This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous w...This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” . After an analytical introduction reminding the existence of a germ from a generic zero lying in , we show through a Weierstrass-Hadamard representation approach of the above germ that non-trivial zeros out of cannot be found.展开更多
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...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 developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.展开更多
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 th...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 calculating problems of summation:展开更多
This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specific...This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.展开更多
文摘This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” . After an analytical introduction reminding the existence of a germ from a generic zero lying in , we show through a Weierstrass-Hadamard representation approach of the above germ that non-trivial zeros out of cannot be found.
文摘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 developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.
基金Project supported by the National Natural Science Foundation of China
文摘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 calculating problems of summation:
文摘This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.
