Approximate Atkin-Serre Conjecture

Abstract

Let $\lambda (m)$ be the $m$th coefficient of a modular form $f(z)={\sum }_{m\ge 1}\lambda (m)q^m$ of weight $k\ge 4$, let $p^n$ be a prime power, and let $\epsilon >0$ be a small number. An approximate of the Atkin-Serre conjecture on the lower bound of the form $\left|\lambda \left(p^n\right)\right|\ge p^{(k-1)n/2-2k+2\epsilon }$ is presented in this note.

Twelve Pages. Keywords: Tau Function; Modular Function; Atkin-Serre Conjecture