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arXiv · 1604.02067

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

Daniel Hast·Vlad Matei
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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.

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Daniel Hast, Vlad Matei. 2016-04-07. Higher moments of arithmetic functions in short intervals: a geometric perspective. https://doi.org/10.1093/imrn%2Frnx310

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