pow:problem5f24 [Department of Mathematics and Statistics, Binghamton University]

Problem 5 (due on Monday, November 4).

Let \[a_n=\sqrt[n^2]{\prod_{i=1}^n\prod_{j=1}^n \gcd(i,j)},\] where $\gcd(i,j)$ is the greatest common divisor of $i$ and $j$. Prove that the sequence $a_n$ converges.

The problem was solved by Prof. Vladislav Kargin and Dr. Mathew Wolak (who submitted two solutions). The submitted solutions are similar to our two in-house solutions. For details see the following link Solution.