Nonzero coefficients of half-integral weight modular forms mod \(\ell \)

  • Joël Bellaïche
  • Ben Green
  • Kannan Soundararajan
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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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Authors and Affiliations

  • Joël Bellaïche
    • 1
  • Ben Green
    • 2
  • Kannan Soundararajan
    • 3
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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