Where can I find the problem by Lagarias?

Question

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.

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