Square-root cancellation for sums of factorization functions over short intervals in function fields

TL;DR

This paper develops geometric methods to estimate sums of divisor and similar functions over short intervals in function fields, achieving near square-root cancellation when the finite field's characteristic is large.

Contribution

It introduces a geometric approach to obtain near square-root cancellation estimates for arithmetic sums in short intervals over function fields, extending previous results.

Findings

Achieves estimates approaching square root cancellation for large characteristic.

Uses geometric analysis of varieties to control sums over short intervals.

Provides applications to moments of L-functions.

Abstract

We present new estimates for sums of the divisor function, and other similar arithmetic functions, in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose $\mathbb F_q$-points control this sum. This has applications to highly unbalanced moments of $L$-functions.