W. Zhai

activity

20082022

Δ (x)

(II)

56 citations · 90 across the 26 of their papers we have counts for

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24 papers · 1 filter

math.NT2022

On some sums involving the counting function of non-isomorphic abelian groups

Haihong Fan, Wenguang Zhai

Let $a (n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivić proved an asymptotic formula of the sum $\sum_{n \leq x} a (n + a (n)) .$ In this paper, we…

math.NT2021

Continued fraction formulae involving ratios of three gamma functions

Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai

Via the MC-algorithm, in this paper we produce seven continued fraction formulae involving products and quotients of three gamma functions with three parameters, and another is an…

math.NT2021

The average size of Ramanujan sums over quadratic number fields(II)

Wenguang Zhai

In this paper we study Ramanujan sums $c_{m} (n)$ , where $m$ and $n$ are integral ideals in an arbitrary quadratic number field. We give some new results about…

math.NT2021★ 1 cited

The Average Size of Ramanujan Sums over Cubic Number Fields

Jing Ma, Huayan Sun, Wenguang Zhai

Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over bo…

math.NT2020

Average of Hardy's function at Gram points

Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai

Let $Z (t) = χ^{- 1/2} (1/2 + i t) ζ (1/2 + i t) = e^{i θ (t)} ζ (1/2 + i t)$ be Hardy's function and $g (n)$ be the $n$ -th Gram points defined by $θ (g (n)) = π n$ . Titchmarsh proved that $\sum_{n \leq N} Z(…

math.NT2020

Some mean value results related to Hardy's function

Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai

Let $ζ (s)$ and $Z (t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_{0}^{T} Z (t) ζ (1/2 + i t) d t$ and $\int_{0}^{T} Z^{2} (t) ζ (1/2 + i t) d t$ .…

math.NT2019

On the divisor problem with congruence conditions

Lirui Jia, Wenguang Zhai, Tianxin Cai

Let $d (n; r_{1}, q_{1}, r_{2}, q_{2})$ be the number of factorization $n = n_{1} n_{2}$ satisfying $n_{i} \equiv r_{i} (mod q_{i})$ ( $i = 1, 2$ ) and $Δ (x; r_{1}, q_{1}, r_{2}, q_{2})$ be the error term of the summ…

math.NT2017

On certain integrals involving the Dirichlet divisor problem

Aleksandar Ivić, Wenguang Zhai

We prove that $\int_{1}^{X} Δ (x) Δ_{3} (x) d x ≪ X^{13/9} lo g^{10/3} X, \int_{1}^{X} Δ (x) Δ_{4} (x) d x ≪_{ε} X^{25/16 + ε},$ where $Δ_{k} (x)$ is the error term in the asy…

math.NT2017★ 11 cited

On multivariable averages of divisor functions

László Tóth, Wenguang Zhai

We deduce asymptotic formulas for the sums $\sum_{n_{1}, \dots, n_{r} \leq x} f (n_{1} \dots n_{r})$ and $\sum_{n_{1}, \dots, n_{r} \leq x} f ([n_{1} \dots n_{r}])$ , where $r \geq 2$ is a fixed integer, $[n…

math.NT2016

On the Sign Changes of a Weighted Divisor Problem

Lirui Jia, Tianxin Cai, Wenguang Zhai

Let $S (x; \frac{a _{1}}{q _{1}}, \frac{a _{2}}{q _{2}}) = \sum^{'}_{mn \leq x} cos (2 π m \frac{a _{1}}{q _{1}}) sin (2 π n \frac{a _{2}}{q _{2}})$ with $x\geq q_1q_2, 1\leq a_i\leq…

math.NT2016★ 1 cited

A Weighted Divisor Problem

Lirui Jia, Wenguang Zhai

We study a weighted divisor function $mn \leq x \sum^{'} cos (2 π m θ_{1}) sin (2 π n θ_{2})$ , where $θ_{i} (0 < θ_{i} < 1)$ is a rational number. By connecting it with the divisor pro…

math.NT2015★ 1 cited

Mean square of the error term in the asymmetric many dimensional divisor problem

Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai

Let $\ba=(a_1,a_2,\ldots,a_k)$, where $a_{j} (j = 1, \dots, k)$ are positive integers such that $a_{1} \leq a_{2} \leq \dots \leq a_{k}$ . Let $d(\ba;n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$…

math.NT2015★ 9 cited

On the Tong-type identity and the mean square of the error term for an extended Selberg class

Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai

In 1956, Tong established an asymptotic formula for the mean square of the error term in the summatory function of the Piltz divisor function $d_{3} (n) .$ The aim of this paper is to…

math.NT2013

On a hybrid fourth moment involving the Riemann zeta-function

Aleksandar Ivić, Wenguang Zhai

We provide explicit ranges for $σ$ for which the asymptotic formula \begin{equation*} \int_0^T|ζ(1/2+it)|^4|ζ(σ+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(σ)\log^k T \quad(j\in\math…

math.NT2013

Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method

Xiaodong Cao, Wenguang Zhai

Let $a, b, c$ and $k$ be positive integers such that $1 \leq a \leq b, a < c < 2 (a + b), c \neq = b$ and $(a, b, c) = 1$ . Define the arithmetic function $f_{k} (a, b; c; n)$ by $$ \sum_{n=1}^{\infty}\fra…

math.NT2012

On the Dirichlet divisor problem in short intervals

Aleksandar Ivic, Wenguang Zhai

We present several new results involving $Δ (x + U) - Δ (x)$ , where $U = o (x)$ and $Δ (x) := n \leq x \sum d (n) - x lo g x - (2 γ - 1) x$ is the error term in the classical Dirichlet divisor prob…

math.NT2011

On the error term in Weyl's law for the Heisenberg manifolds (II)

Wenguang Zhai

In this paper we study the mean square of the error term in the Weyl's law of an irrational $(2 l + 1)$ -dimensional Heisenberg manifold . An asymptotic formula is established.

math.NT2008★ 1 cited

On the mean square of the error term for the two-dimensional divisor problems(II)

Xiaodong Cao, Wenguang Zhai

Let $Δ (a, b; x)$ denote the error term of the general two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value $\sum_{n\leq T}Δ^2(a,…

math.NT2008

On the mean square of the error term for the two-dimensional divisor problem (I)

Wenguang Zhai, Xiaodong Cao

Suppose $a$ and $b$ are two fixed positive integers such that $(a, b) = 1.$ In this paper we shall establish an asymptotic formula for the mean square of the error term $Δ_{a, b} (x)$ o…

math.NT2008★ 1 cited

On the $k$ -free divisor problem

Jun Furuya, Wenguang Zhai

Let $Δ^{(k)} (x)$ denote the error term of the $k$ -free divisor problem for $k \geq 2$ . In this paper we establish an asymptotic formula of the integral $\int_{1}^{T} ∣ Δ^{(k)} (x) ∣^{2} d x$ fo…

math.NT2008★ 8 cited

On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

Yoshio Tanigawa, Wenguang Zhai, Deyu Zhang

Let $Δ_{1} (x; ϕ)$ be the error term of the first Riesz means of the Rankin-Selberg zeta function. We study the higher power moments of $Δ_{1} (x; ϕ)$ and derive an asymptotic formula for…

math.NT2008

On the fourth power moment of $Δ (x)$ and $E (x)$ in short intervals

Yoshio Tanigawa, Wenguang Zhai

Let $Δ (x)$ and $E (x)$ be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper their fourth power moments for sho…

math.NT2008

On higher-power moments of $Δ (x)$ (III)

Wenguang Zhai

Let $Δ (x)$ be the error term of the Dirichlet divisor problem. An asymptotic formula with the error term $O (T^{53/28 + ε})$ is established for the integral $\int_{1}^{T} Δ^{4} (x) d x .$ Simila…

math.NT2008★ 56 cited

On higher-power moments of $Δ (x)$ (II)

Wenguang Zhai

Let $Δ (x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_{1}^{T} Δ^{k} (x) d x$ is established for any integer $3 \leq k \leq 9$ by an unifie…

On some sums involving the counting function of non-isomorphic abelian groups

Continued fraction formulae involving ratios of three gamma functions

The average size of Ramanujan sums over quadratic number fields(II)

The Average Size of Ramanujan Sums over Cubic Number Fields

Average of Hardy's function at Gram points

Some mean value results related to Hardy's function

On the divisor problem with congruence conditions

On certain integrals involving the Dirichlet divisor problem

On multivariable averages of divisor functions

On the Sign Changes of a Weighted Divisor Problem

A Weighted Divisor Problem

Mean square of the error term in the asymmetric many dimensional divisor problem

On the Tong-type identity and the mean square of the error term for an extended Selberg class

On a hybrid fourth moment involving the Riemann zeta-function

Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method

On the Dirichlet divisor problem in short intervals

On the error term in Weyl's law for the Heisenberg manifolds (II)

On the mean square of the error term for the two-dimensional divisor problems(II)

On the mean square of the error term for the two-dimensional divisor problem (I)

On the k-free divisor problem

On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

On the fourth power moment of Δ(x) and E(x) in short intervals

On higher-power moments of Δ(x)(III)

On higher-power moments of Δ(x)(II)

On the $k$ -free divisor problem

On the fourth power moment of $Δ (x)$ and $E (x)$ in short intervals

On higher-power moments of $Δ (x)$ (III)

On higher-power moments of $Δ (x)$ (II)