An induction principle for the Bombieri-Vinogradov theorem over $\mathbb{F}_q[t]$ and a variant of the Titchmarsh divisor problem

TL;DR

This paper extends the Bombieri-Vinogradov theorem to the polynomial ring over finite fields, establishing an induction principle for arithmetic functions and applying it to solve a variant of the Titchmarsh divisor problem in this setting.

Contribution

It introduces a new induction principle for the Bombieri-Vinogradov theorem over $\,\mathbb{F}_q[t]$ and applies it to analyze divisor functions over primes in finite fields.

Findings

Proves that equidistribution results for functions imply similar results for their convolutions.

Provides an asymptotic formula for the divisor function over shifted products of primes in $\,\mathbb{F}_q[t]$.

Extends classical number theory results to the setting of polynomial rings over finite fields.

Abstract

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_{q}$. For arithmetic functions $\psi_{1}, \psi_{2}: \mathbb{F}_{q}[t]\rightarrow\mathbb{C}$, we establish that if a Bombieri-Vinogradov type equidistribution result holds for $\psi_{1}$ and $\psi_{2}$, then it also holds for their Dirichlet convolution $\psi_{1} \ast \psi_{2}$. As an application of this, we resolve a version of the Titchmarsh divisor problem in $\mathbb{F}_{q}[t]$. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in $\mathbb{F}_q[t]$.