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.