Variance of primes in short residue classes for function fields

Abstract

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus $Q$ is a polynomial in ${𝔽}_q[T]$ such that $Q(0)\ne 0$. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.