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

Let 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].