Mertens products in arithmetic progressions over function fields

TL;DR

This paper proves a function field analogue of Mertens' formula for primes in arithmetic progressions over polynomial rings, leveraging Weil's Riemann hypothesis to obtain unconditional results similar to those in number fields.

Contribution

It establishes the first unconditional proof of Mertens' formula analogue in function fields, paralleling results known for integers, without requiring zero correction terms.

Findings

Unconditional proof of Mertens' formula in function fields

Correspondence with integer case results by Languasco and Zaccagnini

Utilizes Weil's Riemann hypothesis for Dirichlet L-functions

Abstract

We establish a function field analogue of Mertens' formula for Euler products restricted to primes in arithmetic progressions over the polynomial ring F_q[t]. Our results are in direct correspondence with those of Languasco and Zaccagnini for arithmetic progressions in the integers. Over function fields, Weil's Riemann hypothesis for Dirichlet L-functions holds unconditionally, and consequently the analogue of the GRH-strength asymptotic is obtained without any exceptional zero correction term.