Sign changes of cusp form coefficients on indices that are sums of two squares

TL;DR

This paper investigates the sign changes of coefficients of holomorphic cusp forms at indices that are sums of two squares, establishing lower bounds on the number of sign changes and improving results under the Generalized Lindelöf Hypothesis.

Contribution

It introduces a new axiomatization for detecting sign changes and provides improved lower bounds on the frequency of sign changes for these coefficients.

Findings

At least $X^{1/4 - ext{epsilon}}$ sign changes in each interval $[X, 2X]$

Number of sign changes can be increased to $X^{1/2 - ext{epsilon}}$ under the Generalized Lindelöf Hypothesis

Develops a variant of an axiomatization for sign change detection

Abstract

We study sign changes in the sequence $\{ A(n) : n = c^2 + d^2 \}$, where $A(n)$ are the coefficients of a holomorphic cuspidal Hecke eigenform. After proving a variant of an axiomatization for detecting and quantifying sign changes introduced by Meher and Murty, we show that there are at least $X^{\frac{1}{4} - \epsilon}$ sign changes in each interval $[X, 2X]$ for $X \gg 1$. This improves to $X^{\frac{1}{2} - \epsilon}$ many sign changes assuming the Generalized Lindel\"{o}f Hypothesis.