ) contains only odd (respectively even) powers of u for odd (respectively even) n. In this case, local limit theorem of $n depends on the parity of n.
3
Central limit theorem
3.1
Lemmas
In this section, we establish some estimates for the function Q(u, e−τ ) (defined in (1)) as τ → 0. We write consistently the complex variable τ in the form τ = r + iy with −π ≤ y ≤ π and r > 0. These estimates are slightly more general than our need for the proof of Theorem 1 since some of them will be required when establishing the corresponding local limit theorem. Let f (u, τ ) = log Q(u, e−τ ): f (u, τ ) =
X
ak log 1 + ue−kτ .
k≥1
6
The sum on the right-hand side being a harmonic sum (see [10]), we have available the Mellin inversion formula: 1 f (u, τ ) = 2πi
Z
α+1+i∞
D(s)Y (u, s)τ −s ds,
(9)
α+1−i∞
for <e τ > 0, where Y (u, s) is the Mellin transform of the function log(1 + ue−x ); see (4). Note that, for |u| ≤ 1 and <e s > 0, Y (u, s) satisfies Y (u, s) = Γ(s)
X (−1)j−1 j≥1
j s+1
uj ,
(10)
a representation no longer useful when |u| > 1. In particular, Y (1, s) = (1 − 2−s )ζ(s + 1)Γ(s), so that κ = AY (1, α). Now by integration by parts, we see that Y (u, s) is related to the Lerch zeta function Φ(z, s, v) by: Y (u, s) = uΓ(s)Φ(−u, s + 1, 1), with Φ defined by (see [6]) Φ(z, s, v) =
X k≥0
zk (k + v)s
for |z| < 1, s ∈ C, and v 6= 0, −1, −2, . . .
Analytic properties of Y (u, s) are summarized in the following lemma. Lemma 1. For each fixed u lying in the cut-plane C \ (−∞, −1], the Mellin transform Y (u, s) can be meromorphically continued into the whole s-plane with simple poles at s = 0, −1, −2, . . . Moreover, Y (u, s) satisfies the estimate |Y (u, σ + it)| e−(π/2−ε)|t|
for any ε > 0 as |t| → +∞,
(11)
uniformly for finite σ and u in the cut-plane. Proof. For completeness, we sketch here a self-contained proof. For further properties, see Erd´elyi [6]. First, by integration by parts, we have u Y (u, s) = s
Z 0
7
∞
xs dx, ex + u
the right-hand side providing a meromorphic continuation of Y to the half-plane <e s > −1. The first assertion of the lemma follows from repeating the same process. As to (11), since log(1 + ue−x ) is an analytic function of x in the half-plane <e x > 0, we have by Cauchy’s theorem Z eiϕ ∞ Y (u, s) = log(1 + ue−x )xs−1 dx for any |ϕ| ≤ π/2 − ε. 0
Thus a change of variable yields Y (u, s) = eiϕs
Z
∞
0
iϕ log 1 + ue−e t ts−1 dt,
from which (11) follows. Remark 1. Since for x ∼ 0 log(1 + ue−x ) = log(1 + u) +
X h≥1
X uxh A(h − 1, j)(−1)h−1−j uj , h!(1 + u)h 0≤j 0. Thus α+` ≤ E1 + E2 , FL−1 (X) − AX (α + `)L−1 9
where E1 E2
FL (X + δX) − FL (X) AX α+` − , = L log(1 + δ) (α + `)L−1 Z X+δX 1 FL−1 (t) − FL−1 (X) = dt. log(1 + δ) X t
Since FL−1 (t) is non-decreasing (the ak being ≥ 0), we have E2 ≤ =
−1 Z X Z X+δX 1 FL−1 (t) 1 FL−1 (t) dt − log dt log(1 + δ) X t 1−δ t X−δX 1 (FL (X + δX) − 2FL (X) + FL (X − δX)) δ + O (max {FL (X + δX) − FL (X), FL (X) − FL (X − δX)}) ,
as δ ∼ 0. From (15) and the estimates (1 + δ)α+` − 1 = (α + `)δ + O(δ 2 ), as δ ∼ 0, it follows that E1 + E2 δX α+` +
RL (X) . δ
Taking δ = X −(α+`)/2 RL (X)1/2 (→ 0+ ) so as to balance the two error terms on the right-hand side, we obtain FL−1 (X) =
AX α+` + O (RL−1 (X)) , (α + `)L−1
where RL−1 (X) = X α/2 (log X)L/2 + X `+(α−α0 )/2 . Repeating the same process, we see that for j = 1, 2, 3, . . . , L FL−j (X) =
AX α+` + O (RL−j (X)) , (α + `)L−j
where RL−j (X) = X (1−2
−j )α
j
(log X)L/2 + X `+(1−2
−j )α−α
0 /2
j
.
This completes the proof. According to (M2), the number c1 (and thus L) depends on the value of α0 . One may choose a suitable α0 (if possible) so that the error terms in (14) are minimized. An interesting consequence of this lemma is the following
10
Corollary 2. As n → +∞, max $n ∼ α−1 (α + 1)α/(α+1) A1/(α+1) nα/(α+1) ,
(16)
where the max is over all distinct partitions of n. Proof. For, by Lemma 3, if n=
X
kak ∼
1≤k≤X
A X α+1 , α+1
then X ∼ ((α + 1)n/A)1/(α+1) ; consequently, max $n =
X
ak ∼
1≤k≤X
A α X , α
from which we obtain (16). Lemma 4. Let u be a positive real number. There exists a constant c5 > 0 such that the inequality |Q(u, e−r−iy )| c5 u 2 ≤ exp − (log(1/r)) (17) Q(u, e−r ) (1 + u)2 holds uniformly for r1+3α/7 ≤ |y| ≤ π as r→ 0+ . Proof. To start with, we observe that |Q(u, e−r−iy )| Q(u, e−r )
ak /2 2ue−kr = 1− (1 − cos ky) (1 + ue−kr )2 k≥1 X u ≤ exp − ak e−kr (1 − cos ky) . (1 + u)2 Y
(18)
k≥1
In view of the assumption (M3), it suffices to show that G(r) := g(r) − <e g(r + iy) ≥ c6 (log(1/r))2 ,
(19)
for r1+3α/7 ≤ |y| ≤ π/2, as r→ 0+ . Consider first the case r ≤ |y| ≤ (log(1/r))−2/α . Using the elementary inequality 1 − cos t ≥
2 2 t π2
for |t| ≤ π,
(20)
we have X
G(r) >
ak e−kr (1 − cos ky) ≥
1≤k≤1/|y|
≥
2Ae−1 π 2 (α + 2)
2 2 −r/|y| y e π2
− ε |y|−α ≥ c6 (log(1/r))2 11
X
k 2 ak
1≤k≤1/|y|
as r→ 0+ ,
by (13). Next, if r1+3α/7 ≤ |y| ≤ r, then G(r) >
X
ak e−kr (1 − cos ky) ≥
2 2 −1 y e π2
≥
π 2 (α + 2)
k 2 ak
1≤k≤1/r
1≤k≤1/r
Ae−1 π α+2
X
y 2 r−α−2 ≥ c7 r−α/7 ≥ c6 (log(1/r))2
as r→ 0+ .
Finally, if y lies in the range 0 < |y| ≤ π/2, then there exists an integer ` such that π ≤ 2` |y| ≤ π. 2 From the elementary inequality 1 − cos θ ≥ 4−` 1 − cos 2` θ , and consequently G(r) ≥ 4−`
X
1 4
(21)
(1 − cos 2θ), we obtain by induction 1 − cos θ ≥
ak e−kr 1 − cos 2` ky
k≥1
2 2 1 3+4/α c2 1 3+4/α ≥ c2 4−` log ≥ 2 y 2 log , r π r
in virtue of (M3) and (21). Thus c2 G(r) ≥ 2 π
1 log r
2
,
for y satisfying (log(1/r))−2/α ≤ |y| ≤ π/2. Taking c5 = min{c2 /π 2 , c6 }, (17) follows.
3.2
The proof of Theorem 1
Throughout this section, u is a positive real number which eventually will be taken to be near 1. Proposition 1. Let δ > 0 be any fixed number in the unit interval. Then we have, uniformly for δ ≤ u ≤ δ −1 , α/(α+1)
Qn (u) = β(u)n−(1+α/2)/(α+1) e(1+1/α)K(u)n
1 + Oδ n−α2 /(α+1) ,
(22)
the O-term holding uniformly in u, where α2 = min{α, α0 , 1}, K(u) = (αAY (u, α))1/(α+1) and s 1/(2α+2) (αAY (u, α)) K(u) p β(u) = (1 + u)D(0) = (1 + u)D(0) . 2π(α + 1) 2π(α + 1) Proof. By Cauchy’s integral formula, Z enr π iny e Q(u, e−r−iy ) dy = enr (I1 + I2 ) , Qn (u) = 2π −π 12
(23)
where I1 and I2 represent the integrals (2π)−1
and (2π)−1
R
|y|≤r 1+3α/7
R
, respectively. Here
r 1+3α/7 0.
(24)
We assume that n is sufficiently large so that r1+3α/7 < π. Consider first I2 , which is bounded above by 2
I2 Q(u, e−r )e−c9 (log(1/r)) eAY (u,α)r
−α −c
10 (log n)
2
,
(25)
in view of (17) and (24). For I1 we have by (12) and a change of variables I1 = (1 + u)D(0)
r 2π
r 3α/7
Z
einry+AY (u,α)r
−α (1+iy)−α
(1 + O (rα1 )) dy
−r 3α/7
= I3 + I4 , where I3 = (1 + u)D(0)
r 2π
Z
r3α/7
einry+AY (u,α)r
−α (1+iy)−α
dy,
−r3α/7
and I4 r
1+α1
Z
r 3α/7
eAY (u,α)r
−α (1+y 2 )−α/2
dy.
−r 3α/7
Using the elementary inequality (1 + y 2 )−α/2 ≤ 1 − (1 − 2−α/2 )y 2
for − 1 ≤ y ≤ 1,
(26)
we obtain I4 r
1+α1 AY (u,α)r−α
e
Z
r−α/7
e−(1−2
−α/2 )AY
(u,α)r−α y 2
dy
−r−α/7
r1+α1 +α/2 eAY (u,α)r It remains to evaluate I3 . Setting B =
−α
.
(27)
p α(α + 1)AY (u, α) and making the change of variables
v 2 = B 2 r−α y 2 , we obtain D(0) r
I3 = (1 + u)
1+α/2 eAY (u,α)r−α
2πB
13
Z
Br−α/7
−Br−α/7
e−v
2 /2
Tr (v) dv,
where 2 X α + j + 1 −iv j v rαj/2 Tr (v) = exp − α(α + 1) j+2 B
j≥1
(α + 2)iv 3 = 1+ p rα/2 + O(v 4 rα ), 6 α(α + 1)AY (u, α) from which we deduce that r1+α/2 eAY (u,α)r
D(0)
I3 = (1 + u)
−α
p
2πα(α + 1)AY (u, α)
(1 + O (rα )) .
(28)
Collecting our results (23)–(28) yields (1 + u)D(0) r1+α/2 enr+AY (u,α)r p Qn (u) = 2πα(α + 1)AY (u, α)
−α
(1 + O (rα2 )) ,
uniformly in u. The relation (22) follows from the above formula using (24). Proof of Theorem 1. (Asymptotic normality) Let Mn (t) = E(e($n −µn )t/σn ), where t is real. Then by (22) Qn (et/σn ) Qn (1) !D(0) !1/(2α+2) et/σn + 1 Y (et/σn , α) eφn (t) 1 + O n−α2 /(α+1) , 2 Y (1, α)
Mn (t) = e−µn t/σn = uniformly in t, where
µn t + 1 + α−1 K(et/σn ) − K(1) nα/(α+1) σn µn t = − + U (t/σn )nα/(α+1) . σn
φn (t) = −
Observe that for <e s > 0 t/σn
Y (e
, s) = Y (1, s) +
Z
∞
0
et/σn − 1 log 1 + x e +1
!
xs−1 dx,
and that as n → +∞ Z 0
∞
et/σn − 1 log 1 + x e +1
!
s−1
x
t t2 dx = h1 + h2 2 + O σn σn
where, in general, (−1)k−1 hk = k
Z
∞
0
14
xs−1 dx. (ex + 1)k
|t|3 σn3
,
Note that hk =
Γ(s) X s(k, j)(1 − 2j−s )ζ(s + 1 − j) k!
for k = 1, 2, 3, . . . ,
(29)
1≤j≤k
where the s(k, j) represent the (signed) Stirling numbers of the first kind (see [3, Ch. 5]). Thus, we have Mn (t) = eφn (t) 1 + O n− min{α/2,α0 ,1}/(α+1) + |t|n−α/(2α+2) , and t/σn
K(e
K(1) ) − K(1) = (α + 1)Y (1, α)
2 h1 t αh21 t 3 −3 + 2h2 − + O |t| σn . σn (α + 1)Y (1, α) 2σn2
Equivalently, this last relation can be written as 3 t U 00 (0)t2 |t| t 0 = U (0) + +O . U 2 σn σn 2σn σ3 From these formulae and the relations (by (29)) µn n
−α/(α+1)
K(1)h1 = αY (1, α)
and
σn2 n−α/(α+1)
= K(1)
2h2 h21 − αY (1, α) (α + 1)Y (1, α)2
,
it follows that 1 + O n−α2 /(α+1) + (|t| + |t|3 )n−α/(2α+2) 2 = et /2 1 + O n− min{α/2,α0 ,1}/(α+1) , 2 /2
Mn (t) = et
(30)
uniformly in t. By the theorem of Curtiss in [4], we conclude that the distribution of the random variable $n is asymptotically Gaussian. (Exponential tails) As to the exponential bounds (3), we observe from the above derivations that (30) remains valid if |t| tends to infinity slowly enough: t = o nα/(6α+6) . We consider only the case when $n∗ ≥ x in the following, the other case −$n∗ ≥ x being similar. From (30), we have for x ≥ 0 ∗
Pr{$n∗ ≥ x} = Pr{e$n t ≥ etx } ≤ e−tx Mn (t) 2 = e−tx+t /2 1 + O n−α2 /(α+1) + (|t| + |t|3 )n−α/(2α+2) .
