Elementary Version of Riemann Hypothesis

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 π be the number of primes less than n. Then the Riemann hypothesis is equivalent to:

for all ε > 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 Hn is the nth Harmonic number (the sum of reciprocals less than or equal to n). It almost looks you could solve it, doesn’t it?

3 thoughts on “Elementary Version of Riemann Hypothesis

  1. 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?

  2. 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.

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