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For s >1 , the Riemann Zeta Fuction is defined as:

[tex]\zeta(s)=\sum_{n=1}^{\infty}n^{-s}[/tex]

I have no problem with this. That series obviously converges for s>1, and will diverge for all other numbers. A Calc II series.

Now, it is my understandind that there are two other important qualities to take notice of from this function. One is that there are zeros at every negative even integer. The second deals with the "critical strip" from 0<s<1. It is here that the Riemann suggested that all "non-trivial" zeros on the complex plane have the real part 1/2.

My two problems with understanding this function and the hypothesis are: (1)How is this function evaluated for s<1 as it diverges? (2)How do zeros occur at negative even integers?

I am very interested in the progress in the proof of the Riemann Hypothesis. I believe I've read online a paper titled "Apology for the Proof of the Riemann Hypothesis" by a Perdue mathematician, but it doesn't seem to have been verified or taken seriously.

All of the information I put on this post was just to make sure I have a proper understanding, not to lecture. Most of it came from mathworld.com, so if any other sources are known, I'd appreciate some input.

Thanks in advance,

Jameson