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Averages of arithmetic functions over polynomials in many variables

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arxiv 2409.18116 v1 pith:Y6QLLB4A submitted 2024-09-26 math.NT

Kevin Destagnol , Efthymios Sofos This is my paper · ORCID
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keywords arithmeticvariablesmanyanalyticapplicationsasymptoticsaverageaverages
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abstract

We estimate the average of any arithmetic function $k$ over the values of any smooth polynomial in many variables provided only that $k$ has a distribution in arithmetic progressions of fixed modulus. We give several applications of this result including the analytic Hasse principle for an intersection of two cubics in 21 variables and asymptotics for the number of integer solutions of a non-algebraic variety.

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  1. Solubility of a family of conics with polynomial coefficients in many variables

    math.NT 2025-11 reject novelty 6.0 of 10

    An asymptotic for the density of soluble conics with polynomial coefficients is proposed, but the sign decomposition (6.5) that bridges the circle-method counts to the true count is false.

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