Exponential sums twisted by general arithmetic functions

TL;DR

This paper develops new bounds for exponential sums twisted by various arithmetic functions, improving results in the Hardy-Littlewood circle method and linking sum bounds to the zeros of the Riemann zeta-function.

Contribution

It introduces a novel strategy to bound exponential sums with general arithmetic functions, enhancing existing minor arc estimates and connecting sum bounds to partition asymptotics and the Riemann zeta zeros.

Findings

Improved bounds on exponential sums for a broad class of arithmetic functions.

Enhanced estimates in the Hardy-Littlewood circle method.

Connection between sum bounds and zeros of the Riemann zeta-function.

Abstract

We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative, additive, or neither. A strategy is shown on how to bound these sums for a wide class of functions $w$ belonging within the same ecosystem. Using this new technology we are able to improve current results on minor arcs that have recently appeared in the literature of the Hardy-Littlewood circle method. Lastly, we show how a bound on $\sum_{n \le X} |\mu(n)| e^{2\pi i\alpha n}$ can be used to study partitions asymptotics over squarefree parts and explain their connection to the zeros of the Riemann zeta-function.