(31)
Let T be any positive quantity tending to infinity with n and satisfying T = o(nα/(6α+6) ). If 0 ≤ x ≤ T then we take (see [27, Ch. III]) t = x in (31) (so as to minimize −tx + t2 /2) and we obtain Pr{$n∗ ≥ x} ≤ e−x
2 /2
1 + O n−α2 /(α+1) + |T |3 n−α/(2α+2) ; 15
and if x ≥ T we have by taking t = T : Pr{$n∗ ≥ x} ≤ e−T x/2 1 + O n−α2 /(α+1) + |T |3 n−α/(2α+2) . Now the estimates (3) follow from choosing T = nα/(6α+6) / log n. (Mean and variance) We still have to prove that the mean and the variance of $n are asymptotic to µn and σn2 , respectively, a result that is not guaranteed by convergence in distribution. Although we may directly evaluate Qu (1, z) and Q00uu (1, z) as in the proof of Proposition 1, the asymptotic form of the variance depends on the values of α and α0 . The following arguments are computationally simpler and are not subject to the values of α. Note that Fn (x), the distribution function of $n∗ , converges pointwise to the standard normal distribution whose mean and variance are 0 and 1, respectively. It suffices to show that E $n∗ = o(1) and Var($n∗ ) = 1 + o(1). For the former we use the representation Z ∞ ∗ E($n ) = (1 − Fn (x) − Fn (−x)) dx, 0
the uniform bounds (3) and Lebesgue’s dominated convergence theorem. The latter follows from Z ∞ E($n∗2 ) = 2x (1 − Fn (x) − Fn (−x)) dx, 0
and similar considerations. Remark 2. A further refinement of the above arguments leads to the better estimates: D(0) (1 − 21−α )ζ(α) + , 2 2(α + 1)(1 − 2−α )ζ(α + 1) D(0) ζ(α − 1)(2α − 4) Var($n ) ∼ σn2 + + 4 (α + 1)ζ(α + 1)(2α − 1) ζ(α)(2α − 2) ζ(α)2 (2α − 2)2 − − , (α + 1)ζ(α + 1)(2α − 1) (α + 1)ζ(α + 1)2 (2α − 1)2 E($n ) ∼ µn +
(32)
(33)
where the convention that (1 − 2−s )ζ(s + 1) = log 2 when s = 0 is assumed.
4
Local limit theorem
4.1
Lemmas
The local behavior of Q(u, e−τ ) as τ → 0 and u → 1 having been made explicit in Lemma 2, we need only consider other ranges of u and τ . As in the last section, we write consistently u = ρeiθ = e%+iθ
and z = e−τ = e−r−iy , 16
where ρ, r > 0, % ∈ R, and −π ≤ θ, y ≤ π. Define Gθ (r) = g(r) − <e eiθ g(r + iy) =
X
ak e−kr (1 − cos(θ − ky)) ,
k≥1
so that G0 (r) = G(r). Then we have, by the derivations for (18), |Q(ρeiθ , e−r−iy )| ρGθ (r) ≤ exp − . Q(ρ, e−r ) (1 + ρ)2 Lemma 5. There exists a positive constant c13 such that the inequality ! |Q(ρeiθ , e−r−iy )| c13 ρ 1 2 < exp − log Q(ρ, e−r ) (1 + ρ)2 r holds uniformly for r1+3α/7 ≤ |y| ≤ π and −π ≤ θ ≤ π, as r→ 0+ . Proof. Consider first the case when r ≤ |y| ≤ (log(1/r))−2/α . If |θ| ≤ X
Gθ (r) >
1 2
then
ak e−kr (1 − cos(θ − ky))
1/|y|≤k≤2/|y|
1 − cos( 12 ) e−2
>
−α
> c14 |y|
≥ c13
X
ak
1/|y|≤k≤2/|y|
1 log r
2
,
by (13) with ` = 0. Next, if
1 2
≤ |θ| ≤ π then X
Gθ (r) >
ak e−kr (1 − cos(θ − ky))
1/(8|y|)≤k≤1/(4|y|)
>
1 − cos( 41 ) e−1/4
X
ak
1/(8|y|)≤k≤1/(4|y|)
1 2 > c15 |y|−α ≥ c13 log , r again by Lemma 3. Now consider the case r1+3α/7 ≤ |y| ≤ r and −π ≤ θ ≤ π. We have X
Gθ (r) >
ak e−kr (1 − cos(θ − ky))
1/r≤k≤2/r
>
1 − cos(r3α/7 ) e−2
X
1/r≤k≤2/r
> c16 r
−α/7
> c13 17
1 log r
2
,
ak
by the inequality (20) and Lemma 3. For the remaining ranges (log(1/r))−2/α ≤ |y| ≤ π, it suffices, by (M3’), to consider the case (log(1/r))−2/α ≤ |y| < π/2 for which we use the same argument as in the proof of Lemma 4. For 0 < |y| < π/2 there exists an integer ` such that π/2 ≤ 2` |y| ≤ π. Using the inequality 1 − cos t ≥ 4−` (1 − cos t) , we obtain Gθ (r) > 4−`
X
ak e−kr 1 − cos 2` θ − 2` ky .
k≥1
Choose integer k such that 2` θ = 2kπ + θ0 where −π ≤ θ0 ≤ π. Thus by (M3’) Gθ (r) ≥ 4−` c3 (log(1/r))2+4/α c3 ≥ (log(1/r))2 , π2
2
≥
c3 2 2 y (log(1/r))2+4/α π2
for (log(1/r))−2/α ≤ |y| < π/2. This completes the proof. Lemma 6. For r3α/7 ≤ |θ| ≤ π and |y| ≤ r1+3α/7 , the estimate |Q(ρeiθ , e−r−iy )| c17 ρ −α/7 < exp − r Q(ρ, e−r ) (1 + ρ)2 holds uniformly in θ and y as r→ 0+ . Proof. We have X
Gθ (r) >
ak e−kr (1 − cos(θ − ky))
1/(3r)≤k≤1/(2r)
>
1 − cos( 12 r3α/7 ) e−1/2
X
ak
1/(3r)≤k≤1/(2r)
> c18 r−α/7 , by the inequality (20) and Lemma 3. We also need the asymptotic behaviors of Y (u, α) as u → ∞ and u→ 0, which are described by the following lemma. Lemma 7. The function Y (u, α) satisfies Y (u, α) = Yu0 (u, α) =
(log u)α+1 1 + O (log u)−2 , α(α + 1) (log u)α 1 + O (log u)−2 , αu
as |u| → +∞ in the sector | arg u| ≤ π − ε. 18
(34) (35)
Proof. By the integral representation u Y (u, α) = α
Z
∞
0
xα dx, ex + u
and the Mellin inversion formula 1 1 = 1+x 2πi
Z
1/2+i∞
1/2−i∞
π x−w dw sin πw
(| arg x| ≤ π − ε),
we obtain Y (u, α) = uΓ(α)
1 2πi
Z
1/2+i∞
1/2−i∞
πu−w dw. (1 − w)α+1 sin πw
Deforming the path of integration into a suitable Hankel-type contour in the style of [11], we deduce the result (34). The formula (35) is derived either in a completely analogous manner or by (34) using Ritt’s theorem (see [26, pp. 9–11]). Corollary 3. The function U (w) satisfies −1 α (α + 1)α/(α+1) A1/(α+1) 1 + w−2 , as w → +∞; 0 U (w) = α−α/(α+1) (AΓ(α))1/(α+1) ew/(α+1) (1 + O (ew )) , as w → −∞.
(36)
Proof. These formulae follow from (35), (10) and the definition of U . The limiting value of U 0 (w) as w → +∞ is a natural one in view of Corollary 2 and the next lemma. We next consider the solution to the system n = αAY (e% , α)r−α−1 m = Ae% Yu0 (e% , α)r−α ,
(37)
which will be needed when applying the two-dimensional saddle-point method. For convenience, set M0 = α−1 (α + 1)α/(α+1) A1/(α+1)
and M1 = α−α/(α+1) (AΓ(α))1/(α+1) .
Lemma 8. For m lying in the range 1 ≤ m ≤ (M0 − ε)nα/(α+1) , there exists a unique solution (%, r) to the system (37) such that r > 0 and % ∈ R. Proof. The solution to the first equation of (37) exists for all finite (and real) % and satisfies r=
αAY (e% , α) n 19
1/(α+1)
> 0.
(38)
Substituting this expression into the second equation of (37) yields m = Ae% Yu0 (e% , α) (αAY (e% , α))−α/(α+1) nα/(α+1) = U 0 (%)nα/(α+1) .
(39)
Thus there exists a unique real solution to (37) whenever m lies in the range (38). Moreover, if ε→ 0+ then by (36) %ε
−1/2
→ +∞
and r = %
A (α + 1)n
1/(α+1)
1 + O %−2
.
On the other hand, if m = o(nα/(α+1) ) then by (36) m m % = (α + 1) log 1 + O , M1 nα/(α+1) nα/(α+1) m αm 1+O . r = n nα/(α+1) In this case, we have % → −∞ and r → 0. Corollary 4. If m = µn + xσn , where x = o(σn ), then the solution (%, r) satisfies X x j X x j , %= %j and r = (αA)1/(α+1) n−1/(α+1) Y (1, α)1/(α+1) + rj σn σn j≥1
j≥1
with
U 0 (w) − U 0 (0) U 00 (0)w
−m
%m =
1 m−1 [w ] m
rm =
1 [wm−1 ]ew Yu0 (ew , α)Y (ew , α)−α/(α+1) m(α + 1)
,
U 0 (w) − U 0 (0) U 00 (0)w
−m
,
for m = 1, 2, 3, . . . The series are convergent. Proof. The relation q m = µn + xσn = U 0 (0)nα/(α+1) + x U 00 (0)nα/(α+1)
(x = o(σn ))
can be written into the more convenient form U 0 (%) − U 0 (0) x = , 00 U (0) σn in view of (39). Thus the solution (%, r) satisfies (40) by the Lagrange inversion formula.
20
(40)
4.2
The proof of Theorem 2
Let q(n, m) denote the number of restricted partitions of n having exactly m parts: q(n, m) = [um z n ]Q(u, z). Proposition 2. If m lies in the range m nα/(α+1)
m ≤ (M0 − ε)nα/(α+1) ,
and
then q(n, m) satisfies (1 + e% )D(0) 1+α −m%+nr+AY (e% ,α)r−α r e (1 + O (rα2 )) , 2πBb p where (%, r) is the unique real solution to the system (37), B = α(α + 1)AY (e% , α) and s αAe2% 00 (e% , α) − b = Ae% Yu0 (e% , α) + Ae2% Yuu Y 0 (e% , α)2 . (α + 1)Y (e% , α) u q(n, m) =
(41)
Proof. We use Cauchy’s integral formula q(n, m) = =
I I 1 u−m−1 z −n−1 Q(u, z) dz du (2πi)2 q(n) |u|=ρ |z|=e−r Z Z e−m%+nr π π −imθ+iny e Q(e%+iθ , e−r−iy ) dy dθ, 4π 2 q(n) −π −π
where (%, r) is chosen to satisfy the system (37). Set r0 = r3α/7 . The ranges of integration are split into three parts: (I)
|θ| ≤ r0 ,
|y| ≤ rr0
(II)
r0 < |θ| ≤ π,
(III)
|θ| ≤ π,
|y| ≤ rr0
rr0 < |y| ≤ π.
By Lemmas 5 and 6, we have ZZ
+
ZZ
(II)
(III)
c19 e% exp − (1 + e% )2
It remains to evaluate the integral J
re−m%+nr := 4π 2 =: J1 + J2 ,
Z
r0
−r0
Z
r0
−r0
RR
(I) .
1 + e%+iθ
1 log r
2 !
(c19 = min{c13 , c17 }).
(42)
By Lemma 2 and a change of variables (α1 = min{1, α0 })
D(0)
%+iθ ,α)(1+iy)−α r −α
e−imθ+inry+AY (e
21
(1 + O (rα1 )) dy dθ
say, where J1 corresponds to the main term in the integrand and Z r0 Z r0 %+iθ −α −α −m%+nr 1+α1 J2 e r eA|Y (e ,α)||1+iy| r dy dθ 0 −r0 Z−r r0 Z r0 %+iθ −α/2 )y 2 )r −α eA|Y (e ,α)|(1−(1−2 dy dθ, ≤ e−m%+nr r1+α1 −r0
−r0
where the inequality (26) was used. By the choice of r0 , we have r02 r−α → +∞. Thus Z r0 %+iθ −α −m%+nr 1+α1 +α/2 J2 e r eA|Y (e ,α)|r dθ. −r0
From the local expansion Y (e%+iθ , α) = Y (e% , α) + iθ
∞
Z 0
as θ ∼ 0, we deduce that |Y
(e%+iθ , α)|
≤Y
e% xα−1 θ2 dx − x % e +e 2 (e% , α)
− c20
θ2
∞
Z 0
e%+x xα−1 dx + O |θ|3 , x % 2 (e + e )
as r→ 0+ . It follows that % ,α)r −α
J2 e−m%+nr r1+α1 +α eAY (e
.
(43)
We now concentrate on the principal part J1 for which we introduce the following abbreviations: Yθ = Y (e%+iθ , α),
Y0 = Y (e% , α),
Y00 = Yu0 (e% , α),
00 Y000 = Yuu (e% , α),
000 Y0000 = Yuuu (e% , α).
Consider first the inner integral of J1 : J3 = Setting B =
eAYθ r 2π
−α
Z
r0
einry+AYθ ((1+iy)
−α −1
)r−α dy.
