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Interpolated Sequences and Critical L-Values of Modular Forms

  • Robert Osburn
  • Armin StraubEmail author
Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

Recently, Zagier expressed an interpolated version of the Apéry numbers for \(\zeta (3)\) 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.

Notes

Acknowledgements

The first author would like to thank the Hausdorff Research Institute for Mathematics in Bonn, Germany for their support as this work began during his stay from January 2–19, 2018 as part of the Trimester Program “Periods in Number Theory, Algebraic Geometry and Physics”. He also thanks Masha Vlasenko for her support and encouragement during the initial stages of this project. The authors are particularly grateful to Wadim Zudilin for sharing his proof of Theorem 2 for sequence \(\varvec{F}\).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity College DublinBelfield, Dublin 4Ireland
  2. 2.Dept. of Mathematics and StatisticsUniversity of South AlabamaMobileUnited States

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