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

TL;DR

This paper extends the understanding of variance in sums of von Mangoldt functions over short intervals in function fields, utilizing recent equidistribution results to generalize previous findings for Galois representations.

Contribution

It introduces a new generalization of variance results for von Mangoldt functions attached to Galois representations in short intervals, based on Sawin's recent equidistribution theorem.

Findings

Generalized variance results for $ ho$-attached von Mangoldt functions

Applied Sawin's equidistribution to short interval analysis

Extended previous work from arithmetic progressions to short intervals

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 $\Lambda_\rho$.