Euler constants from primes in arithmetic progression

Many Dirichlet series of number theoretic interest can be written as a product of generating series ζ d , a ( s ) = ∏ p ≡ a ( m o d d ) ( 1 − p − s ) − 1 \zeta _{\,d,a}(s)=\prod \limits _{p\equiv a\ (\mathrm {mod}\ d)}(1-p^{-s})^{-1} , with p p ranging over all the primes in the primitive residue class modulo a ( m o d d ) a\ (\mathrm {mod}\ d) , and a function H ( s ) H(s) well-behaved around s = 1 s=1 . In such a case the corresponding Euler constant can be expressed in terms of the Euler constants γ ( d , a ) \gamma (d,a) of the series ζ d , a ( s ) \zeta _{\,d,a}(s) involved and the (numerically more harmless) term H ′ ( 1 ) / H ( 1 ) H’(1)/H(1) . Here we systematically study γ ( d , a ) \gamma (d,a) , their numerical evaluation and discuss some examples.