Sums of Cusp Form Coefficients Along Quadratic Sequences

TL;DR

This paper proves an improved asymptotic formula for sums of cusp form coefficients along quadratic sequences, reducing the error term compared to previous results, and includes spectral average bounds.

Contribution

It introduces a refined analysis of shifted convolution sums for cusp forms, achieving a better error term than prior work, and provides stronger spectral average bounds in an appendix.

Findings

Asymptotic sum of cusp form coefficients along quadratic sequences with error term $O(X^{3/4+\, ext{epsilon}})$

Improved error bounds over Blomer's 2008 result

Spectral average bounds established in the appendix

Abstract

Let $f(z) = \sum A(n) n^{(k-1)/2} e(nz)$ be a cusp form of weight $k \geq 3$ on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that $\sum_{n \leq X} A(n^2+h) = c_{f,h} X + O_{f,h,\epsilon}(X^{\frac{3}{4}+\epsilon})$ for $\epsilon>0$, which improves a result of Blomer from 2008 with error $X^{\frac{6}{7}+\epsilon}$. This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages.