On Ikehara type Tauberian theorems with \(O(x^\gamma )\) remainders

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Abstract

Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for \(f:[1,\infty )\rightarrow {\mathbb R}\) non-negative and non-decreasing we prove \(f(x)-x=O(x^\gamma )\) with \(\gamma <1\) under certain assumptions on f. We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more.

Keywords

Tauberian theorems Mellin transform Multiplicative analytic number theory 

Mathematics Subject Classification

40E05 11M45 

Notes

Acknowledgements

I thank G. Tenenbaum for alerting me [8] of a serious error in the first version of this note and for pointing out that Proposition 2.3 is about convergence, not absolute convergence, which led to a simpler proof. I also thank a perspicatious referee for rectifying some (smaller) slips.

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institute for Mathematics, Astrophysics and Particle PhysicsRadboud UniversityNijmegenThe Netherlands

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