A note on Fourier coefficients of Hecke eigenforms in short intervals

Abstract

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length $\frac{x}{(\log x{)}^A}$ for any $A>0$, modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length $x^{1/2+\hat{I}\mu }$ for any $\hat{I}\mu >0$.

To appear in Monatshefte für Mathematik