Exponential sums with multiplicative coefficients and applications

TL;DR

This paper demonstrates that large exponential sums with multiplicative coefficients imply the multiplicative function is pretentious, leading to applications in the circle method and insights into the local-global principle.

Contribution

It establishes a connection between large exponential sums and the pretentiousness of multiplicative functions, providing new applications in analytic number theory.

Findings

Large exponential sums indicate pretentious multiplicative functions.

Applications to the circle method and local-global principles.

Provides a new interpretation of the local-global principle.

Abstract

We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global principle.