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How to Prove Riemann Conjecture by Riemann’s Four Theorems: 《Advances in Pure Mathematics》2024年第8期619-632,共15页Chuanmiao Chen; 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 conjec...; 关键词：Riemann Conjecture ZETA-FUNCTION Xi-Function Functional Equation Product Expression CONTRADICTION

A remark on density theorems for Riemann's zeta-function被引量：1: 《Science China Mathematics》2023年第12期2795-2802,共8页Janos Pintz; supported by the National Research Development and Innovation Office of Hungary,NKFIH,KKP 133819。; The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann's zeta-function which are sufficiently strong to break the density hypothesis for Re s>7/8.Apart from a simp...; 关键词：Riemann's zeta-function density hypothesis density theorems

Multiplicative Functions Resembling the Mobius Function被引量：1: 《Acta Mathematica Sinica,English Series》2023年第12期2316-2328,共13页Qing Yang LIU; Supported by(Grant No.12288201)of the National Natural Science Foundation of 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 p.We prove O-and Ω-results for the summatory function ∑_(n)≤x f(n)f...; 关键词：Mobius function random multiplicative function ZETA-FUNCTION

Generalized Series of Bernoulli Type: 《Advances in Pure Mathematics》2023年第9期537-542,共6页Thomas Beatty Nicholas Bianco Nicole Legge; The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era,...; 关键词：BERNOULLI SERIES CONVERGENCE SUM Recursion Formula ZETA-FUNCTION SINE Maclaurin Series Infinite Product

ON THE EXCEPTIONAL FIELDS FOR A CLASS OF REAL QUADRATIC FIELDS: 《Acta Mathematica Scientia》2011年第3期1179-1188,共10页刘丽陆洪文; supported by the NSFC(10801105;30871444);the Key Project of Natural Science of Anhui Provincial Department of Education(KJ2009A44);the Doctoral Special Fund of Hefei University of Technology(GDBJ2010-012);the Key Project of Science and Technology of Anhui Province(08010302070); 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 the mean-square of the error term related ton ∑_（n≤x）λ~2（nj）被引量：2: 《Science China Mathematics》2011年第5期855-864,共10页LAO HuiXue SANKARANARAYANAN Ayyadurai; supported by National Natural Science Foundation of China (Grant Nos.11026075, 10971119);the Natural Science Foundation of Shandong Province (Grant No. ZR2009AQ007); We prove a non-trivial upper bound for the quantity ∫X2|X ∑_(n≤x)λ~2(nj)-c(j-1)x| 2dx for j=2, 3, 4.; 关键词：Rankin-Selberg zeta-function symmetric square L-functions mean-value theorems

The structural elucidation of Eisenstein's formula: 《Science China Mathematics》2010年第9期2341-2350,共10页HASHIMOTO Masahiro KANEMITSU Shigeru; supported by Natural Science Foundation of Shaanxi Province (Grant Nos.SJ08A22,2010JM1009); 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,a...; 关键词：Eisenstein FORMULA HURWITZ ZETA-FUNCTION POLYLOGARITHM FUNCTION GAUSS FORMULA

Note on the number of integral ideals in Galois extensions被引量：7: 《Science China Mathematics》2010年第9期2417-2424,共8页L GuangShi1,& WANG YongHui2 1School of Mathematics,Shandong University,Jinan 250100,China 2Department of Mathematics,Capital Normal University,Beijing 100048,China; supported in part by National Natural Science Foundation of China(Grant No.10701048);Natural Science Foundation of Shandong Province (Grant No.ZR2009AM007);Independent Innovation Foundation of Shandong University;supported in part by National Basic Research Program of China (973 Program) (Grant No.2007CB807902);National Natural Science Foundation of China (Grant No.10601034); Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...; 关键词：DEDEKIND ZETA-FUNCTION INTEGRAL IDEAL CONGRUENCES

Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function被引量：3: 《Science China Mathematics》2010年第9期2561-2572,共12页TSANG Kai-Man; 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 line.This article is a survey of recent development...; 关键词：DIVISOR PROBLEMS Riemann’s ZETA-FUNCTION mean VALUES

On the Distribution of Integral Ideals and Hecke Grssencharacters: 《Chinese Annals of Mathematics,Series B》2010年第3期385-392,共8页Huixue LAO; Project supported by the National Natural Science Foundation of China (Nos. 10701048, 10971119);the Shandong Provincial Natural Science Foundation of China (No. ZR2009AQ007); The author uses analytic methods to study the distribution of integral ideals and Hecke Grossencharacters in algebraic number fields. Nowak＇s results on the distribution of integral ideals, and Chandrasekharan and G...; 关键词：Dedekind zeta-function Integral ideal Grossencharacters

ZETA-FUNCTION