# Nian Hong ZhouUniversity of Vienna | UniWien · Fakultät für Mathematik

Nian Hong Zhou

PhD in Mathematics

## About

22

Publications

1,095

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

18

Citations

Introduction

**Skills and Expertise**

Education

September 2016 - June 2020

September 2012 - June 2016

## Publications

Publications (22)

Let α>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document} be an irrational number. We establish asymptotic formulas for the number of partitions...

In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties fo...

In this paper, we study properties of the coefficients appearing in the q -series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p , raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle meth...

Let $N_k(m, n)$ denote the number of partitions of $n$ with Garvan $k$-rank $m$. It is well-known that Andrews–Garvan–Dyson’s crank and Dyson’s rank are the $k$-rank for $k = 1$ and $k = 2$, respectively. In this paper, we prove that the sequence $\{N_k(m, n)\}_{|m|≤n−k−71}$ is log-concave for all sufficiently large $n$ and each integer $k$. In par...

An edge-coloring of a graph G is injective if for any two distinct edges e1 and e2, the colors of e1 and e2 are distinct if they are at distance 2 in G or in a common triangle. The injective chromatic index of G, χinj′(G), is the minimum number of colors needed for an injective edge-coloring of G. In this paper, we consider the list version of the...

Let \(F(x) \in \mathbb {Z}[x_1 , x_2 ,\ldots , x_n ]\), \(n\ge 3\), be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $$\begin{aligned} \Sigma _{k,F}(X; {{\mathcal {B}}})=\sum _{\mathbf{x}\in X{\mathcal {B}}\cap {\mathbb {Z}}^{n}}\tau _{k}\left( F(\mathbf{x})...

In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen (Appl Math Comput 283:385–396, 2016).

Let [Formula: see text] be a polynomial with the property that corresponding to every prime [Formula: see text] there exists an integer [Formula: see text] such that [Formula: see text]. In this paper, we establish some equidistributed results between the number of partitions of an integer [Formula: see text] whose parts are taken from the sequence...

Let $\kappa$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be given. Let $p_{\kappa, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(\lfloor \ell^{\kappa}\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hard...

Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations
$$
n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots,
$$
for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\nb\times\{1,2\}$, a generating function and s...

In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coeffic...

Let $\alpha>1$ be an irrational number. We establish asymptotic formulas for the number of partitions of n into summands and distinct summands, chosen from the Beatty sequence $(\lfloor m\alpha\rfloor)_{m\in \mathbb{N}}$. This improves some results of Erd\"{o}s and Richmond established in 1977.

Let $f: \mathbb{Z}_+\rightarrow\mathbb{Z}_+$ be a polynomial with the property that corresponding to
every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results between the number of partitions of integer $n$ whose parts taken from the sequence $\{f(\ell)\}_{\ell=1}^{\infty}$ a...

In this paper, we investigate $\unicode[STIX]{x1D70B}(m,n)$ , the number of partitions of the bipartite number$(m,n)$ into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between $\unicode[STIX]{x1D70B}(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Us...

Let $F({\bf x})\in\mathbb{Z}[x_1,x_2,\dots,x_n]$ be a quadratic polynomial in $n\geq 3$ variables with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $$ \Sigma_{k,F}(X; {\mathcal{B}})=\sum_{{\bf x}\in X\mathcal{B}\cap\mathbb{Z}^{n}}\tau_{k}\left(F({\bf x})\right), $$ for $X$ tending to infinity, wh...

In this paper, we establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions which play a key role in the theory of integer partitions, algebraic geometry, and theoretical physics. The main results improve the recent works of Kathrin Bringmann, Jan Manschot and Jehanne Dousse on this topic.

In this paper we prove a conjecture of Farkas and Kra, which is a modular equation involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjecture holds for all odd integers $k\ge 3$. A new modular equation of Farkas and Kra type is also established.

Let $k$ be a positive integer and $m$ be an integer. Garvan's $k$-rank $N_k(m,n)$ is the number of partitions of $n$ into at least $(k-1)$ successive Durfee squares with $k$-rank equal to $m$. In this paper give some asymptotics for $N_k(m,n)$ with $|m|\ge \sqrt{n}$ as $n\rightarrow \infty$. As a corollary,
we give a more complete answer for the Dy...

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7 (9) (2013), 21...

Let pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_n$$\end{document} denote the n-th prime number, and let dn=pn+1-pn\documentclass[12pt]{minimal} \usepackage{amsm...

Let $(a;q)_{\infty}$ be the $q$-Pochhammer symbol and $\mathrm{li}_2(x)$ be the dilogarithm function. Let $\prod_{\alpha,\beta,\gamma}$ be a finite product with every triple $(\alpha,\beta,\gamma)\in(\mathbb{R}_{>0})^3$ and $S_{\alpha\beta\gamma}\in\mathbb{R}$. Also let the triple $(A,B,v)\in\left(\mathbb{R}_{>0}\times\mathbb{R}^2\right)\cup\left(\...

In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.