−r0
p
α(α + 1)AY0 and carrying out the change of variables y = rα/2 v/B, we obtain
rα/2 eAYθ r J3 = 2πB
−α
Br−α/7
Yθ v 2 (α + 2)Yθ 3 α/2 exp Υiv − + iv r + O rα v4 2Y0 6BY0 −Br−α/7
Z
dv,
where Υ = αAr −α/2 (Y0 − Yθ )/B. Since v 3 rα/2 , v 4 rα → 0 in the range of integration, we have Yθ v 2 (α + 2)Yθ 3 α/2 2 α 4 exp Υiv − + iv r +O r v = eΥiv−v /2 (1 + R1 ) , 2Y0 6BY0 where R1 = −
Y00 % 2 (α + 2)Yθ 3 α/2 e iθv + iv r + O θ2 v2 + θ2 v4 . 2Y0 6BY0
Substituting this estimate into J3 , we obtain −α Z Br −α/7 rα/2 eAYθ r 2 J3 = eΥiv−v /2 (1 + R1 ) dv 2πB −Br−α/7 −α Z ∞ α/2 AY r 1 2 −2α/7 r e θ 2 −α eΥiv−v /2 (1 + R1 ) dv + O r9α/14 eAr <e Yθ − 2 B r = . 2πB −∞ 22
The integral on the right-hand side can be evaluated by Cauchy’s theorem: Z ∞ 2 e−Υ /2 X 1 L (2`)! Υiv−v 2 /2 L e v dv = √ (iΥ)L−2` , 2π −∞ 2` 2` `! 2π 0≤2`≤L
for any L = 0, 1, 2, . . . Thus −α
rα/2 eAYθ r −Υ √ 2π B
J3 =
2 /2
(1 + R2 ) + O(R3 ),
where Y00 e% (α + 2)Yθ 3 iθ(Υ2 − 1) + (Υ − 3Υ)rα/2 , 2Y0 6BY0 −α 2 −α 2 −2α/7 /2 = θ2 rα/2 |Υ|2 + |Υ|4 eAr <e Yθ −<e Υ /2 + r9α/14 eAr <e Yθ −B r ,
R2 = R3
the error term being meaningful as long as Υ=
αA −α/2 r (Y0 − Yθ ) r−α/2 |θ| r−α/7 , B
which is obviously satisfied when |θ| ≤ r0 . Returning to J1 , we have r1+α/2 e−m%+nr √ J1 = 2π 2π B
Z
r0
−r0
1 + e%+iθ
D(0)
eimθ+AYθ r
−α −Υ2 /2
(1 + R2 ) + O(R3 ) dθ.
Using the expansion eimθ+AYθ r
−α −Υ2 /2
= eAY0 r
−α −bθ 2 r −α /2
(1 + R4 ) ,
where iAe% θ3 −α 3α2 A % 0 2 3α2 A 2% 0 00 0 % 00 2% 000 R4 = − r Y0 + 3e Y0 + e Y0 − e Y0 − e Y0 Y0 + O r−α θ4 , 2 2 6 B B we deduce, as the evaluations of J2 and J3 , that J1 =
(1 + e% )D(0) r1+α −m%+nr+AY0 r−α e (1 + O (rα )) . 2πBb
(44)
The formula (41) now follows from (42)–(44). Proof of Theorem 2. Suppose that m = µn + xσn ∈ Z+ , where x = o(σn ). From Proposition 1, we have q(n) =
2D(0) (αAY (1, α))1/(2α+2) −(1+α/2)/(α+1) U (0)nα/(α+1) p n e 1 + O n−α2 /(α+1) . 2πα(α + 1) 23
It follows, by (41), that Pr{$n = m} =
q(n, m) = Ln (%)eHn (%) 1 + O n−α2 , q(n)
where √ (1 + e% )D(0) α + 1 Ln (%) = r1+α n(1+α/2)/(α+1) , √ 1/(2α+2) D(0) 2 2πBb (αAY (1, α)) and α+1 (αAY (1, α))1/(α+1) nα/(α+1) α U (%) − U (0) − %U 0 (%) .
Hn (%) = −m% + nr + AY (e% , α)r−α − = nα/(α+1)
By Lagrange inversion formula, we have U (%) − U (0) − %U 0 (%) = −
U 00 (0) 2 x + ξ(x/σn ). 2σn2
The desired result (5) now follows from expanding Ln (%) at % = 0: Ln (%) =
(αAY (1, α))−α/(2α+2) n−α/(2α+2) r
2π
=
√
AYu0 (1, α)
1 2π σn
00 (1, α) AYuu
+ |x| 1+O , σn
−
αA 0 2 (α+1)Y (1,α) Yu (1, α)
(1 + O(%))
since 00
−α/(α+1)
U (0) = A (αAY (1, α))
AYu0 (1, α)
+
00 AYuu (1, α)
αA 0 2 − Y (1, α) . (α + 1)Y (1, α) u
This completes the proof of Theorem 2.
5
Unrestricted partitions
Recall that each of the p(n) unrestricted partitions of n (into parts λj ) is assumed to be equally likely, and that the random variable ωn represents the number of distinct parts in a random partition of n. The bivariate generating function of ωn satisfies (8). By the equation a j Y a Y 1 + (u − 1)z j j uz j = , 1+ 1 − zj 1 − zj j≥1
j≥1
we obtain the relation for the generating polynomials Pn (u) := p(n)E(uωn ) and Qn (u) = q(n)E(u$n ): Pn (u) =
X
p(j)Qn−j (u − 1)
0≤j≤n
24
for n = 1, 2, 3, . . .
To prove Theorem 3, we proceed along the same line of arguments as in the last section. The analytic properties we need are summarized in Propositions 3 and 4 below. The remaining analysis being parallel to the proof of Theorem 2, we omit the details. Recall that Z(u, s) = Γ(s)ζ(s + 1) + Y (u − 1, s). Proposition 3. Let u ∈ C, | arg u| ≤ π − ε, where ε > 0 being arbitrarily small but fixed number. If τ → 0 in the sector | arg τ | ≤ π/4, then 0
P (u, e−τ ) = eD (0) uD(0) τ −D(0) eAZ(u,α)τ
−α
(1 + O (|τ |α1 )) ,
(45)
uniformly in τ and u, where α1 = min{1, α0 }. Proof. (Sketch) We have −τ
log P (u, e
)=
X k≥1
u ak log 1 + kτ e −1
1 = 2πi
Z
α+1+i∞
D(s)Z(u, s)τ −s ds.
α+1−i∞
The expansion (45) is obtained by shifting the path of integration to the line <e s = −α0 and by computing the residues of the poles encountered (see [1, 25, 34]). To derive uniform estimates for the ratio |P (ρeiθ , e−r−iy )|/P (ρ, e−r ), we use the the following inequalities. Lemma 9. Let u = ρeiθ , where ρ > 0 and |θ| ≤ π. Then the inequalities |P (ρeiθ , e−r−iy )| ≤ e−c21 T (r) P (ρ, e−r )
(46)
|P (ρeiθ , e−r−iy )| < e−c22 G(3r) P (ρ, e−r )
(47)
and
hold for |y| ≤ π and r > 0, where c21 = min{ρ, ρ−1 }, c22 = T (r) =
X
1−e−4 4
min{1, ρ2 }, G(r) = G0 (r) and
2 ak e−kr 1 − e−kr (1 − cos(θ − ky)) .
k≥1
Proof. First, we have 2 2 iθ iθ iθ | iθ ρe ρe 2|ρe ρe 1 + ≤ 1 + 2<e + kr+iky kr+iky + kr+iky kr+iky kr+iky e −1 e |e (e − 1)| e − 1 2ρ 2ρ e2ρ ≤ 1 − kr (1 − cos(θ − ky)) + kr + kr e e − 1 (e − 1)2 ! 2 ρ 2ρ (1 − cos(θ − ky)) = 1 + kr 1− . 2 e −1 ekr (1 + ρ/(ekr − 1)) 25
Thus |P (ρeiθ , e−r−iy )| P (ρ, e−r )
≤
Y
1−
k≥1
≤ exp −ρ
2ρ (1 − cos(θ − ky)) 2
!ak /2
ekr (1 + ρ/(ekr − 1)) X
ak e−kr 1 +
k≥1
ekr
ρ −1
−2
(1 − cos(θ − ky)) .
From the inequalities
ρ 1 + kr e −1
−2
≥
(1 − e−kr )2 , if 0 < ρ ≤ 1; −2 −kr 2 ρ (1 − e ) , if ρ ≥ 1,
the result (46) follows. Next, we have 2 2 ρeiθ ρ 1 + ≤ 1 + kr+iky ekr+iky − 1 |e − 1| 2 ρ ρ2 ρ2 < 1 + kr − − e −1 (ekr − 1)2 |ekr+iky − 1|2 2 ρ ρ2 (ekr − 1)2 = 1 + kr 1 − kr 1 − kr+iky . e −1 (e + ρ − 1)2 |e − 1|2 Using the inequalities (see [17, Eq. (3.14)])
4vX 1+ (X − 1)2
−1
≤ e−4v/X
2e−kr (1 − cos ky) ≤ 4 1 − e−w ≥
(0 ≤ v ≤ 1 < X),
(r > 0, k = 1, 2, 3, . . . ), 1 − e−4 w (0 ≤ w ≤ 4),
1 4
we obtain (ekr − 1)2 1 − kr+iky |e − 1|2
−1 2ekr = 1 − 1 + kr (1 − cos ky) (e − 1)2 ≥ 1 − exp −2e−kr (1 − cos ky) ≥ 12 1 − e−4 e−kr (1 − cos ky) .
Thus 2 2 ρeiθ ρ (1 − e−4 )ρ2 e−kr < 1 + 1 + kr 1− (1 − cos ky) ekr+iky − 1 e −1 2(ekr + ρ − 1)2 2 ρ (1 − e−4 ) min{1, ρ2 } −3kr ≤ 1 + kr 1− e (1 − cos ky) , 2 e −1 26
in virtue of the inequalities kr
(e
2
+ ρ − 1) ≤
e2kr , if 0 < ρ ≤ 1; ρ2 e2kr , if ρ ≥ 1.
It follows that ak /2 Y |P (ρeiθ , e−r−iy )| −3kr < 1 − 2c e (1 − cos ky) , 22 P (ρ, e−r ) k≥1
from which we derive (47). Proposition 4. As r→ 0+ , the inequality ! |P (ρeiθ , e−r−iy )| 1 2 < exp −c23 log P (ρ, e−r ) r holds for (i) r1+3α/7 ≤ |y| ≤ π and −π ≤ θ ≤ π; and (ii) r3α/7 ≤ |θ| ≤ π and |y| ≤ r1+3α/7 . Here c23 can be taken as n o c23 = min c21 c18 (1 − e−1/3 )2 , 12 c22 min{c2 , c6 } . Proof. The result in the range (i) is a direct consequence of (19), (47) and Lemma 4 if we assume (M1) and (M3). For the second range we argue as the proof of Lemma 6. We have for θ and y in range (ii) X
T (r) >
ak e−kr (1 − e−kr )2 (1 − cos(θ − ky))
1/(3r)≤k≤1/(2r)
>
1 − cos( 21 r3α/7 ) e−1/2 (1 − e−1/3 )2
X
ak
1/(3r)≤k≤1/(2r)
1 2 > c24 r−α/7 > c24 log , r by the inequality (20) and Lemma 3, where c24 = c18 (1 − e−1/3 )2 . From (46), we obtain the required inequality. Finally, the asymptotic behaviors of V 0 (w) depend more on the values of α as described in the following result. Lemma 10. If w → +∞ then V 0 (w) satisfies V 0 (w) = α−1 (α + 1)α/(α+1) A1/(α+1) 1 + O w−2 + w−α−1
27
;
and if w → −∞ then 1/(α+1) Aπ αw/(α+1) (1−α)w e 1 + O e , if 0 < α < 1; sin πα 0 V (w) = −A1/2 w(1 − w)−1/2 ew/2 (1 + O (ew )) , if α = 1; α−α/(α+1) (AΓ(α)ζ(α))1/(α+1) ew/(α+1) 1 + O ew + e(α−1)w , if α > 1. Proof. These follow from the definition of Z and properties of Φ(z, s, v) (see [6, §1.11]). Note that U 0 (w) ∼ V 0 (w) as w → +∞, this being intuitively clear in view of (16) and the relation max ωn ∼ max $n .
6
Examples
In general, it is the condition (M3) or (M3’) that is more difficult to check. A sufficient condition for the validity of these two is the following condition of Haselgrove and Temperley (see [15, 31]): there exists a positive constant ϑ < 1 such that |g(r + iy)| < ϑg(r)
for r ≤ |y| ≤ π,
(48)
as r→ 0+ . This condition is satisfied, for example, when λj = j ` (see [15]), where ` is a positive integer. However, as remarked by Richmond [31], (48) is, in general, a difficult condition to work with. Sometimes it is easier to check the following condition: If the abscissa of convergence αϕ of the Dirichlet series Dϕ (s) =
ikϕ k −s k≥1 ak e
P
is < α for each
π/2 ≤ |ϕ| ≤ π, then (M3’) (and a fortiori (M3)) is satisfied. For, by Mellin inversion formula, we deduce g(r + iy) r−α+ε for π/2 ≤ |y| ≤ π, where αy < α − ε < α. Thus g(r) − |g(r + iy)| r−α . (a) Let λj = j ` for j = 1, 2, 3, . . . , where ` is a fixed positive integer. All our theorems apply. Further computations show that the mean and variance of $n satisfy E($n ) = µn + c25 + O n−1/(`+1)
and Var($n ) = σn2 + c26 + O n−1/(`+1) ,
where expressions for the two constants c25 and c26 are given in (32) and (33) (with α there replaced by 1/`). In particular, if ` = 1 we have √ 2 3 log 2 √ 3 log 2 1 −1/2 E($n ) = n+ − + O n , π π2 ! 4 √ √ 3 12 3 36(log 2)2 3 log 2 3 2 √ −1/2 1 Var($n ) = − (log 2) n + − − − + O n , π π3 2π 2 8 π4 π2
28
which improve an old result of Erd˝ os and Lehner in [7]. Likewise, we have E(ωn ) = µ en + c27 + O n−1/(`+1)
and Var(ωn ) = σ en2 + c28 + O n−1/(`+1) ,
where c27 = − 21 +
` (` + 1)ζ(1 + 1/`)
and c28 =
` 1 − 2−1/` − ζ(1 + 1/`)−1 . (` + 1)ζ(1 + 1/`)
(b) Let λj = j ` − 1, ` being a fixed integer ≥ 2 and j = 2, 3, 4, . . . We have (see [10, p. 45]) X X s + j − 1 ` −s D(s) = (j − 1) = (ζ(`s + `j) − 1) , j j≥0
j≥2
the last expression providing a meromorphic continuation of D into the whole s-plane with polynomial growth order at σ ± i∞. To check (M3), we argue as in [23, Example (c), pp. 39–40]. For r > 0 and 0 < |y| ≤ 1 g(r) − <e g(r + 2πiy) ≥
X
e−(j
` −1)r
≥ e−1/2
j≥2 cos 2π(j ` −1)y≤0
X
1.
