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

Nonzero coefficients of half-integral weight modular forms mod $$\ell$$ ℓ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

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

, Volume 5 (1) – Jan 31, 2018
10 pages

/lp/springer_journal/nonzero-coefficients-of-half-integral-weight-modular-forms-mod-ell-EhfpzfCm9U
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
D.O.I.
10.1007/s40687-018-0123-7
Publisher site
See Article on Publisher Site

### 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.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jan 31, 2018

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