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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On convolutions with the Möbius function
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by S. L. Segal PDF
Proc. Amer. Math. Soc. 34 (1972), 365-372 Request permission

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 $(^\ast )$.
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Additional 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