Logarithmic Fourier integrals for the Riemann Zeta Function

2008/04/30 by Matthias Kunik, Kunik, Matthias
Mathematics · #11M06 #11N05 #30D10 #30D50 #42A38 #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions #Number Theory (math.NT) #math.CV #math.NT #msc:11M06 #msc:11N05 #msc:30D10 #msc:30D50 #msc:42A38

paper · pdf · doi:10.48550/arxiv.0804.4829

21 pages

openalex publication_date 2008/04/30 · arxiv created 2009/09/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use symmetric Poisson-Schwarz formulas for analytic functions f in the half-plane Re(s)>\frac12 with f(s)=f(s) in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line Re(s)=\frac12 and Blaschke zeta zeroes.

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