Averages of arithmetic functions over polynomials in many variables

TL;DR

This paper develops a method to estimate the average of arithmetic functions over polynomial values in multiple variables, with applications to solving equations and counting solutions in number theory.

Contribution

It introduces a general approach for averaging arithmetic functions over polynomial values, applicable to various problems in analytic number theory.

Findings

Established bounds for averages of arithmetic functions over polynomial values

Applied results to the analytic Hasse principle for cubic intersections

Derived asymptotics for solutions of non-algebraic varieties

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.