Compute $\prod_{0<i<j<\infty} i^{1/i} - j^{1/j}$. Here are a few steps I tried. Have no clue from which angle to solve it.
Maybe we need to factor the term $i^{1/i} - j^{1/j} = j^{1/j}(i^{1/i}j^{-1/j}-1)=j^{1/j}((i/j)^{1/i}j^{1/i-1/j}-1)$.
Maybe we should rewrite it as a summation of log.
But still lack of ideas.
Hint: What happens when $i=2$ and $j=4$.