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POLES OF zeta functionS OF COMPLETE INTERSECTIONS

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Chinese Annals of Mathematics,Series B 2000年第2期21卷 187-200页

作者： WAN DAQING(Department of Mathematics, University of California, Irvine, CA 92697-3875, USA.) Department of Mathematics University of California Irvine CA 92697-3875 USA

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.

关键词： Pole zeta function Complete intersection

The Infinite Polynomial Products of the Gamma and zeta functions

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Advances in Pure Mathematics 2022年第6期12卷 451-464页

作者： Pál Doroszlai Horacio Keller Independent Researcher Kékkút Hungary

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.

关键词： Gamma function zeta function Critical Line

ON A functionAL EQUATION

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Acta Mathematica Scientia 2009年第2期29卷 225-231页

作者：丁毅 Ding Yi Mathematics Department New Jersey City University

In this article, the author derives a functional equation η（s）=[（π/4）^s-1/2√2/π Г（1-s）sin（πs/2）]η（1-s）（1） of the analytic function η（s） which is defined by η（s）=1^-s-3^-s-5^-s＋7^-s＋… （2） for complex variable s with Re s 〉 1, and is defined by analytic continuation for other values of s. The author proves （1） by Ramanujan identity （see [1], [3]）. Her method provides a new derivation of the functional equation of Riemann zeta function by using Poisson summation formula.

关键词： functional equation zeta function Munts formula

The Riemann Hypothesis-Millennium Prize Problem

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Advances in Pure Mathematics 2016年第12期6卷 915-920页

作者： A. A. Durmagambetov Karaganda State University Karagandy Kazakhstan

This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.

关键词： Euler Chebyshev Dirichlet Riemann Hypothesis zeta function Muntz function Complex Numbers Regular function Integral function Representation Millennium Prize Problem

Selection of Coherent and Concise Formulae on Bernoulli Polynomials-Numbers-Series and Power Sums-Faulhaber Problems

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Applied Mathematics 2022年第10期13卷 799-821页

作者： Do Tan Si The HoChiMinh-City Physical Association Ho Chi Minh City Vietnam

Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1st order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.

关键词： Bernoulli Numbers Bernoulli Polynomials Powers Sums zeta function Faulhaber Conjecture

The Exponential function as Split Infinite Product

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Advances in Pure Mathematics 2022年第4期12卷 308-331页

作者： Pál Doroszlai Horacio Keller Independent Researcher Kékkut Hungary

It is shown that any polynomial written as an infinite product with all positive real roots may be split in two steps into the product of four infinite polynomials: two with all imaginary and two with all real roots. Equations between such infinite products define adjoint infinite polynomials with roots on the adjoint roots (real and imaginary). It is shown that the shifting of the coordinates to a parallel line of one of the adjoint axes does not influence the relative placement of the roots: they are shifted to the parallel line. General relations between original and adjoint polynomials are evaluated. These relations are generalized representations of the relations of Euler and Pythagoras in form of infinite polynomial products. They are inherent properties of split polynomial products. If the shifting of the coordinate system corresponds to the shifting of the imaginary axes to the critical line, then the relations of Euler take the form corresponding to their occurrence in the functional equation of the Riemann zeta function: the roots on the imaginary axes are all shifted to the critical line. Since it is known that the gamma and the zeta functions may be written as composed functions with exponential and trigonometric parts, this opens the possibility to prove the placement of the zeta function on the critical line.

关键词： Infinite Polynomials Shifting to the Critical Line zeta function

Erratum to “The Riemann Hypothesis-Millennium Prize Problem” [Advances in Pure Mathematics 6 (2016) 915-920]

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Advances in Pure Mathematics 2016年第13期6卷 1053-1056页

作者： A. A. Durmagambetov L. N. Gumilyov Eurasian National University

The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mist... 详细信息

关键词： Euler Chebyshev Dirichlet Riemann Hypothesis zeta function Muntz function Complex Numbers Regular function Integral function Representation Millennium Prize Problem

On Computation Formula for a Series

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Chinese Quarterly Journal of Mathematics 2006年第4期21卷 623-628页

作者： YANG Qian-li, LI Hai-long Department of Mathematics Weinan Teacher＇s College Weinan 714000 China

It is one of the most interesting problems in number theory to compute someespecial series by using zeta and Gamma functions, and results have been obtained for someespecial series. In this paper, we give an important... 详细信息

关键词： zeta function Gamma function series