(49)
2≤j≤r−1/` cos 2π(j ` −1)y≤0
By the Weyl criterion (see [22, p. 7]) the sequence ((j ` − 1)y)j≥2 is uniformly distributed mod 1 for irrational y. Thus for any ε > 0 the number of summands in the rightmost summation is at least ( 21 − ε)r−1/` for sufficiently small r. It follows that g(r) − <e g(r + 2πiy) ≥ 13 e−1/2 r−1/` , for irrational y as r→ 0+ . But g(r + 2πiy) is a continuous function of y (actually infinitely differentiable). Thus condition (M3) holds for y in the interval
1 4
≤ y ≤ 34 . Theorem 1 applies. Similarly, it
can be verified that other results are also applicable. (c) Let λj = [j β ], where β > 1 is not an integer. We have X s + m − 1 X D(s) = [j β ]−s = ζ(βs) + ζm (s), m j≥1
where ζm (s) =
P
j≥1 {j
β }m j −β(s+m)
m≥1
for <e s > β −1 − m. Here {t} denotes the fractional part of t.
Thus D admits meromorphic continuation into the half-plane <e s > β −1 − 1 > −1 with a simple pole at s = 1/β. Further analytic properties of D can be derived through those of ζm (s). Conditions (M3) and (M3’) can be checked as in the last example, the uniform distribution modulo 1 being, for example, a consequence of Weyl’s metric theorem (see [22, p. 32]) and the uniform continuity of g. 29
(d) Let λj = h + jd with (h, d) = 1, h, d being positive integers, d ≥ 2 and 1 ≤ h < d. This is an interesting case where the condition (M3) holds (see [25]) but (M3’) is violated. Thus Theorems 1 and 3 apply but Theorem 2 does not. This example describes an interesting “period-hereditary” property of the sequence {λj } on $n , namely, if the sequence {λj } is periodic3 , then $n is of maximum span > 1. On the other hand, the random variables ωn are less sensitive to such a property. (e) λj+`(`−1)/2 = `, j = 1, . . . , `, namely, ak = k and D(s) = ζ(s − 1). All our theorems again apply.
7
Extensions
The results in this paper are susceptible of many different extensions. We only discuss two typical cases in this section. First, let λj = 2j−1 . It is easy to show the relation Y
1 + uz
2j
=1+
j≥0
X
uν(k) z k ,
k≥1
where ν(k) denotes the number of 1’s in the binary representation of k. In this case D(s) = (1−2−s )−1 with a simple pole at s = 0. Despite this degenerate case, it might be possible to extend our results to the case α = 0, as suggested by the example D(s) = 1 + 2−s /(1 − 2−s )2 . On the other hand, a limiting Gaussian law for ωn can be established using the methods of this paper, the regularity conditions requiring somewhat different arguments (see [5]). For further information on Mahler-type partitions, see [29, 35]. Note that if j
Pn (u) = [z n ]
Y
j≥0
uz 2 1+ 1 − z 2j
!
(n = 0, 1, 2, . . . ),
then
P2m (u) = P2m−1 (u) + Pm (u) P2m+1 (u) = P2m−1 (u) + uPm (u)
(m = 1, 2, 3, . . . ),
with P0 (u) = 1 and P1 (u) = u. These relations are useful from a computational point of view. A natural question suggested by the above examples is that between the degenerate limiting behavior of $n and the limiting Gaussian behavior of ωn , from which point on will the “phase change (from a discrete limiting law to a continuous one) occur? More precisely, let q` (n) denote the 3 If there exist two integers c and d such that λj = c + dj with d > 1, then the sequence {λj } is called periodic; otherwise, it is called aperiodic.
30
number of partitions into parts 2j−1 in which each part is allowed to appear at most `-times. Define [`]
$n by 1+
X
n
z q` (n)E(u
[`]
$n
)=
n≥1
Y
1+u z
2j
+z
2·2j
+ ··· + z
`·2j
.
j≥0
[`]
The question is for what values of ` will the distribution of $n be asymptotically normal, an intuition [`]
[`+1]
being that the asymptotic normality of $n would imply the same property for $n ? P Next, take D(s) = p prime p−s . Numerical evidence suggests again that the limiting distributions of $n and of ωn will still be Gaussian. For results on the total number of partitions and the moments of the summands, see Roth and Szekeres [33] and Richmond [30]. As mentioned in the Introduction, the limiting distributions of the number of summands (counted with multiplicities) in unrestricted partitions are non-Gaussian for almost all partitions. However, it was predicted by Haselgrove and Temperley [15] that Gaussian law would appear if α ≥ 2, although a formal proof is still lacking. A concrete example is the generating function Y
1 − uz j
j≥1
−j
,
enumerating the number of plane partitions of n with a given sum of the diagonal parts or with a given trace (see [1, Ch. 11]).
References [1] G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and Its Applications, Volume 2, Edited by Gian-Carlo Rota, Addison-Wesley Publishing Company, 1976. [2] F. C. Auluck, S. Chowla, and H. Gupta, On the maximum value of the number of partitions of n into k parts, Journal of the Indian Mathematical Society, 6 (1942), 105–112. [3] L. Comtet, Advanced combinatorics, the art of finite and infinite expansions, revised and enlarged edition, D. Reidel Publishing Company, Dordrecht-Holland, 1974. [4] J. Curtiss, A note on the theory of moment generating functions, The Annals of Mathematical Statistics, 13 (1942), 430–433. [5] N. G. de Bruijn, On Mahler’s partition problem, Indagationes Mathematicae, 10 (1948), 531– 546.
31
[6] A. Erd´elyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953. [7] P. Erd˝ os and J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Mathematical Journal, 8 (1941), 335–345. [8] P. Erd˝ os and M. Szalay, On the statistical theory of partitions, Colloquiua Mathematica Societatis J´anos Bolyai, 34, Topics in classical number theory, Budapest, 397–450, 1981. [9] P. Erd˝ os and P. Tur´an, On some general problems in the theory of partitions, I, Acta Arithmetica, 18 (1971), 53–62. [10] P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3–58. [11] P. Flajolet and A. M. Odlyzko, Singularity analysis of generating functions, SIAM Journal on Discrete Mathematics, 3 (1990), 216–240. [12] P. Flajolet and M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, Journal of Combinatorial Theory, Series A, 53 (1990), 165–182. [13] B. Fristedt, The structure of random partitions of large integers, Transactions of the American Mathematical Society, 337 (1993), 703–735. [14] W. M. Y. Goh and E. Schmutz, The number of distinct part sizes in a random integer partitions, Journal of Combinatorial Theory, series A, 69 (1995), 149–158. [15] C. B. Haselgrove and H. N. V. Temperley, Asymptotic formulae in the theory of partitions, Proceedings of the Cambridge Philosophical Society, 50 (1954), 225–241. [16] J. Herzog, Weak asymptotic formulas for partitions free of small summands, II, Acta Mathematica Hungarica, 62 (1993), 173–188. [17] A. Hildebrand and G. Tenenbaum, On integers free of large prime factors, Transactions of the American Mathematical Society, 296 (1986), 265–290. [18] H.-K. Hwang, Large deviations for combinatorial distributions. I: central limit theorems, Annals of Applied Probability, 6 (1996), 297–319.
32
[19] H.-K. Hwang, Distribution of integer partitions with large number of summands, Acta Arithmetica, 78 (1997), 351–365. [20] H.-K. Hwang, Large deviations of combinatorial distributions. II. Local limit theorems, Annals of Applied Probability, 8 (1998), 163–181. [21] C. Knessel and J. B. Keller, Partition asymptotics from recursion equations, SIAM Journal on Applied Mathematics, 50 (1990), 323–338. [22] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley & Sons, New York, 1974. [23] D. V. Lee, The asymptotic distribution of the number of summands in unrestricted Λ-partitions, Acta Arithmetica, 65 (1993), 29–43. [24] J. H. Loxton and H.-F. Yeung, Common summands in partitions, Acta Arithmetica, 55 (1992), 308–320. [25] G. Meinardus, Asymptotische Aussagen u ¨ber Partitionen, Mathematische Zeitschrift, 59 (1954), 388–398. [26] F. W. J. Olver, Asymptotics and special functions, Academic Press, Inc., New York, 1974. [27] V. V. Petrov, Sums of independent random variables, Springer-Verlag, Berlin-Heidelberg, NewYork, 1975. Translated from the Russian by A. A. Brown. [28] L. B. Richmond, The moments of partitions, I, Acta Arithmetica, 26 (1975), 411–425. [29] L. B. Richmond, Mahler’s partition problem, Ars Combinatorica, 2 (1976), 169–189. [30] L. B. Richmond, The moments of partitions, II, Acta Arithmetica, 28 (1976), 229–243. [31] L. B. Richmond, Some general problems on the number of parts in partitions, Acta Arithmetica, 64 (1994), 297–313. [32] L. B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae, 49 (1995), 86–97. [33] K. F. Roth and G. Szekeres, Some asymptotic formulae in the theory of partitions, Quartly Journal of Mathematics, Oxford series, 5 (1954), 241–259. 33
[34] E. Schmutz, Part sizes of random integer partitions, Indian Journal of Pure and Applied Mathematics, 25 (1994), 567–575. [35] W. Schwarz, Einige Anwendungen Tauberscher S¨ atze in der Zahlentheorie, C Mahler’s Partitionsproblem, Journal f¨ ur die reine und angewandte Mathematik, 229 (1967), 182–188. [36] G. Szekeres, An asymptotic formula in the theory of partitions, II, Quarterly Journal of Mathematics, Oxford series, 4 (1953), 96–111. [37] G. Szekeres, Asymptotic distributions of the number and size of parts in unequal partitions, Bulletin of the Australian Mathematical Society, 36 (1987), 89–97. [38] G. Tenenbaum, Introduction ` a la th´eorie analytique et probabiliste des nombres, Institut Elie Cartan, Universit´e de Nancy I, Nancy, France, 1990. [39] E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, second edition, 1986. revised by D. R. Heath-Brown. [40] H. Wilf, Three problems in combinatorial asymptotics, Journal of Combinatorial Theory, series A, 35 (1983), 199–207.
Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei 11529 Taiwan e-mail: [email protected]
34
1991 AMS Mathematics subject classification: Primary 11P82; secondary 60F05 60F10. Key words: Integer partitions, central and local limit theorems, large deviations, Meinardus’s scheme, Mellin transform, Lerch’s zeta function, saddle-point method.
Running head: limit theorems for partitions Mailing address: Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei, 115 Taiwan e-mail: [email protected]
Abstract Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cram´er-type large deviations and are proved by Mellin transform and the two-dimensional saddle-point method. Applications of these results include partitions into positive integers, into powers of integers, into integers [j β ], β > 1, into aj + b, etc.
1
Introduction
Let Λ = {λ1 , λ2 , . . . } be a sequence of positive integers satisfying 1 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · and λn → +∞. There is an extensive literature on the asymptotics of the number of partitions of n into parts λj (see Andrews [1] and the references therein). In contrast, considerably fewer results have appeared in the literature on the limiting distribution of the number of summands (or parts) in random restricted (no part being repeated) or unrestricted partitions (repetition allowed). This paper is concerned with this aspect of the theory of partitions. Erd˝ os and Lehner [7] were the first to give a systematic study along this line in the case Λ = Z+ . They showed that the number of summands (counted with multiplicities) in a random unrestricted partition of n follows asymptotically (as n → ∞) an extreme-value distribution, a local version being later derived by Auluck et al. in [2]. Haselgrove and Temperley [15] extended, by a powerful analytic method, the result of Auluck et al. to more general Λ-partitions, their conditions on the given sequence being further extended, in some respects, only recently by Richmond [31]. Weaker results (convergence in distribution) under different analytic settings were derived by Lee [23] by the method of moments. A detailed study on the moments can be found in Richmond [28, 30]. It should be noted that the limiting distributions in these problems are all non-Gaussian. For many other extensions of the original problems, see for example [8, 9, 16, 24] and the references therein. When each part is allowed to appear at most once, Erd˝ os and Lehner [7] derived the asymptotic normality of the number of summands in the case Λ = Z+ (see also [9, 36, 37]). No extension of this result has appeared in the literature. In this paper, we consider a general analytic scheme essentially due to Meinardus (see [25] or [1, Ch. 6]) under which central and local limit theorems will be derived, thus extending Erd˝ os and Lehner’s result. The analytic conditions under which we are developing our arguments are weaker than those of Meinardus. Our analytic method can also be applied to the problem left open in [24, p. 311] concerning the common summands in restricted partitions. It turns out that the limiting law is Gaussian for a large class of partitions. There is another way of counting the number of summands in unrestricted partitions, namely, if the multiplicity of each part is counted only once. Unlike the corresponding counting function (i.e. the ω(n) function) in the theory of primes (see [38]), this problem is rarely discussed in the theory of partitions. It was first briefly mentioned in [7] in the case Λ = Z+ . Wilf [40] introduced the study of distinct components (or sizes of components) in general combinatorial structures. Then Goh and Schmutz [14] derived a central limit theorem for the number of summands for Λ = Z+ . The
1
latter result was then extended by Schmutz [34] to multivariate cases under Meinardus’s scheme. We further improve and extend their results by establishing the corresponding local limit theorem (in univariate case) under weaker conditions. An distinctive feature of integer partitions is that the limiting distribution of the number of summands is non-Gaussian in almost all cases if the multiplicity of each summand is taken into account (see [15, 23, 31]), in contrast to the ubiquitous normal law in a large class of combinatorial structures (see [12, 18]). Intuitively, the former phenomenon may be ascribed to the predominance of small summands when the number of summands becomes large, say, larger than the mean value. However, Gaussian limiting distribution appears if the parts are counted without multiplicity, this being intuitively clear since no single part can contribute preponderantly to the corresponding counting function, in accordance with the classical law of errors. Our results show that the same phenomenon still persists if each part is allowed to occur at most once. For completeness, we add that a formal approach was introduced in Knessel and Keller [21] for characterizing the asymptotic behaviors of many quantities in partition problems satisfying suitable recurrences. Another recent reference on related problems is Fristedt [13], the methods employed there being probabilistic. We state the main results of this paper in § 2. The proof of these results is divided into two parts: central (§ 3) and local (§ 4) limit theorems. In each section, we first derive some necessary estimates and then prove the result in question. Since our assumptions are weaker than those used in [25, 34], some techniques are introduced to justify the regularity conditions (in order to apply the saddle-point method). Unrestricted partitions is treated in § 5. Finally, we discuss some examples in § 6 and conclude with some remarks for further extensions.
