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