A central limit theorem for integer partitions into small powers

The study of the well-known partition function p(n) counting the number of solutions to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = a_{1} + \dots + a_{\ell }$$\end{document}n=a1+⋯+aℓ with integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le a_{1} \le \dots \le a_{\ell }$$\end{document}1≤a1≤⋯≤aℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n=\left\lfloor a_1^\alpha \right\rfloor +\cdots +\left\lfloor a_\ell ^\alpha \right\rfloor \end{aligned}$$\end{document}n=a1α+⋯+aℓαwith \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a_1< \cdots < a_\ell $$\end{document}1≤a1<⋯<aℓ and some fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< \alpha < 1$$\end{document}0<α<1. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.


Introduction
For a positive integer n, let f (n) denote the number of unordered factorizations as products of integer factors greater than 1.Balasubramanian and Luca [1] considered the set In order to provide an upper bound for |F(x)|, they had to analyse the number q(n) of partitions of n of the form , where ⌊x⌋ denotes the integer part of x.Chen and Li [2] proved a similar result, and Luca and Ralaivaosaona [14] refined the previous results to obtain the asymptotic formula q(n) ∼ Kn −8/9 exp 6ζ(3) 1/3 4 2/3 n 2/3 + ζ(2) (4ζ(3)) 1/3 n 1/3 , where K = (4ζ(3)) 7/18 πA 2 √ 12 exp 4ζ(3) − ζ(2) 2 24ζ(3) and A is the Glaisher-Kinkelin constant.Li and Chen [11,12] extended the result to arbitrary powers 0 < α < 1 not being of the form α = 1/m for a positive integer m.Finally, Li and Wu [13] considered the case of α = 1/m: They obtained a complete expansion along lines similar to the one in Tenenbaum, Wu and Li [18] as well as in Debruyne and Tenenbaum [4].
In the present paper, we take a different point of view.For fixed 0 < α < 1, we consider the distribution of the length of restricted α-partitions.A restricted α-partition of n is a representation of n of the form and ℓ is called its length.In contrast to the corresponding counting function ω(n) in the theory of primes, the problem of restricted partitions is rarely discussed in the theory of integer partitions.A short mentioning can be found in Erdős and Lehner [5].The study of distinct components was introduced by Wilf [20].First considerations on the length of partitions of integers can be found in Goh and Schmutz [9].Their result was extended by Schmutz [17] to multivariate cases under the Meinardus' scheme (cf.Meinardus [16]), and Hwang [10] provided an extended version with weaker necessary conditions.Madritsch and Wagner [15] finally considered sets with digital restrictions, which lead to a Dirichlet generating function having equidistant poles along a vertical line in the complex plane.In the present paper, we use a similar method but for the case of multiple poles on the real line.

Main Result
Let 0 < α < 1 be a fixed real number.We let Π(n) denote the set of partitions of a positive integer n into parts ⌊a α j ⌋ where each a j occurs at most once.These partitions are called (restricted) α-partitions for short.Furthermore, let q(n) = |Π(n)| be the cardinality of the set Π(n).Moreover, we let Π(n, k) denote the subset of partitions Π(n) whose length (number of summands) is k and q(n, k) = |Π(n, k)| is its cardinality.
In the present work, we consider the random variable ̟ n counting the number of summands in a random α-partition of n.The probability distribution of ω n is given by P(ω n = k) = q(n, k)/q(n).In order to obtain a central limit theorem for ̟ n , we have to carefully analyse the associated bivariate generating function Q(z, u), which is given by Furthermore, for a fixed integer k ≥ 1, we let g(k) denote the number of integers n ≥ 1 satisfying ⌊n α ⌋ = k, i.e., g(k) := (k + 1) β − k β with β := 1/α.Then the following lemma holds.Lemma 1.With the notation above, we have Furthermore, the mean length µ n = E(̟ n ) and its variance σ 2 n = V(̟ n ) are given by Proof.By the definition of g(k), we have ⌊a α j ⌋ = k for exactly g(k) different integers a j .Thus, it follows that Furthermore, it holds that Next, we turn our attention to the mean length µ n .For the derivative of Q with respect to u we obtain Thus, it follows that The equation for the variance follows similarly.
Now we can state the main theorem of this work as follows.
Theorem 2. Let 0 < α < 1 and let ̟ n be the random variable counting the number of summands in a random partition of n into α-powers.Then ̟ n is asymptotically normally distributed with mean uniformly for all x as n → ∞.The mean µ n and the variance σ 2 n are given by (2.1)

and
(2.2) where η is the implicit solution of .
Explicit formulae for the occurring constants c 1 and c 2 are given in (4.12) and (4.13), respectively.
Finally, the tails of the distribution satisfy the exponential bounds and the analogous inequalities also hold for P ̟−µn σn ≤ −x .
This result fits into the series of other results on partitions in integers of the form ⌊k α ⌋ with k ≥ 1.In particular, if α = 1, then we have the classic case of partitions and Erdős and Lehner [5] showed that µ n ∼ cn 1/2 .For α > 1, not every integer has a representation of the form ⌊k α ⌋ and there are gaps in the set {⌊k α ⌋ | k ∈ N}.This led Hwang [10] to the result µ n ∼ cn 1/(α+1) .Consequently, our result µ n ∼ c 1 n 1/(α+1) seems to be a natural extension of these results.
One of the main difficulties of the case 0 < α < 1 lies in the special structure of the function g(k).In particular, if α = 1/m with m ≥ 2 being an integer, then the parts of the partitions are mth roots and g(k) is given by the polynomial However, in the general case of α ∈ Q, we have an additional error term (see (4.2)) of which no explicit form is known.This makes the analysis more involved.
Finally we want to mention that a local version of this central limit theorem is the topic of a subsequent project.In particular, it seems that the above mentioned error term of the function g(k) needs further considerations in this case.