2
Statement of results
Throughout this paper, the symbols cj always denote absolute positive constants. The symbol ε represents always suitable small quantity whose value may vary from one occurrence to another. Given a sequence of positive integers 1 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · tending to infinity, let Π(n) = ΠΛ (n) be the set of partitions of the positive integer n into distinct parts λj (each λj occurring at most once), j = 1, 2, . . . (in the case when there are more than one λi with the same value, one can think of properly labeling these λi ’s so that each is “different” from the other). Let q(n) = |Π(n)|, the cardinality of the set Π(n). It is more convenient to work with ak , denoting the number of λj ’s such
2
that λj = k. The generating function of q(n) satisfies Q(z) = QΛ (z) = 1 +
X
q(n)z n =
n≥1
Y
j≥1
Y ak 1 + z λj = 1 + zk , k≥1
for |z| < 1. To state our results, we first introduce an analytic scheme essentially due to Meinardus [25] in which the sequence {ak } satisfies the following three conditions. P (M1) The Dirichlet series D(s) = k≥1 ak k −s converges in the half-plane <e s > α > 0, and can be analytically continued into the half-plane <e s ≥ −α0 , for some α0 > 0. In <e s ≥ −α0 , D is analytic except for a simple pole at s = α with residue A1 . (M2) There exists an absolute constant c1 such that2 D(s) |t|c1 uniformly for <e s ≥ −α0 as |t| → +∞. (M3) Define g(τ ) =
−kτ , k≥1 ak e
P
where τ = r + iy with r > 0 and −π ≤ y ≤ π. There exists 2
a positive constant c2 such that g(r) − <e g(τ ) ≥ c2 (log(1/r))2+4/α uniformly for π/2 ≤ |y| ≤ π as r→ 0+ . The assumption (M3) here is much weaker than those used in [25] and [34], the essential difference being that we did not impose a similar estimate for g(r) − <e g(r + iy) in the region r ≤ |y| ≤ π/2, which is established by other assumptions, notably by the growth properties of the sum function P k≤X ak and by (M3). Introducing a uniform probability measure on the set Π(n), we consider the random variable $n , counting the number of summands in a random partition of n. The bivariate generating function of $n satisfies Q(u, z) = 1 +
X
q(n)E(u$n )z n =
n≥1
Y
1 + uz k
ak
,
(1)
k≥1
for finite u and |z| < 1, where E(u$n ) represents the probability generating function of $n . Set κ = AΓ(α)(1 − 2−α )ζ(α + 1), (1 − 21−α )ζ(α) nα/(α+1) , α(1 − 2−α )ζ(α + 1) (1 − 22−α )ζ(α − 1) (1 − 21−α )2 ζ(α)2 1/(α+1) = (κα) − nα/(α+1) . α(1 − 2−α )ζ(α + 1) (α + 1)(1 − 2−α )2 ζ(α + 1)2
µn = (κα)1/(α+1) σn2
Here Γ is the Gamma-function, ζ is Riemann’s zeta function and the factor (1 − 2−s )ζ(s + 1) is defined to be log 2 when s = 0. Note that σn > 0 as can be checked. 1 2
In Meinardus’s original paper, the quantity α0 is assumed to satisfy 0 < α0 < 1. The Vinogradov symbol is the same as the Landau symbol O(.) and is used interchangeably as is convenient.
3
Theorem 1. Suppose that the sequence {ak } satisfies (M1), (M2), and (M3). Set $n∗ = ($n − µn )/σn . Then the random variable $n is asymptotically normally distributed with mean E($n ) ∼ µn and variance Var($n ) ∼ σn2 : Pr{$n∗
1 < x} = √ 2π
x
Z
2 /2
e−t
dt + o(1),
(2)
−∞
uniformly for all x as n → ∞. Moreover, for sufficiently large n, we have the exponential bounds: ( 2 e−x /2 1 + O (log n)−1 , if 0 ≤ x ≤ nα/(6α+6) / log n; ∗ Pr{$n ≥ x} ≤ (3) α/(6α+6) x/(2 log n) e−n 1 + O (log n)−1 , if x ≥ nα/(6α+6) /(log n), and the same inequalities for Pr{$n∗ ≤ −x}. The method of proof consists of analytic and probabilistic parts: the analytic part is based on Mellin transform and the saddle-point method; and the probabilistic part utilizes Curtiss’s theorem [4] for convergence of moment generating functions. It turns out that Lerch’s zeta function (see [6, P §1.11]) Φ(z, s, v) = n≥0 z n (n + v)−s intervenes in a natural way in our analysis. Our application of the saddle-point method differs from that in [25] and [1, Ch. 6] and yields a better error term. We complete the asymptotic normality of $n by its strong concentration property (3) using a simple technique (see [27, Ch. III]) amended from the usual Chernoff bound. We can also derive a local limit theorem in the form of Cram´er-type large deviations (see [20] [27]). It suffices to replace condition (M3) by the following stronger one. (M3’) There exists a fixed constant c3 > 0 such that g(r) − <e eiθ g(r + iy) ≥ c3 (log(1/r))2+4/α
2
uniformly for π/2 ≤ |y| ≤ π and −π ≤ θ ≤ π, as r→ 0+ . Let Y (u, s) be the Mellin transform of the function log(1 + ue−x ): Z ∞ Y (u, s) = xs−1 log(1 + ue−x ) dx for <e s > 0.
(4)
0
As we will see, Y is essentially Lerch’s zeta function. Theorem 2. Assume that the sequence {ak } satisfies (M1), (M2) and (M3’). If m = µn +xσn ∈ Z+ , where x = o nα/(2α+2) , then Pr{$n = m} =
e−x
2 /2+ξ(x/σ
√
α/(α+1) n )n
2π σn
1+O
uniformly in x, where α2 = min{1, α0 , α} and ξ(w) =
|x| nα/(2α+2)
P
j≥3 ξj w
origin whose Taylor coefficients satisfy 0 −1 k−2 00 U (w) − U 0 (0) −k ξk = [w ]U (w) k U 00 (0)w 4
j
+n
−α2 /(α+1)
,
(5)
is analytic in a neighborhood of the
for k = 3, 4, 5, . . .
(6)
with U (w) = (α + 1)α−α/(α+1) A1/(α+1) Y (ew , α)1/(α+1) − Y (1, α)1/(α+1) . Here the symbol [z n ]f (z) denotes the coefficient of z n in the Taylor expansion of f (z). Note that U is convex due to the same property of Y and that µn = U 0 (0)nα/(α+1) and σn2 = U 00 (0)nα/(α+1) . The first two terms of ξk are given by (see [20]) U 000 (0)2 (4) 1 000 1 ξ3 = 6 U (0) and ξ4 = 24 U (0) − 00 . U (0) As an interesting consequence, we state the following Corollary 1. If m = µn + xσn ∈ Z+ , where x = o nα/(6α+6) , then 2
e−x /2 Pr{$n = m} = √ 2π σn
1+O
|x| + |x|3 + n−α2 /(α+1) nα/(2α+2)
,
(7)
uniformly in x. The proof of this theorem utilizes essentially the two-dimensional saddle-point method and is technically more involved. As is usual in the application of the saddle-point method, it is the verification of the regularity conditions to which much of our analysis is devoted. Actually, we prove more (see Proposition 2 below) but content ourselves with the statement of the theorem. Our methods can also be applied to the number of distinct parts in unrestricted partitions (repetition allowed) under the same assumptions (M1)–(M3) as in Theorem 1. e Let Π(n) represent the set of unrestricted partitions of n and let p(n) be its cardinality. Let
ωn be the number of distinct parts (i.e., counted without multiplicities) in a random partition of n, where all p(n) partitions of n are equally likely. The bivariate generating function of ωn satisfies P (u, z) = 1 +
X
ωn
n
p(n)E(u )z =
n≥1
Y
k≥1
uz k 1+ 1 − zk
a k
,
for |z| < 1. Set κ1 = AΓ(α)ζ(α + 1), (κ1 α)1/(α+1) α/(α+1) µ en = AΓ(α)(κ1 α)−α/(α+1) nα/(α+1) = n , αζ(α + 1) α (κ1 α)1/(α+1) 2 −α σ en = 1−2 − nα/(α+1) . αζ(α + 1) (α + 1)ζ(α + 1) 5
(8)
Define Z(u, s) =
Z
∞
0
u log 1 + x e −1
xs−1 dx for <e s > 0 and | arg u| < π.
It is obvious, by (4), that Z(u, s) = Γ(s)ζ(s + 1) + Y (u − 1, s). Theorem 3. Under the assumptions (M1), (M2), and (M3), the random variable ωn satisfies asymptotically E(ωn ) ∼ µ en , Var(ωn ) ∼ σ en2 , and Pr{ωn = µ en + xe σn } =
e−x
2 /2+η(x/e µn )nα/(α+1)
√
2π σ en
1+O
|x| nα/(2α+2)
+n
−α2 /(α+1)
,
uniformly for all x = o nα/(2α+2) such that µ en + xe σn ∈ Z+ . Here α2 is as in Theorem 2 and P η(w) = j≥3 ηj wj is analytic at the origin with coefficients given by −1 k−2 00 ηk = [w ]V (w) k
V 0 (w) − V 0 (0) V 00 (0)w
−k
for k = 3, 4, 5, . . . ,
where V (w) = (α + 1)α−α/(α+1) A1/(α+1) Z(ew , α)1/(α+1) − Z(1, α)1/(α+1) . As the proof of this theorem parallels that of Theorems 1 and 2, only the necessary regularity conditions is worked out in § 5. That the assumptions needed for the local limit theorem of ωn are weaker than those for $n is seen by the following example. Take λj = 2j − 1. Then it is obvious that the span of the random variable $n is 2 whereas that of ωn is 1. More precisely, E(u$n ) contains only odd (respectively even) powers of u for odd (respectively even) n. In this case, local limit theorem of $n depends on the parity of n.
3
Central limit theorem
3.1
Lemmas
In this section, we establish some estimates for the function Q(u, e−τ ) (defined in (1)) as τ → 0. We write consistently the complex variable τ in the form τ = r + iy with −π ≤ y ≤ π and r > 0. These estimates are slightly more general than our need for the proof of Theorem 1 since some of them will be required when establishing the corresponding local limit theorem. Let f (u, τ ) = log Q(u, e−τ ): f (u, τ ) =
X
ak log 1 + ue−kτ .
k≥1
6
The sum on the right-hand side being a harmonic sum (see [10]), we have available the Mellin inversion formula: 1 f (u, τ ) = 2πi
Z
α+1+i∞
D(s)Y (u, s)τ −s ds,
(9)
α+1−i∞
for <e τ > 0, where Y (u, s) is the Mellin transform of the function log(1 + ue−x ); see (4). Note that, for |u| ≤ 1 and <e s > 0, Y (u, s) satisfies Y (u, s) = Γ(s)
X (−1)j−1 j≥1
j s+1
uj ,
(10)
a representation no longer useful when |u| > 1. In particular, Y (1, s) = (1 − 2−s )ζ(s + 1)Γ(s), so that κ = AY (1, α). Now by integration by parts, we see that Y (u, s) is related to the Lerch zeta function Φ(z, s, v) by: Y (u, s) = uΓ(s)Φ(−u, s + 1, 1), with Φ defined by (see [6]) Φ(z, s, v) =
X k≥0
zk (k + v)s
for |z| < 1, s ∈ C, and v 6= 0, −1, −2, . . .
Analytic properties of Y (u, s) are summarized in the following lemma. Lemma 1. For each fixed u lying in the cut-plane C \ (−∞, −1], the Mellin transform Y (u, s) can be meromorphically continued into the whole s-plane with simple poles at s = 0, −1, −2, . . . Moreover, Y (u, s) satisfies the estimate |Y (u, σ + it)| e−(π/2−ε)|t|
for any ε > 0 as |t| → +∞,
(11)
uniformly for finite σ and u in the cut-plane. Proof. For completeness, we sketch here a self-contained proof. For further properties, see Erd´elyi [6]. First, by integration by parts, we have u Y (u, s) = s
Z 0
7
∞
xs dx, ex + u
the right-hand side providing a meromorphic continuation of Y to the half-plane <e s > −1. The first assertion of the lemma follows from repeating the same process. As to (11), since log(1 + ue−x ) is an analytic function of x in the half-plane <e x > 0, we have by Cauchy’s theorem Z eiϕ ∞ Y (u, s) = log(1 + ue−x )xs−1 dx for any |ϕ| ≤ π/2 − ε. 0
Thus a change of variable yields Y (u, s) = eiϕs
Z
∞
0
iϕ log 1 + ue−e t ts−1 dt,
from which (11) follows. Remark 1. Since for x ∼ 0 log(1 + ue−x ) = log(1 + u) +
X h≥1
X uxh A(h − 1, j)(−1)h−1−j uj , h!(1 + u)h 0≤j 0. Thus α+` ≤ E1 + E2 , FL−1 (X) − AX (α + `)L−1 9
where E1 E2
FL (X + δX) − FL (X) AX α+` − , = L log(1 + δ) (α + `)L−1 Z X+δX 1 FL−1 (t) − FL−1 (X) = dt. log(1 + δ) X t
Since FL−1 (t) is non-decreasing (the ak being ≥ 0), we have E2 ≤ =
−1 Z X Z X+δX 1 FL−1 (t) 1 FL−1 (t) dt − log dt log(1 + δ) X t 1−δ t X−δX 1 (FL (X + δX) − 2FL (X) + FL (X − δX)) δ + O (max {FL (X + δX) − FL (X), FL (X) − FL (X − δX)}) ,
as δ ∼ 0. From (15) and the estimates (1 + δ)α+` − 1 = (α + `)δ + O(δ 2 ), as δ ∼ 0, it follows that E1 + E2 δX α+` +
RL (X) . δ
Taking δ = X −(α+`)/2 RL (X)1/2 (→ 0+ ) so as to balance the two error terms on the right-hand side, we obtain FL−1 (X) =
AX α+` + O (RL−1 (X)) , (α + `)L−1
where RL−1 (X) = X α/2 (log X)L/2 + X `+(α−α0 )/2 . Repeating the same process, we see that for j = 1, 2, 3, . . . , L FL−j (X) =
AX α+` + O (RL−j (X)) , (α + `)L−j
where RL−j (X) = X (1−2
−j )α
j
(log X)L/2 + X `+(1−2
−j )α−α
0 /2
j
.
