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On the Order of Growth of Lerch Zeta Functions

Author

Listed:
  • Jörn Steuding

    (Department of Mathematics, Würzburg University, Am Hubland, 97 218 Würzburg, Germany
    These authors contributed equally to this work.)

  • Janyarak Tongsomporn

    (School of Science, Walailak University, Nakhon Si Thammarat 80 160, Thailand
    These authors contributed equally to this work.)

Abstract

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L ( λ , α , 1/2 + it ) ≪ t 13/84+ ϵ as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t ϵ (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros.

Suggested Citation

  • Jörn Steuding & Janyarak Tongsomporn, 2023. "On the Order of Growth of Lerch Zeta Functions," Mathematics, MDPI, vol. 11(3), pages 1-7, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:723-:d:1053486
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