Main Idea, Outline and Tools for the Proof
The proof of our main theorem consists of analytic and probabilistic parts.In the analytic part, we use Mellin transform and the saddle-point method.The probabilistic part is based on the use of Curtiss' theorem for moment-generating functions.Before we give the details of the proof, this section is dedicated to give an overview of the main techniques and tools.
We first note that the central limit theorem for the random variable ̟ n is equivalent to the fact that the normalized moment-generating function M n (t) = E(e (̟n−µn)t/σn ) tends to e t 2 /2 for t → ∞.Consequently, we will show this limit.Furthermore, the mentioned tail estimates can be obtained by Chernov's bound.
In other words, it is sufficient for our purpose to obtain the coefficient of z n in Q(z, u).By Cauchy's integral formula, we derive A standard transformation yields for r > 0 that For the integral in (3.1), we use the well-known saddle-point method, also known as the method of steepest decent.The main application of this method is to obtain estimates for integrals of the form π −π e g(r+it) dt for some suitable function g.We choose t n > 0 in order to split the integral up into two parts, one near the positive real axis and the other one for the rest, i.e., For the second integral, we compare the contribution of the integrand with the contribution from the real line, i.e., we estimate e g(r+it)−Re(g(r)) .This will contribute to the error term.
For the first integral in (3.2), we use a third order Taylor expansion of g(r + it) around t = 0, which is Now we choose r such that the first derivative g ′ (r) vanishes.Then in the integral we are left with |t|<tn e g(r+it) dt = e g(r) Now the integrand is the one of a Gaussian integral and so we add the missing part.The Gaussian integral contributes to the main part, and we need to analyse g ′′ (r) and g ′′′ (r) in order to show that all our transformations and estimates are valid.For more details on the saddle-point method, we refer the interested reader to Flajolet and Sedgewick [8,Chapter VIII].
The estimates for g ′′ (r) and g ′′′ (r) are based on singular analysis using the well-known Mellin transform.The Mellin transform h * (s) of a function h is defined by The most important property for our considerations is the so called rescaling rule, which is given by see [6,Theorem 1].This provides a link between a generating function and its Dirichlet generating function.For a detailed account on this integral transformation, we refer the interested reader to the work of Flajolet, Gourdon and Dumas [6] and to the work of Flajolet, Grabner, Kirschenhofer, Prodinger and Tichy [7].
Let δ > 0. Throughout the rest of our paper we assume δ ≤ u ≤ δ −1 and by "uniformly in u" we always mean "uniformly as δ ≤ u ≤ δ −1 ".The following theorem concerning the Mellin transform will be helpful multiple times in the proof of our main result.
Theorem 3 (Converse Mapping [6,Theorem 4]).Let f (x) be continuous in (0, +∞) with Mellin transform f * (s) having a nonempty fundamental strip α, β .Assume that f * (s) admits a meromorphic continuation to the strip γ, β, for some γ < α with a finite number of poles there, and is analytic on Re(s) = γ.Assume also that there exists a real number η ∈ (α, β) such that for s ∈ γ, α , then an asymptotic expansion of f (x) at 0 is given by Roughly speaking, converse mapping says that the asymptotic expansion of f is determined by the poles of its Mellin transform.
In the present paper, we apply this converse mapping to products of the Gamma function Γ(s), the Riemann zeta function ζ(s) and the polylogarithm Li s (z), which are defined by respectively.To apply converse mapping, we need to show that (3.4) as well as (3.5) are both fulfilled for these functions.Stirling's formula yields for the Gamma function that for a ≤ x ≤ b and |y| ≥ 1.Furthermore, the Riemann zeta function satisfies For the polylogarithm we follow the ideas of Flajolet and Sedgewick [8,VI.8].This is a good application of the converse mapping, so we want to reproduce it here: First of all, let w = − log z and define the function This is a harmonic sum and so we apply Mellin transform theory.The Mellin transform of Λ(w) satisfies for Re(s) > max(0, 1 − α).The Gamma function has simple poles in the negative integers and ζ(s + α) has a simple pole in 1 − α.Thus, the application of converse mapping (Theorem 3) yields Using these estimates, we obtain an asymptotic formula for Q n (u) of the form Recall that .
Using implicit differentiation, we obtain a Taylor expansion for the moment-generating function, which yields proving the central limit theorem for ̟ n .Finally, we will use Chernov's bound for the tail estimates.

