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Some functional relations derived from the Lindelöf-Wirtinger expansion of the Lerch transcendent function


Authors: Luis M. Navas, Francisco J. Ruiz and Juan L. Varona
Journal: Math. Comp. 84 (2015), 803-813
MSC (2010): Primary 41A60; Secondary 11M35, 42A10
DOI: https://doi.org/10.1090/S0025-5718-2014-02864-0
Published electronically: August 19, 2014
MathSciNet review: 3290964
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Abstract | References | Similar Articles | Additional Information

Abstract: The Lindelöf-Wirtinger expansion of the Lerch transcendent function implies, as a limiting case, Hurwitz's formula for the eponymous zeta function. A generalized form of Möbius inversion applies to the Lindelöf-Wirtinger expansion and also implies an inversion formula for the Hurwitz zeta function as a limiting case. The inverted formulas involve the dynamical system of rotations of the circle and yield an arithmetical functional equation.


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Additional Information

Luis M. Navas
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain
Email: navas@usal.es

Francisco J. Ruiz
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, Campus de la Plaza de San Francisco, 50009 Zaragoza, Spain
Email: fjruiz@unizar.es

Juan L. Varona
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Calle Luis de Ulloa s/n, 26004 Logroño, Spain
Email: jvarona@unirioja.es

DOI: https://doi.org/10.1090/S0025-5718-2014-02864-0
Keywords: Lerch transcendent function, M\"obius inversion, Fourier series
Received by editor(s): February 5, 2013
Received by editor(s) in revised form: July 28, 2013
Published electronically: August 19, 2014
Additional Notes: The authors were supported by grant MTM2012-36732-C03-02 of the DGI
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society