Elementary Version of Riemann Hypothesis
July 3rd, 2005 by WaltI 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?
July 4th, 2005 at 1:22 pm
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?
July 4th, 2005 at 8:17 pm
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.
January 20th, 2008 at 11:36 am
[...] back when, I had a post about explaining the Riemann hypothesis in elementary terms. I thought I’d go into some more [...]