This completes the proof. According to (M2), the number c1 (and thus L) depends on the value of α0 . One may choose a suitable α0 (if possible) so that the error terms in (14) are minimized. An interesting consequence of this lemma is the following
10
Corollary 2. As n → +∞, max $n ∼ α−1 (α + 1)α/(α+1) A1/(α+1) nα/(α+1) ,
(16)
where the max is over all distinct partitions of n. Proof. For, by Lemma 3, if n=
X
kak ∼
1≤k≤X
A X α+1 , α+1
then X ∼ ((α + 1)n/A)1/(α+1) ; consequently, max $n =
X
ak ∼
1≤k≤X
A α X , α
from which we obtain (16). Lemma 4. Let u be a positive real number. There exists a constant c5 > 0 such that the inequality |Q(u, e−r−iy )| c5 u 2 ≤ exp − (log(1/r)) (17) Q(u, e−r ) (1 + u)2 holds uniformly for r1+3α/7 ≤ |y| ≤ π as r→ 0+ . Proof. To start with, we observe that |Q(u, e−r−iy )| Q(u, e−r )
ak /2 2ue−kr = 1− (1 − cos ky) (1 + ue−kr )2 k≥1 X u ≤ exp − ak e−kr (1 − cos ky) . (1 + u)2 Y
(18)
k≥1
In view of the assumption (M3), it suffices to show that G(r) := g(r) − <e g(r + iy) ≥ c6 (log(1/r))2 ,
(19)
for r1+3α/7 ≤ |y| ≤ π/2, as r→ 0+ . Consider first the case r ≤ |y| ≤ (log(1/r))−2/α . Using the elementary inequality 1 − cos t ≥
2 2 t π2
for |t| ≤ π,
(20)
we have X
G(r) >
ak e−kr (1 − cos ky) ≥
1≤k≤1/|y|
≥
2Ae−1 π 2 (α + 2)
2 2 −r/|y| y e π2
− ε |y|−α ≥ c6 (log(1/r))2 11
X
k 2 ak
1≤k≤1/|y|
as r→ 0+ ,
by (13). Next, if r1+3α/7 ≤ |y| ≤ r, then G(r) >
X
ak e−kr (1 − cos ky) ≥
2 2 −1 y e π2
≥
π 2 (α + 2)
k 2 ak
1≤k≤1/r
1≤k≤1/r
Ae−1 π α+2
X
y 2 r−α−2 ≥ c7 r−α/7 ≥ c6 (log(1/r))2
as r→ 0+ .
Finally, if y lies in the range 0 < |y| ≤ π/2, then there exists an integer ` such that π ≤ 2` |y| ≤ π. 2 From the elementary inequality 1 − cos θ ≥ 4−` 1 − cos 2` θ , and consequently G(r) ≥ 4−`
X
1 4
(21)
(1 − cos 2θ), we obtain by induction 1 − cos θ ≥
ak e−kr 1 − cos 2` ky
k≥1
2 2 1 3+4/α c2 1 3+4/α ≥ c2 4−` log ≥ 2 y 2 log , r π r
in virtue of (M3) and (21). Thus c2 G(r) ≥ 2 π
1 log r
2
,
for y satisfying (log(1/r))−2/α ≤ |y| ≤ π/2. Taking c5 = min{c2 /π 2 , c6 }, (17) follows.
3.2
The proof of Theorem 1
Throughout this section, u is a positive real number which eventually will be taken to be near 1. Proposition 1. Let δ > 0 be any fixed number in the unit interval. Then we have, uniformly for δ ≤ u ≤ δ −1 , α/(α+1)
Qn (u) = β(u)n−(1+α/2)/(α+1) e(1+1/α)K(u)n
1 + Oδ n−α2 /(α+1) ,
(22)
the O-term holding uniformly in u, where α2 = min{α, α0 , 1}, K(u) = (αAY (u, α))1/(α+1) and s 1/(2α+2) (αAY (u, α)) K(u) p β(u) = (1 + u)D(0) = (1 + u)D(0) . 2π(α + 1) 2π(α + 1) Proof. By Cauchy’s integral formula, Z enr π iny e Q(u, e−r−iy ) dy = enr (I1 + I2 ) , Qn (u) = 2π −π 12
(23)
where I1 and I2 represent the integrals (2π)−1
and (2π)−1
R
|y|≤r 1+3α/7
R
, respectively. Here
r 1+3α/7 0.
(24)
We assume that n is sufficiently large so that r1+3α/7 < π. Consider first I2 , which is bounded above by 2
I2 Q(u, e−r )e−c9 (log(1/r)) eAY (u,α)r
−α −c
10 (log n)
2
,
(25)
in view of (17) and (24). For I1 we have by (12) and a change of variables I1 = (1 + u)D(0)
r 2π
r 3α/7
Z
einry+AY (u,α)r
−α (1+iy)−α
(1 + O (rα1 )) dy
−r 3α/7
= I3 + I4 , where I3 = (1 + u)D(0)
r 2π
Z
r3α/7
einry+AY (u,α)r
−α (1+iy)−α
dy,
−r3α/7
and I4 r
1+α1
Z
r 3α/7
eAY (u,α)r
−α (1+y 2 )−α/2
dy.
−r 3α/7
Using the elementary inequality (1 + y 2 )−α/2 ≤ 1 − (1 − 2−α/2 )y 2
for − 1 ≤ y ≤ 1,
(26)
we obtain I4 r
1+α1 AY (u,α)r−α
e
Z
r−α/7
e−(1−2
−α/2 )AY
(u,α)r−α y 2
dy
−r−α/7
r1+α1 +α/2 eAY (u,α)r It remains to evaluate I3 . Setting B =
−α
.
(27)
p α(α + 1)AY (u, α) and making the change of variables
v 2 = B 2 r−α y 2 , we obtain D(0) r
I3 = (1 + u)
1+α/2 eAY (u,α)r−α
2πB
13
Z
Br−α/7
−Br−α/7
e−v
2 /2
Tr (v) dv,
where 2 X α + j + 1 −iv j v rαj/2 Tr (v) = exp − α(α + 1) j+2 B
j≥1
(α + 2)iv 3 = 1+ p rα/2 + O(v 4 rα ), 6 α(α + 1)AY (u, α) from which we deduce that r1+α/2 eAY (u,α)r
D(0)
I3 = (1 + u)
−α
p
2πα(α + 1)AY (u, α)
(1 + O (rα )) .
(28)
Collecting our results (23)–(28) yields (1 + u)D(0) r1+α/2 enr+AY (u,α)r p Qn (u) = 2πα(α + 1)AY (u, α)
−α
(1 + O (rα2 )) ,
uniformly in u. The relation (22) follows from the above formula using (24). Proof of Theorem 1. (Asymptotic normality) Let Mn (t) = E(e($n −µn )t/σn ), where t is real. Then by (22) Qn (et/σn ) Qn (1) !D(0) !1/(2α+2) et/σn + 1 Y (et/σn , α) eφn (t) 1 + O n−α2 /(α+1) , 2 Y (1, α)
Mn (t) = e−µn t/σn = uniformly in t, where
µn t + 1 + α−1 K(et/σn ) − K(1) nα/(α+1) σn µn t = − + U (t/σn )nα/(α+1) . σn
φn (t) = −
Observe that for <e s > 0 t/σn
Y (e
, s) = Y (1, s) +
Z
∞
0
et/σn − 1 log 1 + x e +1
!
xs−1 dx,
and that as n → +∞ Z 0
∞
et/σn − 1 log 1 + x e +1
!
s−1
x
t t2 dx = h1 + h2 2 + O σn σn
where, in general, (−1)k−1 hk = k
Z
∞
0
14
xs−1 dx. (ex + 1)k
|t|3 σn3
,
Note that hk =
Γ(s) X s(k, j)(1 − 2j−s )ζ(s + 1 − j) k!
for k = 1, 2, 3, . . . ,
(29)
1≤j≤k
where the s(k, j) represent the (signed) Stirling numbers of the first kind (see [3, Ch. 5]). Thus, we have Mn (t) = eφn (t) 1 + O n− min{α/2,α0 ,1}/(α+1) + |t|n−α/(2α+2) , and t/σn
K(e
K(1) ) − K(1) = (α + 1)Y (1, α)
2 h1 t αh21 t 3 −3 + 2h2 − + O |t| σn . σn (α + 1)Y (1, α) 2σn2
Equivalently, this last relation can be written as 3 t U 00 (0)t2 |t| t 0 = U (0) + +O . U 2 σn σn 2σn σ3 From these formulae and the relations (by (29)) µn n
−α/(α+1)
K(1)h1 = αY (1, α)
and
σn2 n−α/(α+1)
= K(1)
2h2 h21 − αY (1, α) (α + 1)Y (1, α)2
,
it follows that 1 + O n−α2 /(α+1) + (|t| + |t|3 )n−α/(2α+2) 2 = et /2 1 + O n− min{α/2,α0 ,1}/(α+1) , 2 /2
Mn (t) = et
(30)
uniformly in t. By the theorem of Curtiss in [4], we conclude that the distribution of the random variable $n is asymptotically Gaussian. (Exponential tails) As to the exponential bounds (3), we observe from the above derivations that (30) remains valid if |t| tends to infinity slowly enough: t = o nα/(6α+6) . We consider only the case when $n∗ ≥ x in the following, the other case −$n∗ ≥ x being similar. From (30), we have for x ≥ 0 ∗
Pr{$n∗ ≥ x} = Pr{e$n t ≥ etx } ≤ e−tx Mn (t) 2 = e−tx+t /2 1 + O n−α2 /(α+1) + (|t| + |t|3 )n−α/(2α+2) .
(31)
Let T be any positive quantity tending to infinity with n and satisfying T = o(nα/(6α+6) ). If 0 ≤ x ≤ T then we take (see [27, Ch. III]) t = x in (31) (so as to minimize −tx + t2 /2) and we obtain Pr{$n∗ ≥ x} ≤ e−x
2 /2
1 + O n−α2 /(α+1) + |T |3 n−α/(2α+2) ; 15
and if x ≥ T we have by taking t = T : Pr{$n∗ ≥ x} ≤ e−T x/2 1 + O n−α2 /(α+1) + |T |3 n−α/(2α+2) . Now the estimates (3) follow from choosing T = nα/(6α+6) / log n. (Mean and variance) We still have to prove that the mean and the variance of $n are asymptotic to µn and σn2 , respectively, a result that is not guaranteed by convergence in distribution. Although we may directly evaluate Qu (1, z) and Q00uu (1, z) as in the proof of Proposition 1, the asymptotic form of the variance depends on the values of α and α0 . The following arguments are computationally simpler and are not subject to the values of α. Note that Fn (x), the distribution function of $n∗ , converges pointwise to the standard normal distribution whose mean and variance are 0 and 1, respectively. It suffices to show that E $n∗ = o(1) and Var($n∗ ) = 1 + o(1). For the former we use the representation Z ∞ ∗ E($n ) = (1 − Fn (x) − Fn (−x)) dx, 0
the uniform bounds (3) and Lebesgue’s dominated convergence theorem. The latter follows from Z ∞ E($n∗2 ) = 2x (1 − Fn (x) − Fn (−x)) dx, 0
and similar considerations. Remark 2. A further refinement of the above arguments leads to the better estimates: D(0) (1 − 21−α )ζ(α) + , 2 2(α + 1)(1 − 2−α )ζ(α + 1) D(0) ζ(α − 1)(2α − 4) Var($n ) ∼ σn2 + + 4 (α + 1)ζ(α + 1)(2α − 1) ζ(α)(2α − 2) ζ(α)2 (2α − 2)2 − − , (α + 1)ζ(α + 1)(2α − 1) (α + 1)ζ(α + 1)2 (2α − 1)2 E($n ) ∼ µn +
(32)
(33)
where the convention that (1 − 2−s )ζ(s + 1) = log 2 when s = 0 is assumed.
4
Local limit theorem
4.1
Lemmas
The local behavior of Q(u, e−τ ) as τ → 0 and u → 1 having been made explicit in Lemma 2, we need only consider other ranges of u and τ . As in the last section, we write consistently u = ρeiθ = e%+iθ
and z = e−τ = e−r−iy , 16
where ρ, r > 0, % ∈ R, and −π ≤ θ, y ≤ π. Define Gθ (r) = g(r) − <e eiθ g(r + iy) =
X
ak e−kr (1 − cos(θ − ky)) ,
k≥1
so that G0 (r) = G(r). Then we have, by the derivations for (18), |Q(ρeiθ , e−r−iy )| ρGθ (r) ≤ exp − . Q(ρ, e−r ) (1 + ρ)2 Lemma 5. There exists a positive constant c13 such that the inequality ! |Q(ρeiθ , e−r−iy )| c13 ρ 1 2 < exp − log Q(ρ, e−r ) (1 + ρ)2 r holds uniformly for r1+3α/7 ≤ |y| ≤ π and −π ≤ θ ≤ π, as r→ 0+ . Proof. Consider first the case when r ≤ |y| ≤ (log(1/r))−2/α . If |θ| ≤ X
Gθ (r) >
1 2
then
ak e−kr (1 − cos(θ − ky))
1/|y|≤k≤2/|y|
1 − cos( 12 ) e−2
>
−α
> c14 |y|
≥ c13
X
ak
1/|y|≤k≤2/|y|
1 log r
2
,
by (13) with ` = 0. Next, if
1 2
≤ |θ| ≤ π then X
Gθ (r) >
ak e−kr (1 − cos(θ − ky))
1/(8|y|)≤k≤1/(4|y|)
>
1 − cos( 41 ) e−1/4
X
ak
1/(8|y|)≤k≤1/(4|y|)
1 2 > c15 |y|−α ≥ c13 log , r again by Lemma 3. Now consider the case r1+3α/7 ≤ |y| ≤ r and −π ≤ θ ≤ π. We have X
Gθ (r) >
ak e−kr (1 − cos(θ − ky))
1/r≤k≤2/r
>
1 − cos(r3α/7 ) e−2
X
1/r≤k≤2/r
> c16 r
−α/7
> c13 17
1 log r
2
,
ak
by the inequality (20) and Lemma 3. For the remaining ranges (log(1/r))−2/α ≤ |y| ≤ π, it suffices, by (M3’), to consider the case (log(1/r))−2/α ≤ |y| < π/2 for which we use the same argument as in the proof of Lemma 4. For 0 < |y| < π/2 there exists an integer ` such that π/2 ≤ 2` |y| ≤ π. Using the inequality 1 − cos t ≥ 4−` (1 − cos t) , we obtain Gθ (r) > 4−`
X
ak e−kr 1 − cos 2` θ − 2` ky .
