It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: Convergence of subseries of the harmonic series and asymptotic densities of sets of integers (1987), which in turn redirects to Šalát's On subseries for a different proof. However, I am convinced that the result is (much?) older than that, and was told by G. Grekos that Šalát himself in his talks presented it as well-known. So my question is:

Do you have any clue about the first (explicit) occurrence of the result in the literature?


I believe this is due to Kronecker. Namely, if you look at theorem 3 here, which is due to Kronecker, and says that if $$\sum_{n=1}^\infty a_n$$ is convergent, and $(p_n)_{n\geq 1}$ is an increasing and unbounded sequence, then $$\lim_{n\rightarrow \infty}\frac{p_1 a_1 + p_2 a_2 + \dotsc + p_n a_n}{p_n} = 0.$$ Now, let your set be $X =\{x_1, \dotsc, x_k, \dotsc\},$ in order. Set $a_n = 1/ x_n,$ while $p_n = x_n,$ your assertion follows.

I should note that in (one of) his papers, Salat attributes the result independently to Leo Moser (Monthly, 1958, DOI: 10.2307/2308884), and Krzyś (Prace Matem 1956) - I could not find the latter paper. Neither can I find the original Kronecker paper.

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    $\begingroup$ Following your hint, I gave a look at Knopp's Theory and Application of Infinite Series (2nd English ed.): The result on the "weighted limit" in your post appears as Theorem 3 in Section 82 (p. 129) of the book, and in a footnote on the same page Knopp provides a reference: L. Kronecker, C. R. Math. Acad. Sci. Paris 103 (1886), p. 980 (no title is provided). This may be the first implicit reference to the result mentioned in the OP, but I am not quite sure/don't know whether the notion of natural density had already been introduced at that time. In any case, +1. $\endgroup$ – Salvo Tringali Jul 9 '15 at 21:35
  • $\begingroup$ For the record, here is the full reference to the papers mentioned in the answer above: J. Krzyś, A theorem of Olivier and its generalizations (in Polish), Prace matem. 2 (1956), 159-164 and L. Moser, On the series $\sum 1/p$, Amer. Math. Monthly 65 (1958), 104-105. @Igor: G. Grekos noted there is a typo in the way the name of Krzyś is spelled in the post, and tried to edit, but I think he can't, since he is not a registered user. $\endgroup$ – Salvo Tringali Jul 10 '15 at 6:57
  • $\begingroup$ In addition, the full reference to the paper of Kronecker mentioned in the above comments is: L. Kronecker, Quelques remarques sur la détermination des valeurs moyennes, C. R. Math. Acad. Sci. Paris 103 (Nov., 1886), 980-987. The paper is available to download from the website of the Bibliotèque nationale de France (gallica.bnf.fr/ark:/12148/cb343481087/date.r=.langEN). $\endgroup$ – Salvo Tringali Jul 10 '15 at 8:23

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