Averages of arithmetic functions over polynomials in many variables

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.