k≥1
Choose integer k such that 2` θ = 2kπ + θ0 where −π ≤ θ0 ≤ π. Thus by (M3’) Gθ (r) ≥ 4−` c3 (log(1/r))2+4/α c3 ≥ (log(1/r))2 , π2
2
≥
c3 2 2 y (log(1/r))2+4/α π2
for (log(1/r))−2/α ≤ |y| < π/2. This completes the proof. Lemma 6. For r3α/7 ≤ |θ| ≤ π and |y| ≤ r1+3α/7 , the estimate |Q(ρeiθ , e−r−iy )| c17 ρ −α/7 < exp − r Q(ρ, e−r ) (1 + ρ)2 holds uniformly in θ and y as r→ 0+ . Proof. We have X
Gθ (r) >
ak e−kr (1 − cos(θ − ky))
1/(3r)≤k≤1/(2r)
>
1 − cos( 12 r3α/7 ) e−1/2
X
ak
1/(3r)≤k≤1/(2r)
> c18 r−α/7 , by the inequality (20) and Lemma 3. We also need the asymptotic behaviors of Y (u, α) as u → ∞ and u→ 0, which are described by the following lemma. Lemma 7. The function Y (u, α) satisfies Y (u, α) = Yu0 (u, α) =
(log u)α+1 1 + O (log u)−2 , α(α + 1) (log u)α 1 + O (log u)−2 , αu
as |u| → +∞ in the sector | arg u| ≤ π − ε. 18
(34) (35)
Proof. By the integral representation u Y (u, α) = α
Z
∞
0
xα dx, ex + u
and the Mellin inversion formula 1 1 = 1+x 2πi
Z
1/2+i∞
1/2−i∞
π x−w dw sin πw
(| arg x| ≤ π − ε),
we obtain Y (u, α) = uΓ(α)
1 2πi
Z
1/2+i∞
1/2−i∞
πu−w dw. (1 − w)α+1 sin πw
Deforming the path of integration into a suitable Hankel-type contour in the style of [11], we deduce the result (34). The formula (35) is derived either in a completely analogous manner or by (34) using Ritt’s theorem (see [26, pp. 9–11]). Corollary 3. The function U (w) satisfies −1 α (α + 1)α/(α+1) A1/(α+1) 1 + w−2 , as w → +∞; 0 U (w) = α−α/(α+1) (AΓ(α))1/(α+1) ew/(α+1) (1 + O (ew )) , as w → −∞.
(36)
Proof. These formulae follow from (35), (10) and the definition of U . The limiting value of U 0 (w) as w → +∞ is a natural one in view of Corollary 2 and the next lemma. We next consider the solution to the system n = αAY (e% , α)r−α−1 m = Ae% Yu0 (e% , α)r−α ,
(37)
which will be needed when applying the two-dimensional saddle-point method. For convenience, set M0 = α−1 (α + 1)α/(α+1) A1/(α+1)
and M1 = α−α/(α+1) (AΓ(α))1/(α+1) .
Lemma 8. For m lying in the range 1 ≤ m ≤ (M0 − ε)nα/(α+1) , there exists a unique solution (%, r) to the system (37) such that r > 0 and % ∈ R. Proof. The solution to the first equation of (37) exists for all finite (and real) % and satisfies r=
αAY (e% , α) n 19
1/(α+1)
> 0.
(38)
Substituting this expression into the second equation of (37) yields m = Ae% Yu0 (e% , α) (αAY (e% , α))−α/(α+1) nα/(α+1) = U 0 (%)nα/(α+1) .
(39)
Thus there exists a unique real solution to (37) whenever m lies in the range (38). Moreover, if ε→ 0+ then by (36) %ε
−1/2
→ +∞
and r = %
A (α + 1)n
1/(α+1)
1 + O %−2
.
On the other hand, if m = o(nα/(α+1) ) then by (36) m m % = (α + 1) log 1 + O , M1 nα/(α+1) nα/(α+1) m αm 1+O . r = n nα/(α+1) In this case, we have % → −∞ and r → 0. Corollary 4. If m = µn + xσn , where x = o(σn ), then the solution (%, r) satisfies X x j X x j , %= %j and r = (αA)1/(α+1) n−1/(α+1) Y (1, α)1/(α+1) + rj σn σn j≥1
j≥1
with
U 0 (w) − U 0 (0) U 00 (0)w
−m
%m =
1 m−1 [w ] m
rm =
1 [wm−1 ]ew Yu0 (ew , α)Y (ew , α)−α/(α+1) m(α + 1)
,
U 0 (w) − U 0 (0) U 00 (0)w
−m
,
for m = 1, 2, 3, . . . The series are convergent. Proof. The relation q m = µn + xσn = U 0 (0)nα/(α+1) + x U 00 (0)nα/(α+1)
(x = o(σn ))
can be written into the more convenient form U 0 (%) − U 0 (0) x = , 00 U (0) σn in view of (39). Thus the solution (%, r) satisfies (40) by the Lagrange inversion formula.
20
(40)
4.2
The proof of Theorem 2
Let q(n, m) denote the number of restricted partitions of n having exactly m parts: q(n, m) = [um z n ]Q(u, z). Proposition 2. If m lies in the range m nα/(α+1)
m ≤ (M0 − ε)nα/(α+1) ,
and
then q(n, m) satisfies (1 + e% )D(0) 1+α −m%+nr+AY (e% ,α)r−α r e (1 + O (rα2 )) , 2πBb p where (%, r) is the unique real solution to the system (37), B = α(α + 1)AY (e% , α) and s αAe2% 00 (e% , α) − b = Ae% Yu0 (e% , α) + Ae2% Yuu Y 0 (e% , α)2 . (α + 1)Y (e% , α) u q(n, m) =
(41)
Proof. We use Cauchy’s integral formula q(n, m) = =
I I 1 u−m−1 z −n−1 Q(u, z) dz du (2πi)2 q(n) |u|=ρ |z|=e−r Z Z e−m%+nr π π −imθ+iny e Q(e%+iθ , e−r−iy ) dy dθ, 4π 2 q(n) −π −π
where (%, r) is chosen to satisfy the system (37). Set r0 = r3α/7 . The ranges of integration are split into three parts: (I)
|θ| ≤ r0 ,
|y| ≤ rr0
(II)
r0 < |θ| ≤ π,
(III)
|θ| ≤ π,
|y| ≤ rr0
rr0 < |y| ≤ π.
By Lemmas 5 and 6, we have ZZ
+
ZZ
(II)
(III)
c19 e% exp − (1 + e% )2
It remains to evaluate the integral J
re−m%+nr := 4π 2 =: J1 + J2 ,
Z
r0
−r0
Z
r0
−r0
RR
(I) .
1 + e%+iθ
1 log r
2 !
(c19 = min{c13 , c17 }).
(42)
By Lemma 2 and a change of variables (α1 = min{1, α0 })
D(0)
%+iθ ,α)(1+iy)−α r −α
e−imθ+inry+AY (e
21
(1 + O (rα1 )) dy dθ
say, where J1 corresponds to the main term in the integrand and Z r0 Z r0 %+iθ −α −α −m%+nr 1+α1 J2 e r eA|Y (e ,α)||1+iy| r dy dθ 0 −r0 Z−r r0 Z r0 %+iθ −α/2 )y 2 )r −α eA|Y (e ,α)|(1−(1−2 dy dθ, ≤ e−m%+nr r1+α1 −r0
−r0
where the inequality (26) was used. By the choice of r0 , we have r02 r−α → +∞. Thus Z r0 %+iθ −α −m%+nr 1+α1 +α/2 J2 e r eA|Y (e ,α)|r dθ. −r0
From the local expansion Y (e%+iθ , α) = Y (e% , α) + iθ
∞
Z 0
as θ ∼ 0, we deduce that |Y
(e%+iθ , α)|
≤Y
e% xα−1 θ2 dx − x % e +e 2 (e% , α)
− c20
θ2
∞
Z 0
e%+x xα−1 dx + O |θ|3 , x % 2 (e + e )
as r→ 0+ . It follows that % ,α)r −α
J2 e−m%+nr r1+α1 +α eAY (e
.
(43)
We now concentrate on the principal part J1 for which we introduce the following abbreviations: Yθ = Y (e%+iθ , α),
Y0 = Y (e% , α),
Y00 = Yu0 (e% , α),
00 Y000 = Yuu (e% , α),
000 Y0000 = Yuuu (e% , α).
Consider first the inner integral of J1 : J3 = Setting B =
eAYθ r 2π
−α
Z
r0
einry+AYθ ((1+iy)
−α −1
)r−α dy.
−r0
p
α(α + 1)AY0 and carrying out the change of variables y = rα/2 v/B, we obtain
rα/2 eAYθ r J3 = 2πB
−α
Br−α/7
Yθ v 2 (α + 2)Yθ 3 α/2 exp Υiv − + iv r + O rα v4 2Y0 6BY0 −Br−α/7
Z
dv,
where Υ = αAr −α/2 (Y0 − Yθ )/B. Since v 3 rα/2 , v 4 rα → 0 in the range of integration, we have Yθ v 2 (α + 2)Yθ 3 α/2 2 α 4 exp Υiv − + iv r +O r v = eΥiv−v /2 (1 + R1 ) , 2Y0 6BY0 where R1 = −
Y00 % 2 (α + 2)Yθ 3 α/2 e iθv + iv r + O θ2 v2 + θ2 v4 . 2Y0 6BY0
Substituting this estimate into J3 , we obtain −α Z Br −α/7 rα/2 eAYθ r 2 J3 = eΥiv−v /2 (1 + R1 ) dv 2πB −Br−α/7 −α Z ∞ α/2 AY r 1 2 −2α/7 r e θ 2 −α eΥiv−v /2 (1 + R1 ) dv + O r9α/14 eAr <e Yθ − 2 B r = . 2πB −∞ 22
The integral on the right-hand side can be evaluated by Cauchy’s theorem: Z ∞ 2 e−Υ /2 X 1 L (2`)! Υiv−v 2 /2 L e v dv = √ (iΥ)L−2` , 2π −∞ 2` 2` `! 2π 0≤2`≤L
for any L = 0, 1, 2, . . . Thus −α
rα/2 eAYθ r −Υ √ 2π B
J3 =
2 /2
(1 + R2 ) + O(R3 ),
where Y00 e% (α + 2)Yθ 3 iθ(Υ2 − 1) + (Υ − 3Υ)rα/2 , 2Y0 6BY0 −α 2 −α 2 −2α/7 /2 = θ2 rα/2 |Υ|2 + |Υ|4 eAr <e Yθ −<e Υ /2 + r9α/14 eAr <e Yθ −B r ,
R2 = R3
the error term being meaningful as long as Υ=
αA −α/2 r (Y0 − Yθ ) r−α/2 |θ| r−α/7 , B
which is obviously satisfied when |θ| ≤ r0 . Returning to J1 , we have r1+α/2 e−m%+nr √ J1 = 2π 2π B
Z
r0
−r0
1 + e%+iθ
D(0)
eimθ+AYθ r
−α −Υ2 /2
(1 + R2 ) + O(R3 ) dθ.
Using the expansion eimθ+AYθ r
−α −Υ2 /2
= eAY0 r
−α −bθ 2 r −α /2
(1 + R4 ) ,
where iAe% θ3 −α 3α2 A % 0 2 3α2 A 2% 0 00 0 % 00 2% 000 R4 = − r Y0 + 3e Y0 + e Y0 − e Y0 − e Y0 Y0 + O r−α θ4 , 2 2 6 B B we deduce, as the evaluations of J2 and J3 , that J1 =
(1 + e% )D(0) r1+α −m%+nr+AY0 r−α e (1 + O (rα )) . 2πBb
(44)
The formula (41) now follows from (42)–(44). Proof of Theorem 2. Suppose that m = µn + xσn ∈ Z+ , where x = o(σn ). From Proposition 1, we have q(n) =
2D(0) (αAY (1, α))1/(2α+2) −(1+α/2)/(α+1) U (0)nα/(α+1) p n e 1 + O n−α2 /(α+1) . 2πα(α + 1) 23
It follows, by (41), that Pr{$n = m} =
q(n, m) = Ln (%)eHn (%) 1 + O n−α2 , q(n)
where √ (1 + e% )D(0) α + 1 Ln (%) = r1+α n(1+α/2)/(α+1) , √ 1/(2α+2) D(0) 2 2πBb (αAY (1, α)) and α+1 (αAY (1, α))1/(α+1) nα/(α+1) α U (%) − U (0) − %U 0 (%) .
Hn (%) = −m% + nr + AY (e% , α)r−α − = nα/(α+1)
By Lagrange inversion formula, we have U (%) − U (0) − %U 0 (%) = −
U 00 (0) 2 x + ξ(x/σn ). 2σn2
The desired result (5) now follows from expanding Ln (%) at % = 0: Ln (%) =
(αAY (1, α))−α/(2α+2) n−α/(2α+2) r
2π
=
√
AYu0 (1, α)
1 2π σn
00 (1, α) AYuu
+ |x| 1+O , σn
−
αA 0 2 (α+1)Y (1,α) Yu (1, α)
(1 + O(%))
since 00
−α/(α+1)
U (0) = A (αAY (1, α))
AYu0 (1, α)
+
00 AYuu (1, α)
αA 0 2 − Y (1, α) . (α + 1)Y (1, α) u
This completes the proof of Theorem 2.
5
Unrestricted partitions
Recall that each of the p(n) unrestricted partitions of n (into parts λj ) is assumed to be equally likely, and that the random variable ωn represents the number of distinct parts in a random partition of n. The bivariate generating function of ωn satisfies (8). By the equation a j Y a Y 1 + (u − 1)z j j uz j = , 1+ 1 − zj 1 − zj j≥1
j≥1
we obtain the relation for the generating polynomials Pn (u) := p(n)E(uωn ) and Qn (u) = q(n)E(u$n ): Pn (u) =
X
p(j)Qn−j (u − 1)
0≤j≤n
24
for n = 1, 2, 3, . . .
