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Explorations in the Theory of Partition Zeta Functions

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Exploring the Riemann Zeta Function

Abstract

We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the analytic continuations of these “partition zeta functions,” find unusual formulas for the Riemann zeta function, prove identities for multiple zeta values, and see that some of the formulas allow for p-adic interpolation. The second family we study was anticipated by Manin and makes use of modular forms, functions which are intimately related to integer partitions by universal polynomial recurrence relations. We survey recent work on these zeta polynomials, including the proof of their Riemann Hypothesis.

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Notes

  1. 1.

    This is a slight renormalization of the period polynomials considered in references such as [15, 32, 38, 50].

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Acknowledgements

The authors thank Armin Straub for useful discussion on the history of Theorem 5. Ken Ono is supported by National Science Foundation and the Asa Griggs Candler Fund. Larry Rolen thanks the support of the DFG and the University of Cologne postdoc program.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Ken Ono & Robert Schneider

  2. Department of Mathematics, The Pennsylvania State University, State College, PA, USA

    Larry Rolen

  3. School of Mathematics, Trinity College, Dublin 2, Ireland

    Larry Rolen

Authors
  1. Ken Ono
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  2. Larry Rolen
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  3. Robert Schneider
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Correspondence to Ken Ono .

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Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Hugh Montgomery

  2. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Ashkan Nikeghbali

  3. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Ono, K., Rolen, L., Schneider, R. (2017). Explorations in the Theory of Partition Zeta Functions. In: Montgomery, H., Nikeghbali, A., Rassias, M. (eds) Exploring the Riemann Zeta Function. Springer, Cham. https://doi.org/10.1007/978-3-319-59969-4_10

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