Products of primes in function fields and number fields

A classical question of Erdős asks whether every reduced residue class modulo q can be represented by a product of two primes bounded by q. We will discuss analogues of this question in function fields and number fields. Over F_q[t], we use Katz’s equidistribution methods to study representations by products of irreducible polynomials. We will also briefly discuss how this fits into the framework of Forey, Fresán, and Kowalski for connected commutative algebraic groups, which may allow an extension of the result to general global function fields. Over number fields, the dense model argument of Matomäki and Teräväinen can be adapted to narrow ray class groups giving analogous results whose strength depends on the available character-sum bounds.