Mertens products in arithmetic progressions over function fields

Abstract

We prove a function field analogue of Mertens' formula for Euler products over prime polynomials in arithmetic progressions in $\mathbb{F}_q[t]$, the counterpart of a formula of Languasco and Zaccagnini over the integers. An elementary argument shows that the product over the prime polynomials of degree at most $n$ equals $e^{-H_n}$, where $H_n$ is the $n$-th harmonic number, up to an exponentially small relative error. Combined with the unconditional Riemann hypothesis for Dirichlet $L$-functions, this gives the main result: the product over a reduced residue class is an explicit Euler-product constant times a power of the normalizing function $\mathcal{L}_q(n)=e^{H_n-\gamma}\log q$, the exponent being the reciprocal of the number of reduced classes, again with an exponentially small error and uniformly for all moduli of degree at most $n$. It is $\mathcal{L}_q(n)$, and not the naive analogue $n\log q$ of $\log x$ under $x=q^{n}$, that is the right normalization; substituting the latter turns the formula into an expansion in powers of $1/n$ whose leading coefficients we compute, the leading correction being combinatorial rather than arithmetic.