# 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.

• 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? Commented Oct 13, 2020 at 22:22
• The Monthly is available through JSTOR jstor.org Commented Oct 13, 2020 at 22:22
• 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. Commented Oct 13, 2020 at 22:29
• Why is there an 'also' in "Lagarias also proved unconditionally …" in the first sentence of your question? Commented Oct 13, 2020 at 22:39
• Please do not vandalize your own questions. Commented Nov 14, 2020 at 17:19

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