Riemann theorem/Euler product

[MisterBitcoin]

Hello guys,

there is an topic in my mind that need a bit more detailed explanation. I´m talking about the connection between the primes and the Riemann-Zeta-Function (also called Euler product)

My head just starts to explodes every time I try to understand these theorem, but there must be a solution.
If the connection between the primes and the zero´s from riemann´s zeta-function are just giving the same results, we should be able to use that as a new prime test, isn´t?

It might be possible that I´m totally wrong with that (most likely, because someone else would have seen that before me. hehe). Anyway, it would be great if some guys could me (or us) more about the euler product an the primes.

[CRGreathouse]

Originally Posted by MisterBitcoin

If the connection between the primes and the zero´s from riemann´s zeta-function are just giving the same results, we should be able to use that as a new prime test, isn´t?

Sure. But it's not even remotely in the neighborhood of being competitive. You'd need tons of zeros, and lots of precision, and horrendous analytic calculations just to be able to compute something like pi(7918.5) and pi(7919.5) to within 1/3 each, just enough to verify that 7919 is prime.

[Nick]

@MisterBitcoin:
To get insight into the Riemann-Zeta function, you need to master the basics of complex analysis first.
A couple of books which teach that and then introduce the Riemann-Zeta function at the end are:

"Complex Analysis" (2nd edition) by Freitag and Busam, published by Springer:
http://www.springer.com/us/book/9783540939825

"Function Theory of One Complex Variable" by Greene and Krantz, published by the AMS:
http://bookstore.ams.org/gsm-40-r

[Dubslow]

Well... you can probably get something out of it with a suitably good understanding of multivariable calculus and a passing knowledge of complex analysis... still quite a bit of underlying theory though.

Reading through Riemann's original paper is a treat (though I wish there was a version that used modern notation to go with the translation for my sorry monolingual brain)

[MisterBitcoin]

Originally Posted by CRGreathouse

Sure. But it's not even remotely in the neighborhood of being competitive. You'd need tons of zeros, and lots of precision, and horrendous analytic calculations just to be able to compute something like pi(7918.5) and pi(7919.5) to within 1/3 each, just enough to verify that 7919 is prime.

Autsch, ok.

@Nick Thanks for the links. :) They might also help with the decision "What shall I study?". Only chemistry, particle physics and math were left.

@dubslow
Reading Riemanns original paper is for me easier, because a german mathematician writes his books in german.