I was looking at the Wikipedia entry for Harmonic number, where I spotted a rather surprising reformulation of the Riemann hypothesis.

The Riemann hypothesis was already known to be equivalent to a not-very-complicated statement about the distribution of primes. Let Ï€ be the number of primes less than *n*. Then the Riemann hypothesis is equivalent to:

for all Îµ > 0. This fact, which goes back at least to Riemann, is the main reason why the Riemann hypothesis is of interest. In 2002, Jeffrey Lagarias found an even more elementary statement.:

where *H*_{n} is the *n*th Harmonic number (the sum of reciprocals less than or equal to *n*). It almost looks you could solve it, doesn’t it?

I am not suprised I have never heard of it, but I do find myself asking the question (yes, I realize I could read the paper, but I figure you aready have ) How sharp is that?

I imagine the bound is pretty sharp. The Riemann hypothesis is supposed to be a best possible kind of result, and I’d think that applies here. Most numbers won’t be close to the bound (for example, for primes the left-hand side is n+1), but for a strictly-increasing right-hand side, it’s probably the best you can do.