Euler constants from primes in arithmetic progression

TL;DR

This paper investigates Euler constants associated with primes in arithmetic progressions, providing a systematic study, numerical evaluations, and examples to deepen understanding of their properties and relationships.

Contribution

It introduces a systematic analysis and numerical evaluation of Euler constants for Dirichlet series involving primes in arithmetic progressions, with new explicit examples.

Findings

Explicit formulas for Euler constants in terms of prime distributions.

Numerical evaluations of these constants for various residue classes.

Illustrative examples demonstrating the properties of these constants.

Abstract

Many Dirichlet series of number theoretic interest can be written as a product of generating series $\zeta_{\,d,a}(s)=\prod\limits_{p\equiv a\pmod{d}}(1-p^{-s})^{-1}$, with $p$ ranging over all the primes in the primitive residue class modulo $a\pmod{d}$, and a function $H(s)$ well-behaved around $s=1$. In such a case the corresponding Euler constant can be expressed in terms of the Euler constants $\gamma(d,a)$ of the series $\zeta_{\,d,a}(s)$ involved and the (numerically more harmless) term $H'(1)/H(1)$. Here we systematically study $\gamma(d,a)$, their numerical evaluation and discuss some examples.