# Otto-von-Guericke-Universität Magdeburg

### 08-01

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

Preprint series: 08-01, Preprints

MSC:
11M06 $zeta (s)$ and $L(s, chi)$
11N05 Distribution of primes
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30D10 Representations of entire functions by series and integrals
30D50 Blashke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part

Abstract: We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane $\mbox{Re}(s)>\frac12$ with $\overline{f(\overline{s})}=f(s)$ 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 $\mbox{Re}(s)=\frac12$ and Blaschke zeta zeroes.

Keywords: Zeta function, explicit formulas, Fourier analysis,symmetric Poisson-Schwarz formulas.

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Letzte Änderung: 10.02.2016 - Ansprechpartner: Pierre Krenzlin