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Posted by Mark on August 29, 2001 03:14:52 UTC

My first questions, (and this doesn't necessarily pertain to the zeta function specificly), are: is there a difference between a simple pole and a singularity? And what makes a simple pole simple?
Are there more complicated poles?

Secondly: Euler stated that zeta(s) diverges for all s < 1 and converges for all s > 1. Then how does it make sence that the trivial zeros of the Riemann Zeta Function are of negative value...i.e. (-2, -4, -6)...?? These values are clearly less than 1, and does it make sence to say that a particular summation converges on zero? How are zeros derived?

An answer to any of these questions would be greatly appreciated.


[Side note]...

I would also like to note to any readers that the Riemann Zeta Function is the underlying foundation to prime number theory. And anybody who has been aquainted with the distribution of primes, is well aware of the random patterns that occur. The behavior of the Riemann Zeta Function bears a striking resemblence to the distribution of masses amongst elementary particles and behavioral statistics of gasses. It's very interesting to realize the almost magical relationship between the completely abstact & esoteric prime numbers, and their physical "partical mass" counterparts. Prime numbers are the atoms of the real numbers; elementary particles are the atoms of matter. They both share in their underlying nature. Once again I must state that our universe is a mathematical reallity, where the presence of mathematical rules emerge in the behavior of the universe.

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