Several Sequence and New Conjectures of Zeros of Riemann Zeta Functions

Exploring Zeros and Summation Formulas of Riemann Zeta Functions

Abstract

It is well known that zeroes of Riemann's zeta functionencode a lot of numbertheoretical information, in particular, about the distri­butionof prime numbers via Riemann's and von Mangpldt's formulas forandThe goal of this paper is to presentnumerical evi­dence for a (presumably new and not yet proved) method forrevealing all divisors of all natural numbers from the zeroes of the zetafunction. This text is essentially a written version of t he talkgiven by the author at the Department of Mathematics of University ofLeicester, UK on June 18, '2012. This talk was based on more intensive compu­tationsmade after previous author's talk on the same subject given originally at theMathematical Institute of the University of Oxford on January 26, 2012. The newnumerical data indicate that some of conjectures stated in Oxford are, mostlikely, wrong. We propose and develop yet another approach tothe problem of summation of series involving the Riemann Zeta functionthe (Hurwitz's) generalized Zeta functionthe Polygamma functionand the polylogarithmic functionThe key ingredients in our approachinclude certain known integral representations forand The method developed in this paper isillustrated by numerous examples of closed-form evaluations of series of theaforementioned types; in particular, has been implemented in Mathamatica.Many of the resulting summation formulas are believed to be new.