To prove Theorem 3, we proceed along the same line of arguments as in the last section. The analytic properties we need are summarized in Propositions 3 and 4 below. The remaining analysis being parallel to the proof of Theorem 2, we omit the details. Recall that Z(u, s) = Γ(s)ζ(s + 1) + Y (u − 1, s). Proposition 3. Let u ∈ C, | arg u| ≤ π − ε, where ε > 0 being arbitrarily small but fixed number. If τ → 0 in the sector | arg τ | ≤ π/4, then 0
P (u, e−τ ) = eD (0) uD(0) τ −D(0) eAZ(u,α)τ
−α
(1 + O (|τ |α1 )) ,
(45)
uniformly in τ and u, where α1 = min{1, α0 }. Proof. (Sketch) We have −τ
log P (u, e
)=
X k≥1
u ak log 1 + kτ e −1
1 = 2πi
Z
α+1+i∞
D(s)Z(u, s)τ −s ds.
α+1−i∞
The expansion (45) is obtained by shifting the path of integration to the line <e s = −α0 and by computing the residues of the poles encountered (see [1, 25, 34]). To derive uniform estimates for the ratio |P (ρeiθ , e−r−iy )|/P (ρ, e−r ), we use the the following inequalities. Lemma 9. Let u = ρeiθ , where ρ > 0 and |θ| ≤ π. Then the inequalities |P (ρeiθ , e−r−iy )| ≤ e−c21 T (r) P (ρ, e−r )
(46)
|P (ρeiθ , e−r−iy )| < e−c22 G(3r) P (ρ, e−r )
(47)
and
hold for |y| ≤ π and r > 0, where c21 = min{ρ, ρ−1 }, c22 = T (r) =
X
1−e−4 4
min{1, ρ2 }, G(r) = G0 (r) and
2 ak e−kr 1 − e−kr (1 − cos(θ − ky)) .
k≥1
Proof. First, we have 2 2 iθ iθ iθ | iθ ρe ρe 2|ρe ρe 1 + ≤ 1 + 2<e + kr+iky kr+iky + kr+iky kr+iky kr+iky e −1 e |e (e − 1)| e − 1 2ρ 2ρ e2ρ ≤ 1 − kr (1 − cos(θ − ky)) + kr + kr e e − 1 (e − 1)2 ! 2 ρ 2ρ (1 − cos(θ − ky)) = 1 + kr 1− . 2 e −1 ekr (1 + ρ/(ekr − 1)) 25
Thus |P (ρeiθ , e−r−iy )| P (ρ, e−r )
≤
Y
1−
k≥1
≤ exp −ρ
2ρ (1 − cos(θ − ky)) 2
!ak /2
ekr (1 + ρ/(ekr − 1)) X
ak e−kr 1 +
k≥1
ekr
ρ −1
−2
(1 − cos(θ − ky)) .
From the inequalities
ρ 1 + kr e −1
−2
≥
(1 − e−kr )2 , if 0 < ρ ≤ 1; −2 −kr 2 ρ (1 − e ) , if ρ ≥ 1,
the result (46) follows. Next, we have 2 2 ρeiθ ρ 1 + ≤ 1 + kr+iky ekr+iky − 1 |e − 1| 2 ρ ρ2 ρ2 < 1 + kr − − e −1 (ekr − 1)2 |ekr+iky − 1|2 2 ρ ρ2 (ekr − 1)2 = 1 + kr 1 − kr 1 − kr+iky . e −1 (e + ρ − 1)2 |e − 1|2 Using the inequalities (see [17, Eq. (3.14)])
4vX 1+ (X − 1)2
−1
≤ e−4v/X
2e−kr (1 − cos ky) ≤ 4 1 − e−w ≥
(0 ≤ v ≤ 1 < X),
(r > 0, k = 1, 2, 3, . . . ), 1 − e−4 w (0 ≤ w ≤ 4),
1 4
we obtain (ekr − 1)2 1 − kr+iky |e − 1|2
−1 2ekr = 1 − 1 + kr (1 − cos ky) (e − 1)2 ≥ 1 − exp −2e−kr (1 − cos ky) ≥ 12 1 − e−4 e−kr (1 − cos ky) .
Thus 2 2 ρeiθ ρ (1 − e−4 )ρ2 e−kr < 1 + 1 + kr 1− (1 − cos ky) ekr+iky − 1 e −1 2(ekr + ρ − 1)2 2 ρ (1 − e−4 ) min{1, ρ2 } −3kr ≤ 1 + kr 1− e (1 − cos ky) , 2 e −1 26
in virtue of the inequalities kr
(e
2
+ ρ − 1) ≤
e2kr , if 0 < ρ ≤ 1; ρ2 e2kr , if ρ ≥ 1.
It follows that ak /2 Y |P (ρeiθ , e−r−iy )| −3kr < 1 − 2c e (1 − cos ky) , 22 P (ρ, e−r ) k≥1
from which we derive (47). Proposition 4. As r→ 0+ , the inequality ! |P (ρeiθ , e−r−iy )| 1 2 < exp −c23 log P (ρ, e−r ) r holds for (i) r1+3α/7 ≤ |y| ≤ π and −π ≤ θ ≤ π; and (ii) r3α/7 ≤ |θ| ≤ π and |y| ≤ r1+3α/7 . Here c23 can be taken as n o c23 = min c21 c18 (1 − e−1/3 )2 , 12 c22 min{c2 , c6 } . Proof. The result in the range (i) is a direct consequence of (19), (47) and Lemma 4 if we assume (M1) and (M3). For the second range we argue as the proof of Lemma 6. We have for θ and y in range (ii) X
T (r) >
ak e−kr (1 − e−kr )2 (1 − cos(θ − ky))
1/(3r)≤k≤1/(2r)
>
1 − cos( 21 r3α/7 ) e−1/2 (1 − e−1/3 )2
X
ak
1/(3r)≤k≤1/(2r)
1 2 > c24 r−α/7 > c24 log , r by the inequality (20) and Lemma 3, where c24 = c18 (1 − e−1/3 )2 . From (46), we obtain the required inequality. Finally, the asymptotic behaviors of V 0 (w) depend more on the values of α as described in the following result. Lemma 10. If w → +∞ then V 0 (w) satisfies V 0 (w) = α−1 (α + 1)α/(α+1) A1/(α+1) 1 + O w−2 + w−α−1
27
;
and if w → −∞ then 1/(α+1) Aπ αw/(α+1) (1−α)w e 1 + O e , if 0 < α < 1; sin πα 0 V (w) = −A1/2 w(1 − w)−1/2 ew/2 (1 + O (ew )) , if α = 1; α−α/(α+1) (AΓ(α)ζ(α))1/(α+1) ew/(α+1) 1 + O ew + e(α−1)w , if α > 1. Proof. These follow from the definition of Z and properties of Φ(z, s, v) (see [6, §1.11]). Note that U 0 (w) ∼ V 0 (w) as w → +∞, this being intuitively clear in view of (16) and the relation max ωn ∼ max $n .
6
Examples
In general, it is the condition (M3) or (M3’) that is more difficult to check. A sufficient condition for the validity of these two is the following condition of Haselgrove and Temperley (see [15, 31]): there exists a positive constant ϑ < 1 such that |g(r + iy)| < ϑg(r)
for r ≤ |y| ≤ π,
(48)
as r→ 0+ . This condition is satisfied, for example, when λj = j ` (see [15]), where ` is a positive integer. However, as remarked by Richmond [31], (48) is, in general, a difficult condition to work with. Sometimes it is easier to check the following condition: If the abscissa of convergence αϕ of the Dirichlet series Dϕ (s) =
ikϕ k −s k≥1 ak e
P
is < α for each
π/2 ≤ |ϕ| ≤ π, then (M3’) (and a fortiori (M3)) is satisfied. For, by Mellin inversion formula, we deduce g(r + iy) r−α+ε for π/2 ≤ |y| ≤ π, where αy < α − ε < α. Thus g(r) − |g(r + iy)| r−α . (a) Let λj = j ` for j = 1, 2, 3, . . . , where ` is a fixed positive integer. All our theorems apply. Further computations show that the mean and variance of $n satisfy E($n ) = µn + c25 + O n−1/(`+1)
and Var($n ) = σn2 + c26 + O n−1/(`+1) ,
where expressions for the two constants c25 and c26 are given in (32) and (33) (with α there replaced by 1/`). In particular, if ` = 1 we have √ 2 3 log 2 √ 3 log 2 1 −1/2 E($n ) = n+ − + O n , π π2 ! 4 √ √ 3 12 3 36(log 2)2 3 log 2 3 2 √ −1/2 1 Var($n ) = − (log 2) n + − − − + O n , π π3 2π 2 8 π4 π2
28
which improve an old result of Erd˝ os and Lehner in [7]. Likewise, we have E(ωn ) = µ en + c27 + O n−1/(`+1)
and Var(ωn ) = σ en2 + c28 + O n−1/(`+1) ,
where c27 = − 21 +
` (` + 1)ζ(1 + 1/`)
and c28 =
` 1 − 2−1/` − ζ(1 + 1/`)−1 . (` + 1)ζ(1 + 1/`)
(b) Let λj = j ` − 1, ` being a fixed integer ≥ 2 and j = 2, 3, 4, . . . We have (see [10, p. 45]) X X s + j − 1 ` −s D(s) = (j − 1) = (ζ(`s + `j) − 1) , j j≥0
j≥2
the last expression providing a meromorphic continuation of D into the whole s-plane with polynomial growth order at σ ± i∞. To check (M3), we argue as in [23, Example (c), pp. 39–40]. For r > 0 and 0 < |y| ≤ 1 g(r) − <e g(r + 2πiy) ≥
X
e−(j
` −1)r
≥ e−1/2
j≥2 cos 2π(j ` −1)y≤0
X
1.
(49)
2≤j≤r−1/` cos 2π(j ` −1)y≤0
By the Weyl criterion (see [22, p. 7]) the sequence ((j ` − 1)y)j≥2 is uniformly distributed mod 1 for irrational y. Thus for any ε > 0 the number of summands in the rightmost summation is at least ( 21 − ε)r−1/` for sufficiently small r. It follows that g(r) − <e g(r + 2πiy) ≥ 13 e−1/2 r−1/` , for irrational y as r→ 0+ . But g(r + 2πiy) is a continuous function of y (actually infinitely differentiable). Thus condition (M3) holds for y in the interval
1 4
≤ y ≤ 34 . Theorem 1 applies. Similarly, it
can be verified that other results are also applicable. (c) Let λj = [j β ], where β > 1 is not an integer. We have X s + m − 1 X D(s) = [j β ]−s = ζ(βs) + ζm (s), m j≥1
where ζm (s) =
P
j≥1 {j
β }m j −β(s+m)
m≥1
for <e s > β −1 − m. Here {t} denotes the fractional part of t.
Thus D admits meromorphic continuation into the half-plane <e s > β −1 − 1 > −1 with a simple pole at s = 1/β. Further analytic properties of D can be derived through those of ζm (s). Conditions (M3) and (M3’) can be checked as in the last example, the uniform distribution modulo 1 being, for example, a consequence of Weyl’s metric theorem (see [22, p. 32]) and the uniform continuity of g. 29
(d) Let λj = h + jd with (h, d) = 1, h, d being positive integers, d ≥ 2 and 1 ≤ h < d. This is an interesting case where the condition (M3) holds (see [25]) but (M3’) is violated. Thus Theorems 1 and 3 apply but Theorem 2 does not. This example describes an interesting “period-hereditary” property of the sequence {λj } on $n , namely, if the sequence {λj } is periodic3 , then $n is of maximum span > 1. On the other hand, the random variables ωn are less sensitive to such a property. (e) λj+`(`−1)/2 = `, j = 1, . . . , `, namely, ak = k and D(s) = ζ(s − 1). All our theorems again apply.
7
Extensions
The results in this paper are susceptible of many different extensions. We only discuss two typical cases in this section. First, let λj = 2j−1 . It is easy to show the relation Y
1 + uz
2j
=1+
j≥0
X
uν(k) z k ,
k≥1
where ν(k) denotes the number of 1’s in the binary representation of k. In this case D(s) = (1−2−s )−1 with a simple pole at s = 0. Despite this degenerate case, it might be possible to extend our results to the case α = 0, as suggested by the example D(s) = 1 + 2−s /(1 − 2−s )2 . On the other hand, a limiting Gaussian law for ωn can be established using the methods of this paper, the regularity conditions requiring somewhat different arguments (see [5]). For further information on Mahler-type partitions, see [29, 35]. Note that if j
Pn (u) = [z n ]
Y
j≥0
uz 2 1+ 1 − z 2j
!
(n = 0, 1, 2, . . . ),
then
P2m (u) = P2m−1 (u) + Pm (u) P2m+1 (u) = P2m−1 (u) + uPm (u)
(m = 1, 2, 3, . . . ),
with P0 (u) = 1 and P1 (u) = u. These relations are useful from a computational point of view. A natural question suggested by the above examples is that between the degenerate limiting behavior of $n and the limiting Gaussian behavior of ωn , from which point on will the “phase change (from a discrete limiting law to a continuous one) occur? More precisely, let q` (n) denote the 3 If there exist two integers c and d such that λj = c + dj with d > 1, then the sequence {λj } is called periodic; otherwise, it is called aperiodic.
30
number of partitions into parts 2j−1 in which each part is allowed to appear at most `-times. Define [`]
$n by 1+
X
n
z q` (n)E(u
[`]
$n
)=
n≥1
Y
1+u z
2j
+z
2·2j
+ ··· + z
`·2j
.
j≥0
[`]
The question is for what values of ` will the distribution of $n be asymptotically normal, an intuition [`]
[`+1]
being that the asymptotic normality of $n would imply the same property for $n ? P Next, take D(s) = p prime p−s . Numerical evidence suggests again that the limiting distributions of $n and of ωn will still be Gaussian. For results on the total number of partitions and the moments of the summands, see Roth and Szekeres [33] and Richmond [30]. As mentioned in the Introduction, the limiting distributions of the number of summands (counted with multiplicities) in unrestricted partitions are non-Gaussian for almost all partitions. However, it was predicted by Haselgrove and Temperley [15] that Gaussian law would appear if α ≥ 2, although a formal proof is still lacking. A concrete example is the generating function Y
1 − uz j
j≥1
−j
,
enumerating the number of plane partitions of n with a given sum of the diagonal parts or with a given trace (see [1, Ch. 11]).
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Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei 11529 Taiwan e-mail: [email protected]
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