Continued fraction expression of the Mathieu series
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
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arXiv subjects
Wenguang Zhai
Publications and source records attributed to Wenguang Zhai.
The fastest possible continued fraction approximations of a class of functions
The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are proposed.
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On some mean value results for the zeta-function and a divisor problem II
Let $d(n)$ be the number of divisors of $n$, let $γ$ denote Euler's constant and $$ Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1) $$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. It is shown that $$ \int_0^TΔ(t)|ζ(1/2+it)|^2\,dt \ll T(\log T)^{4}. $$ Further, if $2\le k\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \int_1^{T}Δ^{k}(t)|ζ(1/2+it)|^2\,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-η_k+\varepsilon}), $$ where $c_1(k)$ and $c_2(k)$ are explicit constants, and where $$η_2= 3/20, η_3= η_4=1/10,\ η_5=3/80,\ η_6=35/4742,\ η_7=17/6312,\ η_8=8/9433.$$ The results depend on the power moments of $Δ(t)$ and $E(T)$, the classical error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$.
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Mean square of the error term in the asymmetric many dimensional divisor problem
Let $\ba=(a_1,a_2,\ldots,a_k)$, where $a_j \ (j=1,\ldots,k)$ are positive integers such that $a_1 \leq a_2 \leq \cdots \leq a_k$. Let $d(\ba;n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and $Δ(\ba;x)$ be the error term of the summatory function of $d(\ba;n)$. In this paper we show an asymptotic formula of the mean square of $Δ(\ba;x)$ under a certain condition. Furthermore, in the cases $k=2$ and 3, we give unconditional asymptotic formulas for these mean squares.
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On the Tong-type identity and the mean square of the error term for an extended Selberg class
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 generalize Tong's method to a class of Dirichlet series that satisfy a functional equation. As an application, we can establish the asymptotic formulas for the mean square of the error terms for a class of functions in the well-known Selberg class. The Tong-type identity and formula established in this paper can be viewed as an analogue of the well-known Voronoï's formula.
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On a hybrid fourth moment involving the Riemann zeta-function
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\mathbb N) \end{equation*} holds as $T\rightarrow \infty$, when $1\leq j \leq 6$, where $ζ(s)$ is the Riemann zeta-function. The obtained ranges improve on an earlier result of the authors [Annales Univ. Sci. Budapest., Sect. Comp. {\bf38}(2012), 233-244]. An application to a divisor problem is also given
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Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method
Let $a, b,c $ and $k$ be positive integers such that $1\leq a\leq b,a 1.$$ Let $Δ_k(a,b;c;x)$ denote the error term of the summatory function of the function $f_k(a,b;c;n).$ IN this paper we shall give two expressions of $Δ_k(a,b;c;x)$. As applications, we study the so-called $(l,r)$-integers, the generalized square-full integers, the $e-r$-free integers, the divisor problem over $r$-free integers, the $e$-square-free integers. An important tool is a generalization of a method of H. L. Montgomery and R. C. Vaughan.
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On the Dirichlet divisor problem in short intervals
We present several new results involving $Δ(x+U)-Δ(x)$, where $U = o(x)$ and $$ Δ(x):=\sum_{n\le x}d(n)-x\log x-(2γ-1)x $$ is the error term in the classical Dirichlet divisor problem.
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On the error term in Weyl's law for the Heisenberg manifolds (II)
In this paper we study the mean square of the error term in the Weyl's law of an irrational $(2l+1)$-dimensional Heisenberg manifold . An asymptotic formula is established.
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Higher moments of the error term in the divisor problem
It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}Δ^4_k(x)\d x \ll_εX^ε\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $Δ_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\"ı}--type formula for $Δ_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $Δ_2(x)$.
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On the mean square of the error term for the two-dimensional divisor problems(II)
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 $\sum_{n\leq T}Δ^2(a,b;n)$ and the continuous mean value $\int_1^TΔ^2(a,b;x)dx$.
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On the mean square of the error term for the two-dimensional divisor problem (I)
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 $Δ_{a,b}(x)$ of the general two-dimensional divisor problem.
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On the error term in Weyl's law for the Heisenberg manifolds
For a fixed integer $l\geq 1$, let $R(t)$ denote the error term in the Weyl's law of a $(2l+1)$-dimensional Heisenberg manifold with the metric $g_l.$ In this paper we shall prove the asymptotic formula of the $k$-th power moment for any integers $3\leq k\leq 9.$ We shall also prove that the function $t^{-(l-1/4)}R(t)$ has a distribution function.
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On the $k$-free divisor problem
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 $\int_1^T|Δ^{(k)}(x)|^2dx$ for each $k\geq 4.$
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On the Rankin-Selberg problem: higher power moments of the Riesz mean error term
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 3-rd, 4-th and 5-th power moments by using Ivić 's large value arguments.
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On higher-power moments of $Δ(x)$(III)
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 $\int_1^TΔ^4(x)dx.$ Similar results are also established for some other well-known error terms in the analytic number theory .
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On the fourth power moment of $Δ(x)$ and $E(x)$ in short intervals
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 short intervals of Jutila's type are considered. We get an asymptotic formula for $U$ in some range.
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On higher-power moments of $Δ(x)$(II)
Let $Δ(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_1^TΔ^k(x)dx$ is established for any integer $3\leq k\leq 9$ by an unified method. Similar results are also established for some other well-known error terms in the analytic number theory.
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