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 ${\prod }_{p\text{mod}q\in \mathcal{A}}(1-p^{-s})$ 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.

Better phrasing and Proposition 7.3 counting the number of cyclic subgroups