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

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