### 2008-01

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

Series: 2008-01, 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

Abstract:
br />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, expl