Accurate Computations of Euler Products over Primes in Arithmetic Progressions

TL;DR

This paper develops explicit truncated formulas with error bounds for calculating Euler products over primes in arithmetic progressions, including products involving two-variable polynomials, enhancing computational accuracy.

Contribution

It introduces new explicit formulas with error estimates for Euler products over primes in arithmetic progressions and for products involving polynomial functions of prime reciprocals.

Findings

Derived truncated formulas with explicit error terms

Extended formulas to products involving two-variable polynomials

Improved computational methods for Euler products

Abstract

This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$ when $F$ is a two-variable polynomial with coefficients in $\mathbb{C}$ and satisfying some restrictive conditions.