On the Order of Growth of Lerch Zeta Functions

Jörn Steuding and Janyarak Tongsomporn ()
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Jörn Steuding: Department of Mathematics, Würzburg University, Am Hubland, 97 218 Würzburg, Germany
Janyarak Tongsomporn: School of Science, Walailak University, Nakhon Si Thammarat 80 160, Thailand

Mathematics, 2023, vol. 11, issue 3, 1-7

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.

Keywords: Lerch zeta function; Hurwitz zeta function; (approximate) functional equation; order of growth; exponent pairs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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