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

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.

To appear in Mathematika