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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An Abel-Tauber theorem on convolutions with the Möbius function


Author: J. L. Geluk
Journal: Proc. Amer. Math. Soc. 77 (1979), 201-209
MSC: Primary 40E05; Secondary 40G99
MathSciNet review: 542085
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Abstract: Suppose $ n:{R^ + } \to {R^ + }$ and $ n(x)/x$ is integrable on $ (0,\infty )$. For $ s > 0$ we define

$\displaystyle \tilde n(s) = s\int_0^\infty {\frac{{{e^{ - us}}}}{{1 - {e^{ - us}}}}} n(u)du.$

In this paper an Abel-Tauber theorem is proved concerning this transform. Moreover the relation between $ \tilde n(s)$ and $ {\Sigma _{m \leqslant s}}n(s/m)/m$ is studied.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0542085-7
PII: S 0002-9939(1979)0542085-7
Keywords: Abel-Tauber theorems, regular variation
Article copyright: © Copyright 1979 American Mathematical Society