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

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.