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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References
Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing Company, Boston (1976)
Bellman, R.: A Brief Introduction to Theta Functions. Holt, Rinehart and Winston Inc., New York (1961)
Datskovsky, B., Guerzhoy, P.: P-adic interpolation of Taylor coefficients of modular forms. Math. Ann. 340(2), 465–476 (2008)
Dummit, D.S., Foote, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken (2004)
Finkelstein, H.: Solving equations in groups: a survey of Frobenius’ theorem. Period. Math. Hungarica 9(3), 187–204 (1978)
Larson, H., Smith, G.: Congruence properties of Taylor coefficients of modular forms. Int. J. Number Theory 10, 1501–1518 (2014)
Mihet, D.: Legendre’s and Kummer’s theorems again. Resonance 10(2), 62–71 (2005)
OEIS Foundation Inc. (2019). http://oeis.org/A317651
Romik, D.: The Taylor coefficients of the Jacobi theta constant \(\theta _3.\) Ramanujan J. https://doi.org/10.1007/s11139-018-0109-5 (2019)
Zagier, D.: Elliptic Modular Forms and Their Applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms, pp. 1–103. Springer, New York (2008)
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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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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DOI: https://doi.org/10.1007/s11139-019-00216-2