A note on Fourier coefficients of Hecke eigenforms in short intervals

TL;DR

This paper explores the distribution of large prime factors of Fourier coefficients of Hecke eigenforms within short intervals, employing explicit Chebotarev density results and bounds on prime factors.

Contribution

It introduces an explicit Chebotarev density theorem in very short intervals and strengthens existing bounds on prime factors of Fourier coefficients.

Findings

Derived an explicit Chebotarev density theorem for intervals of length x/(log x)^A.

Established lower bounds for the largest prime factors of Fourier coefficients in intervals of length x^{1/2 + ε}.

Extended previous work by Balog, Ono, Rouse, and Thorner to shorter 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 $x^{1/2 + \epsilon}$ for any $\epsilon >0$.