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Note on the absence of remainders in the Wiener-Ikehara theorem


Authors: Gregory Debruyne and Jasson Vindas
Journal: Proc. Amer. Math. Soc. 146 (2018), 5097-5103
MSC (2010): Primary 11M45, 40E05, 44A10
DOI: https://doi.org/10.1090/proc/14193
Published electronically: August 14, 2018
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Abstract: We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.


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Additional Information

Gregory Debruyne
Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, B 9000 Gent, Belgium
Email: gregory.debruyne@ugent.be

Jasson Vindas
Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, B 9000 Gent, Belgium
Email: jasson.vindas@ugent.be

DOI: https://doi.org/10.1090/proc/14193
Keywords: Complex Tauberians, Wiener-Ikehara theorem, analytic continuation, Laplace transform, Mellin transform, remainders
Received by editor(s): January 10, 2018
Received by editor(s) in revised form: April 8, 2018, and April 11, 2018
Published electronically: August 14, 2018
Additional Notes: The first author gratefully acknowledges support by Ghent University through a BOF Ph.D. grant.
The work of the second author was supported by Ghent University through the BOF grant 01J11615 and by the Research Foundation–Flanders through the FWO grant 1520515N
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2018 American Mathematical Society

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