In this paper, we determine zeta-functions of some curves of genus 3 over finite fields by gluing three elliptic curves based on Xing＇s research, and the examples show that there exists a maximal curve of genus 3 ove...In this paper, we determine zeta-functions of some curves of genus 3 over finite fields by gluing three elliptic curves based on Xing＇s research, and the examples show that there exists a maximal curve of genus 3 over F49.展开更多
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...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).展开更多
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.
This paper discusses two problems. Firstly the authors give the Schwarz formula for a holomorphic function in unit disc when the boundary value of its real part is in the class H of generalized functions in the sense ...This paper discusses two problems. Firstly the authors give the Schwarz formula for a holomorphic function in unit disc when the boundary value of its real part is in the class H of generalized functions in the sense of Hua. Secondly the authors use the classical Schwarz formula to give a new proof of the zero free region of the Riemann zeta-function.展开更多
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,...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, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.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
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...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 simple but ingenious idea of Halász the proof uses only classical knowledge about the zeta-function,results known for at least hundred years.展开更多
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,...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,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.展开更多
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...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)for a class of these f,and the point is that these O-results demonstrate cancellations better than the square-root saving.It is proved in particular that the summatory function is O(x^(1/3+ε))under the Riemann Hypothesis.On the other hand it is proved to be Ω(x^(1/4))unconditionally.It is interesting to compare these with the corresponding results for the Mobius function.展开更多
USING the method of probabilistic number-theory, De Koninck, and De Koninck and Galambos studied the reciprocal sum of the additive function f（n） satisfying f（n）≥t<sub>0</sub>】0 （n≥2） and f（p）≡...USING the method of probabilistic number-theory, De Koninck, and De Koninck and Galambos studied the reciprocal sum of the additive function f（n） satisfying f（n）≥t<sub>0</sub>】0 （n≥2） and f（p）≡1 and obtained an asymptotic formula, where t<sub>0</sub> is an absolute positive constant. Let B（x） denote the number of n≤x not satisfying the inequality loglogn-R（x）≤f（n）≤loglogn+R（x）, （1） where R（x） is a function tending to infinity. Then in refs. [1, 2], it is proved that if R（x）=o（loglogx） and B（x）=o（x/loglogx）,展开更多
Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx...Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx=x<sup>0.33314…</sup>logx as x→∞,it was proved by A,Ivic thatA<sub>k</sub>（x, h）=（d<sub>k</sub>+o（1））h, （1）whered<sub>k</sub>=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e<sup>ikt g<sub>t</sub>（n）dt≥0</sup>)),g<sub>t</sub>（n）=sum from n=d/n to (μ（n/d）e<sup>ita</sup>（d）).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x<sup>877/2653</sup>（logx）<sup>c</sup>=x<sup>0.3305…</sup>（logx）<sup>c</sup>,then Eq.（1）is true, where C is a computable constant. Based on the estimate for △（1, 2, 2;x） in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.展开更多
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...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 series.We 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.展开更多
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...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 Good＇s results on the distribution of Hecke GrSssencharacters are improved.展开更多
基金Supported by Innovation Fund of Shanghai University (Grant No.A.10-0101-08-407) Shanghai Education Commission,Foundation for Excellent Young High Education Teacher of China (Grant No.B.37-0101-08-006)
文摘In this paper, we determine zeta-functions of some curves of genus 3 over finite fields by gluing three elliptic curves based on Xing＇s research, and the examples show that there exists a maximal curve of genus 3 over F49.
文摘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).
基金supported by the NSFC(1080110530871444)+2 种基金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)
文摘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.
文摘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.
文摘This paper discusses two problems. Firstly the authors give the Schwarz formula for a holomorphic function in unit disc when the boundary value of its real part is in the class H of generalized functions in the sense of Hua. Secondly the authors use the classical Schwarz formula to give a new proof of the zero free region of the Riemann zeta-function.
文摘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, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.
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:
基金supported by the National Research Development and Innovation Office of Hungary,NKFIH,KKP 133819。
文摘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 simple but ingenious idea of Halász the proof uses only classical knowledge about the zeta-function,results known for at least hundred years.
基金supported in part by National Natural Science Foundation of China(Grant No.10701048)Natural Science Foundation of Shandong Province (Grant No.ZR2009AM007)+2 种基金Independent Innovation Foundation of Shandong Universitysupported in part by National Basic Research Program of China (973 Program) (Grant No.2007CB807902)National Natural Science Foundation of China (Grant No.10601034)
文摘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,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.
基金supported by National Natural Science Foundation of China (Grant Nos.11026075, 10971119)the Natural Science Foundation of Shandong Province (Grant No. ZR2009AQ007)
文摘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.
基金Supported by(Grant No.12288201)of the National Natural Science Foundation of China。
文摘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)for a class of these f,and the point is that these O-results demonstrate cancellations better than the square-root saving.It is proved in particular that the summatory function is O(x^(1/3+ε))under the Riemann Hypothesis.On the other hand it is proved to be Ω(x^(1/4))unconditionally.It is interesting to compare these with the corresponding results for the Mobius function.
文摘USING the method of probabilistic number-theory, De Koninck, and De Koninck and Galambos studied the reciprocal sum of the additive function f（n） satisfying f（n）≥t<sub>0</sub>】0 （n≥2） and f（p）≡1 and obtained an asymptotic formula, where t<sub>0</sub> is an absolute positive constant. Let B（x） denote the number of n≤x not satisfying the inequality loglogn-R（x）≤f（n）≤loglogn+R（x）, （1） where R（x） is a function tending to infinity. Then in refs. [1, 2], it is proved that if R（x）=o（loglogx） and B（x）=o（x/loglogx）,
文摘Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx=x<sup>0.33314…</sup>logx as x→∞,it was proved by A,Ivic thatA<sub>k</sub>（x, h）=（d<sub>k</sub>+o（1））h, （1）whered<sub>k</sub>=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e<sup>ikt g<sub>t</sub>（n）dt≥0</sup>)),g<sub>t</sub>（n）=sum from n=d/n to (μ（n/d）e<sup>ita</sup>（d）).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x<sup>877/2653</sup>（logx）<sup>c</sup>=x<sup>0.3305…</sup>（logx）<sup>c</sup>,then Eq.（1）is true, where C is a computable constant. Based on the estimate for △（1, 2, 2;x） in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.
基金supported by Natural Science Foundation of Shaanxi Province (Grant Nos.SJ08A22,2010JM1009)
文摘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 series.We 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.
基金Project supported by the National Natural Science Foundation of China (Nos. 10701048, 10971119)the Shandong Provincial Natural Science Foundation of China (No. ZR2009AQ007)
文摘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 Good＇s results on the distribution of Hecke GrSssencharacters are improved.
