Variance of primes in short residue classes for function fields

TL;DR

This paper extends previous work on the variance of primes in function fields by analyzing the intersection of arithmetic progressions and short intervals, using involutions to derive new asymptotic formulas.

Contribution

It introduces a novel approach applying involutions to both short intervals and arithmetic progressions, advancing the understanding of prime distribution in these intersections.

Findings

Derived asymptotic formulas for variance in prime distributions in intersections

Applied involution technique to dual arithmetic progressions in function fields

Discussed conditions for relaxing polynomial restrictions on the modulus

Abstract

Keating and Rudnick 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 $\mathbb{F}_q[T]$ such that $Q(0)\not=0$. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.