Fast multi-precision computation of some Euler products

Abstract

For every modulus q ≥ 3 q\ge 3 , we define a family of subsets A \mathcal {A} of the multiplicative group ( Z / q Z ) × (\mathbb {Z}/q\mathbb {Z})^\times for which the Euler product ∏ p + q Z ∈ A ( 1 − p − s ) \prod _{p+q\mathbb {Z}\in \mathcal {A}}(1-p^{-s}) can be computed with high numerical precision, where s > 1 s>1 is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products ∏ p + q Z ∈ A F ( 1 / p s ) / H ( 1 / p s ) \prod _{p+q\mathbb {Z}\in \mathcal {A}}F(1/p^s)/H(1/p^s) where F F and H H are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products occurring in number theory.