Euler constants from primes in arithmetic progression

Abstract

Many Dirichlet series of number theoretic interest can be written as a product of generating series $\hat{I}{\P }_{d,a}(s)=\prod _{p\equiv a(\mathrm{mod}d)}(1-p^{-s}{)}^{-1}$, with $p$ ranging over all the primes in the primitive residue class modulo $a(\mathrm{mod}d)$, 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 ${\hat{I}}^3(d,a)$ of the series $\hat{I}{\P }_{d,a}(s)$ involved and the (numerically more harmless) term $H^{\prime }(1)/H(1)$. Here we systematically study ${\hat{I}}^3(d,a)$, their numerical evaluation and discuss some examples.

22 pages, one table, one appendix; in the new version Remark 2 was inserted