An induction principle for the Bombieri-Vinogradov theorem over Fq[t] and a variant of the Titchmarsh divisor problem

Dey, SampaSampaDeySavalia, AditiAditiSavalia2025-08-312025-08-312023-05-1510.1016/j.jmaa.2022.1269282-s2.0-85145689452https://d8.irins.org/handle/IITG2025/26799Let F<inf>q</inf>[t] be the polynomial ring over the finite field F<inf>q</inf>. For arithmetic functions ψ<inf>1</inf>,ψ<inf>2</inf>:F<inf>q</inf>[t]→C, we establish that if a Bombieri-Vinogradov type equidistribution result holds for ψ<inf>1</inf> and ψ<inf>2</inf>, then it also holds for their Dirichlet convolution ψ<inf>1</inf>⁎ψ<inf>2</inf>. As an application of this, we resolve a version of the Titchmarsh divisor problem in F<inf>q</inf>[t]. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in F<inf>q</inf>[t].falseBombieri-Vinogradov theorem | Divisor function | Finite fields | Function fields | Large sieve inequality | Titchmarsh divisor problemAn induction principle for the Bombieri-Vinogradov theorem over Fq[t] and a variant of the Titchmarsh divisor problemArticle1096081315 May 20230126928arJournal0