Posted by Danny Tarlow
Some (non-mathematician) friends and I had an email discussion going on tangentially related to the Riemann hypothesis a while back, in the context of what you can say about infinite products over p in prime numbers of 1 / (1 - p^2). Strangely enough, it is equal to the Riemann zeta function (in the specific case where s = 2):
http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
It's pretty interesting stuff. Using the connection to the Riemann zeta function, you can show that our infinite product converges to pi^2 / 6. It's pretty amazing to me that there is this relationship between pi and the prime numbers:
http://en.wikipedia.org/wiki/Basel_problem
But anyway, there is a (possible) development on the Riemann hypothesis front. John Cook says it better than I would:
http://www.johndcook.com/blog/2009/02/05/proposed-riemann-hypothesis-proof-by-julio-alcantara-bode/
Aside from satiating our interest, I'm still not clear on whether there would be any practical implications of proving the Riemann hypothesis. At least from my limited understanding of the problem, it sounds like most people are pretty convinced that it's true, based on lots of empirical evaluation. We just don't have a formal argument of why.
Update: Thanks, Carlos. Abstract: This paper has been withdrawn by the author, due to a crucial error in page 5.(http://arxiv.org/abs/0902.0617)
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