A very simple question

[Kyriakos]

Hi, i have a very simple question (easy to state, i meant...)

-Can someone explain, in the simplest possible way, why the complex numbers were chosen as a basis for the infinite series of the Zeta function?

(in other words: why did Riemann build up his hypothesis of his Zeta function's non-trivial zero rational parts, in the idea of numbers which are partly imaginary too, ie x+iy)?

I ask because, ideally, i would not want to spend huge amounts of time making a connection which probably is quite simple in regards to examining the hypothesis itself. I can suspect that Riemann formed the hypothesis as a (particular type of) series based on complex numbers due to the ability (as he does in his hypothesis) to split part of the function into one about the purely rational part (which can be zero) and one about the complex number as a whole (the series is based on the latter). Also i can already guess that two qualities of numbers play a crucial role here:

1) the different, but simple to compute, cases where the difference between positive and negative rational numbers results in negative or positive for them raised in powers, if the powers are even or odd.

2) The trigonometric examination of the actual complex value of the complex number itself, which is part of the series but not in the same way part of the function Zeta itself.

3) Finally, some other, most probably very easy to compute, relation between the complex values, and the complex number, which plays a role in the specific series by Riemann as a supposed to means to calculate the number of primes up to a specific point (eg number).


(ps: the question was simple, and its answer probably is even simpler. The means to go about trying to examine if the hypothesis of Riemann is correct, is obviously an entirely other issue ).

http://en.wikipedia.org/wiki/Riemann_hypothesis

[mathematic]

"Can someone explain, in the simplest possible way, why the complex numbers were chosen as a basis for the infinite series of the Zeta function?"

 

If you stick to the real line, the definition is only good for s > 1. You need to use analytic continuation (in the complex plane) to get around the singularity at s = 1. Otherwise it is very uninteresting.

Edited September 25, 2013 by mathematic