Approximate Atkin-Serre Conjecture

TL;DR

This paper presents an approximate lower bound for the coefficients of modular forms at prime powers, providing a near-verified version of the Atkin-Serre conjecture with explicit bounds.

Contribution

It introduces an approximate lower bound for the Atkin-Serre conjecture on modular form coefficients at prime powers, advancing understanding of their size.

Findings

Provides an explicit approximate lower bound for mbda(p^n)

Extends the understanding of coefficient growth in modular forms

Offers a near-verified form of the Atkin-Serre conjecture

Abstract

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