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

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by SpringerNature
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

References

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