Explicit formulae for averages of Goldbach representations

J. Brüdern, J. Kaczorowski, A. Perelli

Published 2017-12-03Version 1

We prove an explicit formula, analogous to the classical explicit formula for $\psi(x)$, for the Ces\`aro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.