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Explorations in the Theory of Partition Zeta Functions

In: Exploring the Riemann Zeta Function

Author

Listed:
  • Ken Ono

    (Emory University, Department of Mathematics and Computer Science)

  • Larry Rolen

    (The Pennsylvania State University, Department of Mathematics
    School of Mathematics)

  • Robert Schneider

    (Emory University, Department of Mathematics and Computer Science)

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.

Suggested Citation

  • Ken Ono & Larry Rolen & Robert Schneider, 2017. "Explorations in the Theory of Partition Zeta Functions," Springer Books, in: Hugh Montgomery & Ashkan Nikeghbali & Michael Th. Rassias (ed.), Exploring the Riemann Zeta Function, pages 223-264, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-59969-4_10
    DOI: 10.1007/978-3-319-59969-4_10
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