An Abel-Tauber theorem on convolutions with the Möbius function

Geluk, J. L.

doi:10.1090/S0002-9939-1979-0542085-7

An Abel-Tauber theorem on convolutions with the Möbius function
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by J. L. Geluk

Proc. Amer. Math. Soc. 77 (1979), 201-209

DOI: https://doi.org/10.1090/S0002-9939-1979-0542085-7

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Abstract:

Suppose $n:{R^ + } \to {R^ + }$ and $n(x)/x$ is integrable on $(0,\infty )$. For $s > 0$ we define \[ \tilde n(s) = s\int _0^\infty {\frac {{{e^{ - us}}}}{{1 - {e^{ - us}}}}} n(u)du.\] 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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