Now showing 1 - 4 of 4
  • Publication
    A modular supercongruence for 6F5: An apéry-like story
    (Centre Mersenne, Annales de l'Institut Fourier, 2018-01-01) ; ;
    We prove a supercongruence modulo p3 between the pth Fourier coefficient of a weight 6 modular form and a truncated 6F5-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to δ(3) to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence relating the Apéry numbers to another Apéry-like sequence.
      3Scopus© Citations 10
  • Publication
    Interpolated Sequences and Critical L-Values of Modular Forms
    (Springer, 2019-01-31) ;
    Recently, Zagier expressed an interpolated version of the Apéry numbers for 𝜁(3) in terms of a critical L-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier’s six sporadic sequences are essentially critical L-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown’s cellular integrals and critical L-values of modular forms of odd weight.
      4
  • Publication
    Sequences, modular forms and cellular integrals
    (Cambridge University Press, 2020-03) ; ;
    It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.
      5Scopus© Citations 6
  • Publication
    Supercongruences for sporadic sequences
    (Cambridge University Press, 2016-05) ; ;
    We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.
    Scopus© Citations 24  425