Jeffrey Lagarias proved, unconditionally, that: $$ \sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1 $$

This was posed as a problem in:

  • J. C. Lagarias, Problem 10949: A generous bound for divisor sums, American Math. Monthly 109 no 6 (2002) 569, doi:10.2307/2695448

(edit: with solution given in AMM 111 no 3 (2004), 264–265, doi:10.2307/4145148)

Where can I find this problem? Or, any other links that shows how the inequality has been derived would be greatly appreciated.

EDIT: I will also accept the answer if anyone can outline the steps, how Lagarias derived his criterion.

  • 4
    $\begingroup$ It's not a paper. It's a problem Jeff submitted to the Monthly, and you have the citation for the issue and page where the solution was published. What's the problem? $\endgroup$ Commented Oct 13, 2020 at 22:22
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    $\begingroup$ The Monthly is available through JSTOR jstor.org $\endgroup$
    – Stopple
    Commented Oct 13, 2020 at 22:22
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    $\begingroup$ You likely were looking at the monthly issue in 2002 (which solely states the problem). The solution can be found in the 2004 issue found here. $\endgroup$ Commented Oct 13, 2020 at 22:29
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    $\begingroup$ Why is there an 'also' in "Lagarias also proved unconditionally …" in the first sentence of your question? $\endgroup$
    – LSpice
    Commented Oct 13, 2020 at 22:39
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    $\begingroup$ Please do not vandalize your own questions. $\endgroup$
    – Wojowu
    Commented Nov 14, 2020 at 17:19

1 Answer 1


The easiest way is to use Ivic's inequality

$\sigma(n)<2.59n \log \log n, n \ge 7$

Then $H_n>\log n +\gamma$ so $e^{H_N}>e^{\gamma}n$ and $\log H_n > \log \log n$ for $n \ge 3$ hence:

$2\exp(H_n)\log(H_n)>2e^{\gamma}n \log \log n>2.59 n \log \log n > \sigma(n), n \ge 7$ and check the cases $n=2,...,6$ by hand

  • $\begingroup$ Thank you @Conrad $\endgroup$ Commented Oct 13, 2020 at 22:55
  • $\begingroup$ happy to be of help $\endgroup$
    – Conrad
    Commented Oct 13, 2020 at 22:58

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