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Around the Lipschitz Summation Formula

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  • Wenbin Li
  • Hongyu Li
  • Jay Mehta

Abstract

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.

Suggested Citation

  • Wenbin Li & Hongyu Li & Jay Mehta, 2020. "Around the Lipschitz Summation Formula," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-16, April.
    • Handle: RePEc:hin:jnlmpe:5762823
    DOI: 10.1155/2020/5762823
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    Cited by:

    1. Guodong Liu & Kalyan Chakraborty & Shigeru Kanemitsu, 2023. "A Unifying Principle in the Theory of Modular Relations," Mathematics, MDPI, vol. 11(3), pages 1-32, January.

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