Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

Special Values for the Riemann Zeta Function

The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression for representing the zeta function at the odd integer values 2n+1 for n a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers E2n and the polygamma functions ψ(2n)(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.

Modified Double Zeta Function and Its Properties

The present paper aims at introducing and investigating a new class of generalized double zeta function i.e. modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta function and Bin-Saad generalized double zeta function as particular cases. The results are obtained by suitably applying Riemann-Liouville type and Tremblay fractional integral and differential operators. We derive the expansion formula for the proposed function with some of its properties via fractional operators and discuss the link with known results.

The Infinite Polynomial Products of the Gamma and Zeta Functions

Starting with the binomial coefficient and using its infinite product representation, the infinite product representation of the gamma function and of the zeta function are composed of an exponential and of a trigonometric component and proved. It is proved, that all these components define imaginary roots on the critical line, if written in the form as they are in the functional equation of the zeta function.

Fast Converging Series for Riemann Zeta Function

Riemann zeta function has a key role in number theory and in its applications. In this paper we present a new fast converging series for . Applications of the series include the computation of the and recursive computation of , and generally . We discuss on the production of irrational number sequences e.g. for encryption coding and zeta function maps for analysis and synthesis of log-time sampled signals.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.

Zeta Functions of the Complement and xyz-Transformations of a Regular Graph

Let Z(λ,G)denote the zeta function of a graph G.In this paper the complement G^Cand the G^(xyz)-transformation G^(xyz)of an r-regular graph G with n vertices and m edges for x,y,z∈{0,1,+,-},are considerd.The relationship between Z(λ,G)and Z(λ,G^C)is obtained.For all x,y,z∈{0,1,+,-},the explicit formulas for the reciprocal of Z(λ,G^(xyz))in terms of r,m,n and the characteristic polynomial of G are obtained.Due to limited space,only the expressions for G^(xyz)with z=0,and xyz∈{0++,+++,1+-}are presented here.

A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.

Accelerated Series for Riemann Zeta Function at Odd Integer Arguments

Riemann zeta function is an important tool in signal analysis and number theory. Applications of the zeta function include e.g. the generation of irrational and prime numbers. In this work we present a new accelerated series for Riemann zeta function. As an application we describe the recursive algorithm for computation of the zeta function at odd integer arguments.

On the Higher Moments of Coefficients Attached to Dedekind Zeta Function

Let K_(3) be a non-normal cubic extension over Q.In this paper,we investigate the higher moments of the coefficients a_(K_3)(n) of Dedekind zeta function over sum of two squares of the following types■ where l≥9 is any fixed positive integer,which generalizes the results in [Front.Math.China,2020,15(1):57-67].

基于Zeta电位和FTIR的中高阶煤与天然焦表面电性研究

通过对中高阶煤与天然焦表面电性的研究,可为揭示煤与天然焦电磁辐射机理,判断矿物可浮性和实现不同矿物分离提供基础。为深入了解中高阶煤与天然焦表面电性的差异,分析煤与天然焦表面官能团组成的差异,采用Zeta电位与FTIR分析方法对样品的表面特性进行综合分析,研究其表面电性与主要官能团的耦合关系。结果表明:①煤样的Zeta电位总体为负,天然焦样品的Zeta电位为正,同时随着变质程度的升高,样品表面负电官能团减少,Zeta电位增大;②脱灰后,样品表面负电官能团被破坏,煤与天然焦的Zeta电位均为正,且随着煤变质程度的增加,Zeta电位值减小;③羟基、COOH、C=O是影响Zeta电位的主要官能团,其中,C=O对Zeta电位的影响大于COOH;羟基-π是羟基结构中影响Zeta电位的主要官能团。研究成果为进一步探究煤与天然焦电磁辐射机理与浮选性提供基础。

基于Zeta函数的Mellin变换及其应用

Mellin变换是一种以幂函数为积分核的积分变换,常用于广义留数定理、信号处理和渐进展开理论的相关研究中.从Mellin变换理论出发,基于Mellin变换与Fourier变换、Zeta函数的相互关系,推导了Zeta函数与Gamma函数相乘的Mellin变换形式,结合Euler乘积公式详细讨论了Mellin变换在Zeta函数变换中的应用方法和技巧.

On Functional Relations for the Alternating Analogues of Tornheim's Double Zeta Function

In this paper,new proofs of two functional relations for the alternating analogues of Tornheim's double zeta function are given.Using the functional relations,the author gives new proofs of some evaluation formulas found by Tsumura for these alternating series.

On the Absence of Zeros of Riemann Zeta-Function Out of ℜ(z) = 1/2

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.

Trace of heat kernel,spectral zeta function and isospectral problem for sub-laplacians

In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

A vanishing theorem is proved for e-adic cohomology with compact support on an affine (singular) complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibiliy of the poles or equivalently an improvement of the polar part of the AxKatz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.

Two Theorems on the Zero Density of the Riemann Zeta Function

§1.IntroductionLet N(σ,T) be the number of zeros of the Riemann zeta funetion ζ(s) in theregion σ≤Re(s)≤1,|Im(s)|≤T.For σ>(3/4),by using the Halász-Montgomerymethod,one can get somo results which are better than the classical result givenby Ingham.For example,Jutila proved that the zero density hypothesisN(σ,T)T2-3σ+

A new zeta function for number fields

According to the idea of class field theory, the possible absolutely normal number fieldsare restricted in nature by the arithmetical properties of rational number field .Becausethe fundamental arithmetical property of is the distributive law of the prime numbers,

On spectral theory of the Riemann zeta function

Every nontrivial zero of the Riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a Hilbert-P′olya space. It has geometric multiplicity one. A relation between nontrivial zeros of the zeta function and eigenvalues of the convolution operator is given. It is an analogue of the Selberg transform in Selberg’s trace formula. Elements of the Hilbert-P′olya space are characterized by the Poisson summation formula.

On two recurrence formulas for two kinds of identities of Riemann Zeta function

For any complex s, let ζ（s） denote Riemann Zeta function. We have ζ（s）=sum from n=1 to ∞ （1/ns） when Re（s）】1. Now we define A（n,k,l）=sum from α1+α2+……+αk=n to （（α1α2…αk）1ζ（2α1）ζ（2α2）…ζ（2αk））, where n≥k is a positive integer, α+α2+…αk=n denotes the summation for k-dimensional group of positive integers （α1, α2,…, αk）which satisfies this formula. In this note, our main purpose is to discuss computing problem of summation on equation （1）.