Beyond the Weyl barrier for $\mathrm{GL}(2)$ exponential sums

TL;DR

This paper develops advanced analytical techniques to establish non-trivial bounds for $ ext{GL}(2)$ exponential sums with phases beyond the classical Weyl barrier, improving understanding of their behavior in number theory.

Contribution

The paper introduces new variants of the van der Corput method combined with the Bessel δ-method to surpass the Weyl barrier for $ ext{GL}(2)$ exponential sums.

Findings

Non-trivial bounds for sums with phase $eta < 1.63651$

Surpassing the Weyl barrier at $eta=3/2$

Enhanced analytical methods for exponential sum estimation

Abstract

In this paper, we use the Bessel $\delta$-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for $\mathrm{GL}(2)$ exponential sums beyond the Weyl barrier. More explicitly, for sums of $\mathrm{GL}(2)$ Fourier coefficients twisted by $e(f(n))$, with length $N$ and phase $f(n)=N^{\beta} \log n / 2\pi$ or $a n^{\beta}$, non-trivial bounds are established for $ \beta < 1.63651... $, which is beyond the Weyl barrier at $\beta = 3/2$.