The Euler-Maclaurin-Siegel and Abel-Plana summation formulae for the entire Riemann functional equation to handle the Riemann hypothesis

摘要

In this article, with the aid of the entire Riemann functional equation (ERFE), defined by xi (s) = 1/2s (s-1) pi (-s/2) Gamma (s/2) zeta (s) , where s is a complex variable, Gamma (s) is the Euler's gamma function, and zeta (s) is the Riemann Zeta function (RZF), the Euler-Maclaurin-Siegel summation formula (EMSSF) and the Abel-Plana summation formula (APSF) are addressed to prove the Riemann hypothesis (RH) for the first time. The theorems for the ERFE are presented, and the complex zeros for the ERFE and RZF are also discussed in detail. The presented results are accurately and efficiently proposed to find the critical line of the ERFE. (C) 2019 Elsevier B.V. All rights reserved.