Exponential sums, twisted multiplicativity and moments

TL;DR

This paper investigates the behavior of exponential sums with polynomial phases over squarefree moduli, providing bounds on moments and evidence of their uncorrelation, using analytic and algebraic methods.

Contribution

It introduces new bounds on moments of exponential sums and demonstrates uncorrelation for sums associated with unrelated polynomials, combining analytic and algebraic techniques.

Findings

Upper bounds on moments of exponential sums

Evidence of un-correlation between sums of unrelated polynomials

Integration of analytic and algebraic methods in proofs

Abstract

We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different suitably unrelated and generic polynomials. The proofs combine analytic arguments with the algebraic interpretation of exponential sums and their monodromy groups.