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 Lindelöf Hypothesis.

14 pages