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

Xiao-Jun Yang

Physica A: Statistical Mechanics and its Applications, 2019, vol. 525, issue C, 1203-1211

Abstract: In this article, with the aid of the entire Riemann functional equation (ERFE), defined by ξs=12ss−1π−s2Γs2ςs, where s is a complex variable, Γs is the Euler’s gamma function, and ςs 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.

Keywords: Euler–Maclaurin–Siegel summation formula; Abel–Plana summation formula; Entire Riemann functional equation; Riemann Zeta function; Critical line; Riemann hypothesis (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:525:y:2019:i:c:p:1203-1211

DOI: 10.1016/j.physa.2019.04.063

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