Computing the Mertens and Meissel–Mertens Constants for Sums over Arithmetic Progressions

Abstract

We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for $\sum_{\substack{p\le x\\ p\equiv a \operatorname{mod} q}} 1/p$ and, as a byproduct, for the Meissel-Mertens constant defined as $\sum_{p\equiv a \operatorname{mod}q} (\log(1-1/p) + 1/p)$, for $q \in \{3,\dots,100\}$ and $(q, a) = 1$. The complete set of results is available online (http://www.math.unipd.it/~languasc/Mertens-comput.html).