On asymptotics for lacunary partition functions
Alexander E Patkowski
本研究针对缺项分拆函数相关幂级数开展渐近分析，同时提供若干基本超几何级数的新分拆理论阐释作为实例，丰富了分拆函数的渐近研究内容。
我们对与缺项分拆函数相关的幂级数进行了渐近分析，并给出了部分基本超几何级数的新分拆理论解释作为示例。
We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples.
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An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Ok…
A power series is called lacunary if `almost all' of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han's extension of the Nekrasov-Okounkov formula. More precisely, we consider the modular forms \[F_{a,b,c}(z) := \frac{η(24az)^a η(24acz)^{b-a}}{η(24z)},\] defined in terms of the Dedekind $η$-function, for integers $a,c \geq 1$ where $b \geq 1$ is odd throughout. Serre determined the lacunarity of the series when $a = c = 1$. Later, Clader, Kemper, and Wage extended this result by allowing $a$ to be general, and completely classif…
For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case of a conjecture of Keith and Zanello on self-similarities of $b_3(n)$ modulo $2$. We next prove that the series $\sum_{n=0}^{\infty}b_9(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$. We also prove that the series $\sum_{n=0}^{\infty}b_9(4n)q^n$ is lacunary modulo $2$.
Let $f$ be a measurable function satisfying $$f(x+1)=f(x), \qquad \int_0^1 f(x) dx=0, \qquad \textrm{Var} ~f < + \infty,$$ and let $(n_k)_{k\ge 1}$ be a sequence of integers satisfying $n_{k+1}/n_k \ge q >1$ $(k=1, 2, \ldots)$. By the classical theory of lacunary series, under suitable Diophantine conditions on $n_k$, $(f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $(n_k)_{k\ge 1}$ as well, but as Fukuyama (2009) showed, the behavior of $f(n_kx)$ is generally not permutation-i…
Maximal volume of convex bodies avoiding random points and lacunary dilates
该研究确定了缺项序列下凸体避开随机点的最大体积渐近公式，并证明缺项性条件不可被稀疏性替代。
We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary $q$-series, and to a new class of composition-theoretic Dirichlet series.
It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutat…
Arithmetic properties of $5$-regular partitions into distinct parts
本文研究 $5$-正则分拆为不同部分的计数函数 $b'_5(n)$ 的算术性质，给出奇偶性完整刻画、模 $4$ 同余式，并证明相关生成函数的缺项性。
Moment generating functions and moderate deviation principles for lacunary trigonometric sums
本文研究缺项三角和的矩生成函数，确定算术效应起作用的精确阈值，证明其中偏差原理，证实尾概率在中心极限定理与大偏差原理区间内符合高斯行为。
Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits
本文研究稀疏三角求和的累积增长，发现其渐进行为对序列的算术结构敏感，超越数比极限情况下累积量线性增长，而代数比极限时可能呈现不同增长模式。
Asymptotic formulas of the number of various partitions are studied, like 3-colored partitions, concave partitions, certain plane partitions, partitions without small parts, the number of p-rings.
By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{…
We deduce from the strong form of the Hardy--Ramanujan asymptotics for the partition function $p(n)$ an asymptotics for $p_{-S}(n)$, the number of partitions of $n$ that do not use parts from a finite set $S$ of positive integers. We apply this to construct highly oscillating partition ideals.
An asymptotic formula for the number of partitions into p-cores is derived. As a byproduct some integer valued trigonometric sums are found
On the asymptotics of certain colored partitions
本文证明了某些双色分区对数的无限族渐近公式，其中无限子族验证了Guadalupe提出的猜想。
The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these polynomials inside the unit disk using the circle method.
In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.
In this paper, we investigate $π(m,n)$, the number of partitions of the \emph{bipartite number} $(m,n)$ into \emph{steadily decreasing} parts, introduced by L.Carlitz ['A problem in partitions', Duke Math Journal 30 (1963), 203--213]. We give a relation between $π(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, some uniform asymptotic formulas for $π(m,n)$ are established.
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<\dots <s_k$ and $\ell_1, \ell_2,\dots , \ell_k$. Let $g(\mathbf{s}, \mathbf{l}, n)$ be the number of $\ell$-colored pa…
In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $ n $ large enough, the two quantities are different and which of the two is bigger depends on the parity of $ n. $ This solves a recent conjecture formulated by Bringmann and Mahlburg (2012).
