Fast multi-precision computation of some Euler products

Abstract

For every modulus $q\\ge3$, we define a family of subsets $\\mathcal{A}$ of the multiplicative group $(\\mathbb{Z}/{q}\\mathbb{Z})^\\times$ for which the Euler product $\\prod_{p\\text{mod}q\\in\\mathcal{A}}(1-p^{-s})$ can be computed in double exponential time, where $s>1$ is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products $\\prod_{p\\in\\mathcal{A}}F(1/p)/G(1/p)$ where $F$ and $G$ are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products intervening in Number Theory.