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?


1 Answer 1


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.

  • 2
    $\begingroup$ 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. $\endgroup$ Jul 9, 2015 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. $\endgroup$ Jul 10, 2015 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). $\endgroup$ Jul 10, 2015 at 8:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.