The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function.
Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.
We study the Dyson rank function $N(r,t;n)$, the number of partitions with rank congruent to $r$ modulo $t$. We first show that it is monotonic in $n$, and then show that it equidistributed as $n \rightarrow \infty$. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of $N(r,t;n)$.
In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, they introduce generalized partition functions $p(n)_{\textbf{e}}$. The group theory motivates the study of the asymptotics for these functions. Moreover, Bacher and de la Harpe also conjecture over 200 congruences for these functions which are analogous to the Ramanujan congruences for the unrestricted partition function $p(n)$. We obtain asymptotic formulas for all of the $p(n)_{\textbf{e}…
Improved asymptotics for moments of reciprocal sums for partitions into distinct parts
针对互异整数分拆中的倒数和与倒数平方和，本文改进了 $s_1(n)$ 和 $s_2(n)$ 的渐近估计，并处理了非模、非乘积生成函数带来的技术困难。
Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We also study the Witten zeta function $ζ_{\mathfrak{so…
Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under hypotheses on the roots of $f-f(0)$, we use the Hardy--Littlewood circle method, a polylogarithm identity, and the Matsumoto--Weng zeta function to derive asymptotic formulae for $p_{\mathcal{A}_f}(n)$ as $n$ tends to infinity. This generalises asymptotic formulae for the number of partitions into perfect $d$th powers, established by Vaughan for $d=2$, and Gafni …
We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two- and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in w…
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.
We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our formulas for the $M_2$-rank at roots of unity lead to asymptotics for certain combinations of $N2(r,m,n)$ (the number of partitions without repeated odd parts of $n$ with $M_2$-rank congruent to $r$ modulo $m$). This allows us to deduce inequalities among certain combinations of $N2(r,m,n)$. In particular, we resolve a few conjectured inequalities of Mao.
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm.
Simple error bounds for an asymptotic expansion of the partition function
本文为整数划分函数 $p(n)$ 的 Poincaré 型简化渐近展开建立了可计算的误差界，该界比近期文献更尖锐且推导更简单。
We introduce a notion of a group-partition for a finite Abelian group, which is a generalized notion of the standard partition. To obtain asymptoticdistributions of group-partition, we study the Dirichlet series for group-partitions by employing the generating function of the plane partition.
The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. These techniques are also shown to give the asymptotics of Sylvester waves.
Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases, and our main result states that while the two families of moment functions are asymptotically equal, the crank moments are always asymptotically larger than the rank moments. Andrews, Chan, and Kim primarily focused on one case, and proved the stronger result that the first crank moment is strictly larger than the first rank moment for all partitions by…
For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for $p_{\mathcal A}(n)$, where $\mathcal A$ is the set of $k$-th powers of primes (for fixed $k$). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into $k$-th powers. This new asymptotic formula is an extension of a pattern indicated by several results about …
The paper presents a discussion on the asymptotic formula for the number of plane partitions of a large positive integer.
We establish the asymptotic behavior of the Andrews $G_k(q)$ function as $q\to 1.$
Let A be a set of positive integers with gcd(A) = 1, and let p_A(n) be the partition function of A. Let c = π\sqrt(2/3). Let α> 0. It is proved that log p_A(n) ~ c\sqrt(αn) if and only if the set A has asymptotic density α.
Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.
Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions
本文通过推导ℓ-正则分拆与ℓ-互异分拆中钩长为1、2、3的钩的总数的渐近公式，建立了这两类分拆之间及其内部的钩长不等式，并证明相应钩数比值趋于依赖ℓ和t的常数。
Inequalities and asymptotics for hook numbers in restricted partitions
本文研究了受限划分中钩数的渐进行为，通过渐近公式证明了奇数部分划分在平均意义上比不同部分划分有更多钩数。
Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large $n$, and the stronger conjectures of Ulas are proved. The asymptotics of Wright's generalized Bessel functions are also treated.
Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of $F_n(x)$ have two scales of orders $n$ and $\sqrt{n}$ and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.
