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

Let ${\mathbb{F}}_q[t]$ be the polynomial ring over the finite field ${\mathbb{F}}_q$. For arithmetic functions ${\psi }_1,{\psi }_2:{\mathbb{F}}_q[t]\to \mathbb{C}$, we establish that if a Bombieri-Vinogradov type equidistribution result holds for ${\psi }_1$ and ${\psi }_2$, then it also holds for their Dirichlet convolution ${\psi }_1\ast {\psi }_2$. As an application of this, we resolve a version of the Titchmarsh divisor problem in ${\mathbb{F}}_q[t]$. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in ${\mathbb{F}}_q[t]$.