Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions

TL;DR

This paper computes highly precise numerical values of the Mertens and Meissel-Mertens constants for sums over primes in arithmetic progressions with moduli from 3 to 100, providing valuable data for number theory research.

Contribution

It provides explicit 100-decimal digit values of these constants for various moduli, a novel high-precision computation in the context of primes in arithmetic progressions.

Findings

Precise numerical values of Mertens constants for q=3 to 100.

Precise numerical values of Meissel-Mertens constants for q=3 to 100.

Enhanced data for analytic number theory applications.

Abstract

We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for $\sum\limits_{\substack{p\leq x p\equiv a \bmod{q}}}1/p$ and, as a by-product, for the Meissel-Mertens constant defined as $\sum_{p\equiv a \bmod{q}} (\log(1-1/p)+1/p)$, for $q \in \{3$, ..., $100\}$ and $(q, a) = 1$.