Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant \(\theta _3\)

Abstract

Recently, Romik determined (in: Ramanujan J, 2019, https://doi.org/10.1007/s11139-018-0109-5) the Taylor expansion of the Jacobi theta constant \(\theta _3\), around the point \(x=1\). He discovered a new integer sequence, \((d(n))_{n=0}^\infty =1,1,-1,51,849,-26199,\ldots \), from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences. In this paper, we prove some of these conjectures, for example that \(d(n)\equiv (-1)^{n+1}\) (mod 5) for all \(n\ge 1\), and that for any prime \(p\equiv 3\) (mod 4), d(n) vanishes modulo p for all large enough n.

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Fig. 1
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Acknowledgements

The author would like to especially thank Dan Romik for suggesting the topic of this paper, for sharing a version of Fig. 1, and for many helpful consultations. The author would also like to thank an anonymous referee for suggesting improvements to an earlier version of this paper, and the author would like to thank Tanay Wakhare for suggesting an improvement to the earlier version’s proof in Sect. 3.

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Correspondence to Robert Scherer.

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This material is based upon work supported by the National Science Foundation under Grant No. DMS-1800725.

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Scherer, R. Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant \(\theta _3\). Ramanujan J 54, 427–448 (2021). https://doi.org/10.1007/s11139-019-00216-2

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Keywords

  • Theta function
  • Jacobi theta constant
  • Modular form

Mathematics Subject Classification

  • 11B83
  • 11F37
  • 14K25