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On convolutions with the Möbius function

Author: S. L. Segal
Journal: Proc. Amer. Math. Soc. 34 (1972), 365-372
MSC: Primary 10K20
Erratum: Proc. Amer. Math. Soc. 39 (1973), 652.
MathSciNet review: 0299572
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Abstract: By using the results of [6], it is proved that for an extensive class of increasing functions h,

$\displaystyle \sum\limits_{1 \leqq d \leqq x} {\frac{{\mu (d)}}{d}h\left( {\frac{x}{d}} \right)} \sim xh'(x)\quad {\text{as}}\;x \to \infty$ ($ *$)

where $ \mu $ denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand $ ( ^\ast )$ may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of $ (^\ast)$.

References [Enhancements On Off] (What's this?)

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Keywords: Möbius function, Dirichlet convolution, Tauberian theorems, Riemann zeta-function, Ingham summability
Article copyright: © Copyright 1972 American Mathematical Society

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