Logarithmic Fourier integrals for the Riemann Zeta Function

Details Logarithmic Fourier integrals for the Riemann Zeta Function Logarithmic Fourier integrals for the Riemann Zeta Function

2008-04-30

arxiv.org/abs/0804.4829v2

Mathematics

Complex Variables

21 pages

Scientific paper

We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ 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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