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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  ℓ -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.

Higher Moments of Arithmetic Functions in Short Intervals: A Geometric Perspective