Fast multi-precision computation of some Euler products

Abstract:

For every modulus $q\ge 3$, we define a family of subsets $\mathcal {A}$ of the multiplicative group $(\mathbb {Z}/q\mathbb {Z})^\times$ for which the Euler product $\prod _{p+q\mathbb {Z}\in \mathcal {A}}(1-p^{-s})$ can be computed with high numerical precision, where $s>1$ is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products $\prod _{p+q\mathbb {Z}\in \mathcal {A}}F(1/p^s)/H(1/p^s)$ where $F$ and $H$ 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.