An expression for Riemann Siegel function

2024/06/25 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99

paper · pdf · doi:10.48550/arxiv.2406.17365

openalex publication_date 2024/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There are many analytic functions U(t) satisfying Z(t)=2\Re\ e^iϑ(t)U(t)\. Here, we consider an entire function \mathop\mathcal L(s) such that U(t)=\mathop\mathcal L(\frac12+it) is one of the simplest among them. We obtain an expression for the Riemann-Siegel function Z(t) in terms of the zeros of \mathop\mathcal L(s). Implicitly, the function \mathop\mathcal L(s) is considered by Riemann in his paper on Number Theory. Riemann spoke of having used an expression for Ξ(t) in his demonstration that most of the non-trivial zeros of the zeta function lie on the critical line. Therefore, any expression deserves a study.

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