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

Daniel Hast, Vlad Matei

Published 2016-04-07Version 1

We study the distribution in short intervals of certain arithmetic functions, including the von Mangoldt function and the M\"obius function, on 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, yielding an asymptotic bound on each moment for fixed degree $n$ in the limit as $q \to \infty$.