EMIS ELibM Electronic Journals PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.)
Vol. 50(64) Preimenovati datoteke, proveriti paginaciju!!!, pp. 60--70 (1991)

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Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers

Barry J. Powell and Tibor {\v S}al{á}t

Barry J. Powell\hfill(Received 02 10 1990) 230 Market Street Kirkland, Washington 98033, U.S.A. \msk Tibor {\v S}al{á}t Univerzita Komenskeho Matematicke-fyzikalna fakulta Katedra algebry a t{é}orie {\v c}{í}siel Bratislava-Mlynsk{á} dolina 842 15, Czechoslovakia

Abstract: We investigate the relation between the convergence of subseries $\sum_{n=1}^{\bb} m_n^{-1}$ of the harmonic series $\sum_{n=1}^{\bb}n^{-1}$ and the asymptotic densities $d(M)$ of sets $M=\za{m_10$. In Theorem 8 we give a new proof of the known result that $\sum_{m\in M}m^{-1}<+\bb$ if and only if $\sum_{n=1}^{\bb} M(n)/n^2<+\bb$. We thus give new formulations of well-known principles of analytic number theory. Numerous remarks and examples are provided throughout the paper in supplement to and clarification of the main Theorems.

Classification (MSC2000): 11B99

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