Nonzero coefficients of half-integral weight modular forms mod $$\ell $$ ℓ

Abstract

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $$\ell $$ l . Among the applications of this we show that there are $$\gg \sqrt{X}/\log \log X$$ ≫ X / log log X integers $$n \le X$$ n ≤ X for which the partition function p(n) is not divisible by $$\ell $$ l , and that there are $$\gg \sqrt{X}/\log \log X$$ ≫ X / log log X values of $$n \le X$$ n ≤ X for which c(n), the nth Fourier coefficient of the j-invariant, is odd.