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

Abstract

We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius 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.

arXiv comment25 pages

References in corpus (5)

• Representation stability in cohomology and asymptotics for families of varieties over finite fields
• Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
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• Arithmetic functions in short intervals and the symmetric group
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Cited by in corpus (2)

• Correlation of arithmetic functions over ${\mathbb{F}}_q[T]$
• Cubic twin prime polynomials are counted by a modular form