On sign changes for almost prime coefficients of half-integral weight modular forms

TL;DR

This paper proves that the Fourier coefficients of certain half-integral weight modular forms change sign infinitely often among numbers with limited prime factors, assuming a conjecture related to Ramanujan's conjecture.

Contribution

It establishes the infinite sign change of almost prime coefficients of half-integral weight modular forms under a Ramanujan-type assumption.

Findings

Sign changes occur infinitely often for coefficients with bounded prime factors.

Results depend on an assumed Ramanujan conjecture for half-integral weight forms.

Extends understanding of sign behavior in modular form coefficients.

Abstract

For a half-integral weight modular form $f = \sum_{n=1}^{\infty} a_f(n)n^{\frac{k-1}{2}} q^n$ of weight $k = l +\frac{1}{2}$ on $\Gamma_0(4)$ such that $a_f(n)$ ($n$ $\in$ $\mathbb{N}$) are real, we prove for a fixed suitable natural number $r$ that $a_f(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for half-integral weight forms.