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
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
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
In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial repr...In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial representation of ζK(-1): ζK(-1) = 1/45(26n3 -41n± 9),n = ±2(mod 5), where K = Q(√5q), prime q = 4n2 + 1, and the class number of quadratic number field K2 = Q(vq) is 1.展开更多
Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli ...Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.展开更多
In a fractal zeta universe of bifurcated, ripped spacetime, the Millikan experiment, the quantum Hall effect, atmospheric clouds and universe clouds are shown to be self-similar with mass ratio of about 1020. Chaotic ...In a fractal zeta universe of bifurcated, ripped spacetime, the Millikan experiment, the quantum Hall effect, atmospheric clouds and universe clouds are shown to be self-similar with mass ratio of about 1020. Chaotic one-dimensional period-doublings as iterated hyperelliptic-elliptic curves are used to explain n-dim Kepler- and Coulomb singularities. The cosmic microwave background and cosmic rays are explained as bifurcated, ripped spacetime tensile forces. First iterated binary tree cloud cycles are related to emissions 1…1000 GHz. An interaction-independent universal vacuum density allows to predict large area correlated cosmic rays in quantum Hall experiments which would generate local nuclear disintegration stars, enhanced damage of layers and enhanced air ionization. A self-similarity between conductivity plateau and atmospheric clouds is extended to correlations in atmospheric layer, global temperature and climate.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
The exponential transform or modified Fourier transform is an integral transform where the Kernel function is Ka(ξ,t)=a−iξt, and a∈] 1,+∞ [. This is a general case of the Fourier transform where the Kernel functio...The exponential transform or modified Fourier transform is an integral transform where the Kernel function is Ka(ξ,t)=a−iξt, and a∈] 1,+∞ [. This is a general case of the Fourier transform where the Kernel function is of the form Ke(ξ,t)=exp(−iξt). Joseph Fourier, the famous French Mathematician and Engineer, was the pioneer and he studied the properties in his seminal works. This important tool created an avenue of research later and it is very important in tackling problems in strong differential form and studying spectral properties of various ordinary and partial differential operators. In our article, we will obtain explicit bounds for the modified Fourier transform and derive some corollaries using in the Kernel function a multiple of the Euler-Mascheroni constant γ. The bounds obtained involve the Riemann zeta function for positive integers at the right-hand side.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:
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 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.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10171076).
文摘In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial representation of ζK(-1): ζK(-1) = 1/45(26n3 -41n± 9),n = ±2(mod 5), where K = Q(√5q), prime q = 4n2 + 1, and the class number of quadratic number field K2 = Q(vq) is 1.
文摘Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.
文摘In a fractal zeta universe of bifurcated, ripped spacetime, the Millikan experiment, the quantum Hall effect, atmospheric clouds and universe clouds are shown to be self-similar with mass ratio of about 1020. Chaotic one-dimensional period-doublings as iterated hyperelliptic-elliptic curves are used to explain n-dim Kepler- and Coulomb singularities. The cosmic microwave background and cosmic rays are explained as bifurcated, ripped spacetime tensile forces. First iterated binary tree cloud cycles are related to emissions 1…1000 GHz. An interaction-independent universal vacuum density allows to predict large area correlated cosmic rays in quantum Hall experiments which would generate local nuclear disintegration stars, enhanced damage of layers and enhanced air ionization. A self-similarity between conductivity plateau and atmospheric clouds is extended to correlations in atmospheric layer, global temperature and climate.
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).
文摘The exponential transform or modified Fourier transform is an integral transform where the Kernel function is Ka(ξ,t)=a−iξt, and a∈] 1,+∞ [. This is a general case of the Fourier transform where the Kernel function is of the form Ke(ξ,t)=exp(−iξt). Joseph Fourier, the famous French Mathematician and Engineer, was the pioneer and he studied the properties in his seminal works. This important tool created an avenue of research later and it is very important in tackling problems in strong differential form and studying spectral properties of various ordinary and partial differential operators. In our article, we will obtain explicit bounds for the modified Fourier transform and derive some corollaries using in the Kernel function a multiple of the Euler-Mascheroni constant γ. The bounds obtained involve the Riemann zeta function for positive integers at the right-hand side.
