Euler constants from primes in arithmetic progression

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.