On Multiple Shifted Convolution Sums

TL;DR

This paper establishes strong estimates for averages of shifted convolution sums involving quadratic twists of GL(2) L-functions, utilizing advanced analytic techniques like the circle method and Voronoi summation.

Contribution

It introduces new bounds for shifted convolution sums of quadratic twists, combining classical tools with innovative analytic methods.

Findings

Derived strong bounds for quadratic twist sums

Applied circle method with Voronoi and reciprocity techniques

Enhanced understanding of shifted convolution sums in automorphic forms

Abstract

We prove strong estimates for averages of shifted convolution sums consisting of quadratic twists of $\mathrm{GL}_{2}$ $L$-functions. The key input involves the circle method together with standard tools such as Vorono\u{\i}, quadratic reciprocity, amplification, and divisor switching.