# Where can I find the problem by Lagarias?

Jeffrey Lagarias proved, unconditionally, that: $$\sigma(n)1$$

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.

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• 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? – Gerry Myerson Oct 13 at 22:22
• The Monthly is available through JSTOR jstor.org – Stopple Oct 13 at 22:22
• I can't find the solution. just the problem 10949 – The Company Oct 13 at 22:23
• 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. – Mark Oct 13 at 22:29
• Why is there an 'also' in "Lagarias also proved unconditionally …" in the first sentence of your question? – LSpice Oct 13 at 22:39

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

• Thank you @Conrad – The Company Oct 13 at 22:55
• happy to be of help – Conrad Oct 13 at 22:58