Variance of sums in short intervals and $L$-functions in $\mathbb{F}_q[t]$

Abstract

Keating and Rudnick studied the variance of the polynomial von Mangoldt function Λ : F q [t] → C in arithmetic progressions and short intervals using two equidistribution results by Katz. Hall, Keating and Roditty-Gershon then generalised the result for arithmetic progressions for a von Mangoldt function Λ ρ attached to a Galois representation ρ : Gal F q (t)/F q (t) → GL m (Q ℓ ). We employ a recent equidistribution result by Sawin in order to generalise the corresponding result for short intervals for Λ ρ .