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 $\Lambda \colon \mathbb{F}_q[t] \rightarrow \mathbb{C}$ 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 $\Lambda_\rho$ attached to a Galois representation $\rho \colon \mathrm{Gal} ( \overline{\mathbb{F}_q(t)}/\mathbb{F}_q(t) ) \rightarrow \mathrm{GL}_m(\overline{\mathbb{Q}}_\ell)$. We employ a recent equidistribution result by Sawin in order to generalise the corresponding result for short intervals for $\Lambda_\rho$.