by Matthias Kunik.

**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