Asymptotics of partition parts in arithmetic progressions
研究算术级数中分拆部分的分布，通过分析生成函数中的非模艾森斯坦级数，获得捕获所有指数增长项的新近结果。
Asymptotics for $d$-fold partition diamonds and related infinite products
本文证明了$n$的$d$重分拆钻石及其Schmidt型对应物数量的渐近公式，通过研究无穷乘积的渐近行为，并指出了与数学物理和Bloch群的潜在联系。
A sharp threshold for arithmetic effects on the tail probabilities of lacunary sums
本文研究间隙和尾部概率的算术效应，证明存在尖锐阈值：当丢番图方程解数满足$L(N,a,b,c) \le N / g_N$时，间隙和尾部概率在$t$≤$\sqrt{2 \log g_N}$时呈高斯行为，且该准则最优。
Asymptotics and inequalities for the distinct partition function
本文研究了偏移不同划分函数的渐近分析与不等式，推导出四次二元形式不变量及相关不等式的精确阈值。
A remark on decay rates of odd partitions: An application of spectral asymptotics of the Neumann--Poincaré operators
本研究借助Neumann--Poincaré算子的谱渐近性质，以该算子为紧线性算子工具，证明了一个与分拆问题相关的、此前仅能通过紧算子特征值渐近行为验证的定理。
Let $j,n$ be even positive integers, and let $\overline{p}_j(n)$ denote the number of partitions with BG-rank $j$, and $\overline{p}_j(a,b;n)$ to be the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. We give asymptotics for both statistics, and show that $\overline{p}_j(a,b;n)$ is asymptotically equidistributed over the congruence classes modulo $b$. We also show that each of $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy all higher-order Turán inequalities.
Asymptotics of parity biases for partitions into distinct parts via Nahm sums
本文推广了Kim等人的奇偶偏差研究，利用圆法给出互异分拆的模N偏差渐近结果，同时提供了分析Nahm型q-超几何级数渐近性的独立有用技术。
For primes $p$, we study the maximal possible size of a $p$-core $p'$-partition (a partition with no hook lengths or parts divisible by $p$). McDowell recently proved that the maximum is attained by a unique partition, say $Λ_p$. Using his graph theoretic description of $Λ_p$, we prove for $p > 10^6$ that \[\frac{1}{24}p^6 - p^5\sqrt{p} < |Λ_p| < \frac{1}{24}p^6 - \frac{1}{200}p^5\sqrt{p},\] which shows that $|Λ_p| \sim p^6/24$ as $p \to \infty$.
Let m be a positive integer, and let A be the set of all positive integers that belong to a union of r distinct congruence classes modulo m. We assume that the elements of A are relatively prime, that is, gcd(A) = 1. Let p_A(n) denote the number of partitions of n into parts belonging to A. We obtain the asymptotic formula log p_A(n) ~ π\sqrt(2rn/3m). The proof is based on Erdos's elementary method to obtain the asymptotic formula for the usual partition function p(n).
Asymptotics for moments of the minimal partition excludant in congruence classes
本文研究整数划分中频率小于给定数的最小正整数的矩，针对特定同余类建立渐近公式，并证明固定模数下各矩渐近相等。
On the asymptotic behavior for partitions separated by parity
本文运用 Ingham 的 Tauberian 定理，为 Andrews 在奇偶分隔分拆研究中提出的八个生成函数分别给出渐近主项。
We prove asymptotic formulas for the complex coefficients of $(ζq;q)_\infty^{-1}$, where $ζ$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions of $n$ with largest part congruent $a$ modulo $b$. Our results imply that, as $n \to \infty$, the difference $p(a_1,b,n)-p(a_2,b,n)$ for $a_1 \neq a_2$ oscillates like a cosine, when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of $\{p(a,b,n)\}_{1 \leq a \leq b}$.
Asymptotics of Reciprocal Supernorm Partition Statistics
本文研究整数划分上超范数倒数之和的统计量，在大小、周长和最大部分三个系综下，证明其累积统计量均渐近于 e^γ log n。
Asymptotics for partitions over the Fibonacci numbers and related sequences
本文结合经典分拆理论与 Fibonacci Dirichlet 级数的新结果，首次给出整数表示为非互异斐波那契数之和的方式数 $p_F(n)$ 的渐近公式，并将类似结果推广到一般线性递推序列。
We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple l…
Asymptotic Formula for $(t+1)$-Regular Partitions
本文基于 saddle point 方法，推导了 (t+1) 正则划分数 p(N,t) 的渐近公式，并讨论了其在对称群特征表零点估计中的应用。
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