Proof of the Main Result
To prove Theorem 2, we apply the method we have outlined in the previous section.As indicated above, we choose r = r(n, u) such that the first derivative in (3.3) vanishes, i.e., Since the sum is decreasing in r, we see that this equation has a unique solution, which is the saddle point.The main value of the integral in (3.1) lies around the positive real axis.We set t n = r 1+3β/7 and split the integral into two ranges, namely into where and 4.1.Estimate of I 1 .We start our considerations with the central integral I 1 and show the following lemma on its asymptotic behavior.
Proof.It holds that (4.1) Let m be the integer with m < β = 1/α ≤ m + 1.Since For j ≥ 0, we analogously obtain (4.4) All infinite sums in (4.4) are of the form with γ > 0. Let H γ,j (s, u) denote the Mellin transform of h γ,j (τ, u) with respect to τ , then H γ,j (s, u) is given by The function H γ,j (s, u) converges for Re s > γ + j + 2 and its only pole in the range j + 1 2 < Re s ≤ γ +j +2 is the one from ζ(s−γ −j) in s = γ +j +1.The Riemann zeta function and the polylogarithm grow only polynomially, whereas the Gamma function decreases exponentially on every vertical line in the complex plane; see Section 3. Thus, we can apply converse mapping (Theorem 3) and obtain By plugging everything into (4.4),we obtain (4.5) For the saddle point n, this results in whereas the second derivative is given by For the third derivative occurring in the error term in (4.1), we have We estimate this expression following [15]: Let k 0 = r −(1+c) for some constant c > 0 and write v = u −1 for short.We split the sum into two parts, according to whether k ≤ k 0 or not.For the sum over large k, we obtain (4.7) −i For the remaining sum we note that Therefore, we get Using the Mellin transform and converse mapping, this results in By combining this with (4.7), we obtain All in all, this leads to the expansion For the integral I 1 , we thus obtain Finally, we change the integral to a Gaussian integral and get (4.9) 4.2.Estimate of I 2 .Next, we prove the following asymptotic upper bound for the integral I 2 .
Lemma 5.For I 2 , it holds that where c 3 is a constant uniformly in u.
For the proof of this estimate, we need the following two lemmas.The first lemma provides an upper bound for some exponential that will occur later on, whereas the second one says that |Q(e −r−it , u)| is small compared to Q(e r , u).These results are the main ingredients for the proof of Lemma 5.
Proof.First of all, we note that By the mean value theorem, there exists ξ ∈ (k, k + 1) such that which leads to see also Li and Chen [12, Proof of Lemma 2.6].Moreover, we have Using this, it holds that where the last estimate follows by Lemma 6.Following the lines of Li and Chen [12] again, we further have ) and e −r = 1 − r + O(r 2 ), we find This further implies This lower bound results in An application of Lemma 7 thus yields for a certain constant c 3 uniformly in u, as stated.4.3.Estimate of Q n (u) and Moment-Generating Function.After estimating the main term and the contribution away from the real axis, we put (4.9) and (4.10) together and get ) .
The program for the last part of the proof is to consider the moment-generating function using this asymptotic expansion.This will prove the central limit theorem.Finally at the end of this section we use Chernov's bound in order to obtain the desired tail estimates.Now we will consider the moment-generating function for the random variable ̟ n (the number of summands in a random partition of n).To this end, let M n (t) = E(e (̟n−µn)t/σn ), where t is real and µ n and σ n are the mean and the standard deviation as defined in (2.1) and (2.2), respectively.Then the following estimation holds.Lemma 8.For bounded t, it holds that as n → ∞.
Proof.First of all, we observe that ), e t/σn ) exp − µ n t σ n + nr(n, e t/σn ) + f (r(n, e t/σn ), e t/σn ) − nr(n, 1) − f (r(n, 1), 1) + O(r 2β/7 ) .and similarly as well as We now need estimates for the partial derivatives of f .Estimates for partial derivatives with respect to τ follow from our considerations in Section 4.1.For derivatives with respect to u, we take the derivative of the corresponding Mellin transform and then obtain the estimate via the converse mapping again.Let us exemplarily illustrate this approach for f uτ : By (4.5), f τ is given by The Mellin transform of h β−ν,1 is given by Taking the derivative of H β−ν,1 with respect to u thus yields Consequently, converse mapping implies for f τ u that In a similar manner, we determine estimates for the other partial derivatives and obtain From these estimates it follows that r u , r uu , r uuu ≪ n −1/(β+1) uniformly in u.Expanding r(n, e t/σn ) and f (r(e t/σn , n), e t/σn ) around t = 0 yields In a similar way we get that B 2 (r(n, 1), 1) B 2 (r(n, e t/σn ), e t/σn ) = 1 + O t σ n .

(4. 11 )
Instead of representing the function with respect to u, we interpret r = r(n, u) as a function of n and u and use implicit differentiation on (4.6) as in Madritsch and Wagner[15] and obtainr u = r u (n, u) = − f uτ (r, u) f τ τ (r, u) =k≥1 kg(k)e kr (e kr +u) 2 k≥1 uk 2 g(k)e kr (e kr +u) 2