Archive for December, 2009

Studying the Riemann Hypothesis

December 22, 2009

This post serves only one purpose: a place for me to keep links, and reveal my (lack of) knowledge on this subject.  If you can help, feel free to leave a comment…

I wanted to see if the “Riemann hypothesis” could mean more to me than “an unsolved mathematical puzzle.”  (Some may say this is not a proper entry in the “Comp. Sci.” category, but I’m interested in the opportunity to hone/refresh my math skills.)

Background: A good explanation is at the Physics Forums, under number theory: “Riemann Zeta Function?” It’s the first I found, with a link to another, better discussion, where some more elaboration is given — also a link to American Scientist article on the subject (an interesting read itself — giving some of the relationships to various applications).

In short, from the above paragraph: Riemann extended the Euler summation function (equiv. to the Euler product) using analytic continuation to the zeta function for all values (since Euler’s sum only converges for values greater than 1).

Less helpful by itself, but good reference, also giving related functions, and a couple of nice graphs, is the Wolfram Mathworld entry on the Riemann hypothesis.

Current State: It’s clear that, almost by defn. the equation on page two of this AMS article is equivalent to eq. 13 and 14 of the MathWorld entry;
I am trying to figure out if the equation in the fourth post of the 2nd thread at the Physics Forums is the same (they must be, but I can’t quite work it out…).

This doesn’t help me generate values, but I’m looking at that thread, or others for more info.  If I figure out the algebra, I’ll let you know.