Higher moments of arithmetic functions in short intervals: a geometric perspective

TL;DR

This paper explores the distribution of arithmetic functions over polynomials in finite fields through a geometric lens, connecting algebraic geometry with number theory to analyze moments and asymptotic behavior.

Contribution

It introduces a geometric approach using algebraic varieties and cohomology to interpret and bound moments of arithmetic functions in short intervals over finite fields.

Findings

Computed parts of the $mbda$-adic cohomology of associated varieties.

Provided asymptotic bounds on moments as the size of the finite field grows.

Established a geometric framework explaining known analytic results.

Abstract

We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the M\"obius function, in short intervals of polynomials over a finite field $\mathbb{F}_q$. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the $\ell$-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree $n$ in the limit as $q \to \infty$. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.