On convolutions with the Möbius function
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- by S. L. Segal
- Proc. Amer. Math. Soc. 34 (1972), 365-372
- DOI: https://doi.org/10.1090/S0002-9939-1972-0299572-2
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Erratum: Proc. Amer. Math. Soc. 39 (1973), 652.
Abstract:
By using the results of [6], it is proved that for an extensive class of increasing functions h, \begin{equation}\tag {$*$}\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 \end{equation} 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
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Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 34 (1972), 365-372
- MSC: Primary 10K20
- DOI: https://doi.org/10.1090/S0002-9939-1972-0299572-2
- MathSciNet review: 0299572