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 1 4 −ǫ sign changes in each interval [X, 2X] for X ≫ 1. This improves to X 1 2 −ǫ many sign changes assuming the Generalized Lindelöf Hypothesis.