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

DOI10.1016/J.JMAA.2022.126928arXiv2301.12669WikidataQ122441711 ScholiaQ122441711MaRDI QIDQ6424762

Publication date: 30 January 2023

Abstract: Let

m a t h b b F_{q} [t]

be the polynomial ring over the finite field

m a t h b b F_{q}

. For arithmetic functions

p s i_{1}, p s i_{2} : m a t h b b F_{q} [t] i g h t a r r o w m a t h b b C

, we establish that if a Bombieri-Vinogradov type equidistribution result holds for

p s i_{1}

and

p s i_{2}

, then it also holds for their Dirichlet convolution

p s i_{1} a s t p s i_{2}

. As an application of this, we resolve a version of the Titchmarsh divisor problem in

m a t h b b F_{q} [t]

. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in

m a t h b b F_{q} [t]

.

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Applications of sieve methods (11N36) Estimates on character sums (11L40) Primes in congruence classes (11N13) Sieves (11N35)

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