Identities for series of the type $\Sigma f(n)\mu (n)n^{-s}$
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- by Tom M. Apostol PDF
- Proc. Amer. Math. Soc. 40 (1973), 341-345 Request permission
Abstract:
Identities are obtained relating the series of the title with $\sum {f(n)\mu (n)\mu } (p,n){n^{ - s}}$ where $f$ is completely multiplicative, $|f(n)| \leqq 1$, and $p$ is prime. Applications are given to vanishing subseries of $\sum {\mu (n)/n}$.References
- Tord Hall, Some relations in connection with the Möbius $\mu$-function, Nordisk Mat. Tidskr. 20 (1972), 34–36, 68 (Swedish, with English summary). MR 337735 J. C. Kluyver, Series derived from the series $\sum {\mu (m)/m}$, Koningl. Akad. Wetensch. Amsterdam Proc. Sect. Sci. 6 (1904), 305-312. E. Landau, Neuer Beweis der Gleichung $\sum \nolimits _{k = 1}^\infty \mu (k)/k = 0$, Inauguraldissertation, Berlin, 1899. —, Remarks on the paper of Mr. Kluyver on page 305 of Vol. VI: Series derived from the series $\sum \mu (m)/m$, Koningl. Akad. Wetensch. Amsterdam Proc. Sect. Sci. 7 (1905), 66-77. —, Über die Equivalenz zweier Hauptsätze der analytische Zahlentheorie, S.-B. Akad. Wiss. Wien Nat. Kl. 120 (1911), 973-988. —, Handbuch der Lehre von der Verteilung der Primzahlen, Zweiter Band, Teubner, Berlin, 1909. H. Von Mangoldt, Beweis der Gleichung $\sum \nolimits _{k = 1}^\infty {\mu (k)/k} = 0$, S.-B. Konigl. Preuss. Akad. Wiss. Berlin 1897, pp. 835-852.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 341-345
- MSC: Primary 10A20; Secondary 10H15
- DOI: https://doi.org/10.1090/S0002-9939-1973-0319868-6
- MathSciNet review: 0319868