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

2023/01/30 by Sampa Dey, Dey, Sampa, Aditi Savalia +1
Computer Science · Mathematics · #Coding theory and cryptography #Analytic Number Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2301.12669

Abstract

Let \mathbbF_q[t] be the polynomial ring over the finite field \mathbbF_q. For arithmetic functions ψ₁, ψ₂: \mathbbF_q[t]→ℂ, we establish that if a Bombieri-Vinogradov type equidistribution result holds for ψ₁ and ψ₂, then it also holds for their Dirichlet convolution ψ₁ ∗ ψ₂. As an application of this, we resolve a version of the Titchmarsh divisor problem in \mathbbF_q[t]. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in \mathbbF_q[t].

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