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Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

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Science China Mathematics 2010年第9期53卷 2561-2572页

作者： TSANG Kai-Man Department of Mathematics The University of Hong Kong

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical *** article is a survey of recent developments on the research of these famous error terms in number *** include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.

关键词： divisor problems Riemann’s zeta-function mean values

ON THE SEVERAL IDENTITIES OF RIEMANN zeta-function

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Chinese Science Bulletin 1991年第22期36卷 1852-1856页

作者：张文鹏 Depart ment of Mathe matics Northwest University Xi'an 710069. PRC

Ⅰ. INTRODUCTION For any complex number s, let ζ(s)denote Riemann zeta-function defined by ζ(s)=sum from n=1 to ∞ 1/n^s for Re (s)>1 and by its analytic continuation. The main purpose of this report is to study the... 详细信息

关键词： Riemann zeta-function identity series

How to Prove Riemann Conjecture by Riemann’s Four Theorems

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Advances in Pure Mathematics 2024年第8期14卷 619-632页

作者： Chuanmiao Chen School of Mathematics and Statistics Central South University Changsha China College of Mathematics and Statistics Hunan Normal University Changsha China

Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).

关键词： Riemann Conjecture zeta-function Xi-function functional Equation Product Expression Contradiction

WEAK CONVERGENCE OF SOME DIRICHLET SERIES

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维普期刊数据库

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Acta Mathematica Scientia 2001年第4期21卷 433-439页

作者：丁夏畦罗佩珠 Institute of Applied Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China

This paper treats Dirichlet series from the point of view developed in [1],[2]. Especially it is found that the kernal function is closely related with the Riemann zeta-function.

关键词： weak Dirichlet series zeta-function

ON THE EXCEPTIONAL FIELDS FOR A CLASS OF REAL QUADRATIC FIELDS

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Acta Mathematica Scientia 2011年第3期31卷 1179-1188页

作者：刘丽陆洪文 School of Mathematics Hefei University of Technology Department of Mathematics Tongji University

In this paper, we give a lower bound exp（2.2 × 10~8 ） for those discriminants of real quadratic fields Q（√ d） with d= N^2-4 and h（d）=1.

关键词： quadratic field class number discriminant zeta-function lower bound

ON RITZ METHOD OF ANINTEGRO-DIFFERENTIAL EQUAITON

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Acta Mathematica Scientia 2009年第3期29卷 687-696页

作者：丁夏畦罗佩珠 Institute of Applied Mathematics Academy of Mathematics and Systems Science The Chinese Academy of SciencesBeijing 100190 China

This paper deals with the Ritz method of an integro-differential equation related with Riemann zeta-function.

关键词： zeta-function Ritz method explicit formula

Multiplicative functions Resembling the Mobius function

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Acta Mathematica Sinica,English Series 2023年第12期39卷 2316-2328页

作者： Qing Yang LIU School of Mathematics Renmin University of ChinaBeijing 100872P.R.China Institute of Mathematics AMSSThe Chinese Academy of SciencesBeijing 100080P.R.China

A multiplicative function f is said to be resembling the Mobius function if f is supported on the square-free integers,and f(p)=±1 for each prime *** prove O-and Ω-results for the summatory function ∑_(n)≤x f(n)for a class of these f,and the point is that these O-results demonstrate cancellations better than the square-root *** is proved in particular that the summatory function is O(x^(1/3+ε))under the Riemann *** the other hand it is proved to be Ω(x^(1/4))*** is interesting to compare these with the corresponding results for the Mobius function.

关键词： Mobius function random multiplicative function zeta-function

The structural elucidation of Eisenstein's formula

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Science China Mathematics 2010年第9期53卷 2341-2350页

作者： HASHIMOTO Masahiro KANEMITSU Shigeru Department of Mathematics Wakayama College of TechnologyNoda-machi Gobo-shi 649-15Japan Graduate School of Advanced Technology Kinki University IizukaFukuoka 820-8555Japan

In this paper,we shall give a complete structural description of generalizations of the classical Eisenstein formula that expresses the first periodic Bernoulli polynomial as a finite combination of cotangent values,as a relation between two bases of the vector space of periodic Dirichlet *** shall also determine the limiting behavior of them,giving rise to Gauss' famous closed formula for the values of the digamma function at rational points on the one hand and elucidation of Eisenstein-Wang's formulas in the context of Kubert functions on the other.W shall reveal that most of the relevant previous results are the combinations of the generalized Eisenstein formula and the functional equation.

关键词： Eisenstein formula Hurwitz zeta-function polylogarithm function Gauss formula

Note on the number of integral ideals in Galois extensions

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Science China Mathematics 2010年第9期53卷 2417-2424页

作者： L GuangShi1,& WANG YongHui2 1School of Mathematics,Shandong University,Jinan 250100,China 2Department of Mathematics,Capital Normal University,Beijing 100048,China School of Mathematics Shandong University Jinan China Department of Mathematics Capital Normal University Beijing China

Let K be an algebraic number field of finite degree over the rational filed *** ak be the number of integral ideals in K with norm *** this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and *** an application we consider the number of solutions of polynomial congruences.

关键词： Dedekind zeta-function integral ideal congruences