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 \(\ell \). Among the applications of this we show that there are \(\gg \sqrt{X}/\log \log X\) integers \(n \le X\) for which the partition function p(n) is not divisible by \(\ell \), and that there are \(\gg \sqrt{X}/\log \log X\) values of \(n \le X\) for which c(n), the nth Fourier coefficient of the j-invariant, is odd.
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Notes
Weakly holomorphic allows for polar singularities at the cusps; for this and other basic definitions, we refer the reader to [17, Chapter 1].
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Bellaïche, J., Green, B. & Soundararajan, K. Nonzero coefficients of half-integral weight modular forms mod \(\ell \). Res Math Sci 5, 6 (2018). https://doi.org/10.1007/s40687-018-0123-7
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DOI: https://doi.org/10.1007/s40687-018-0123-7