Can someone explain zeros and zeta function for Riemann Hypothesis? (Yr13)

CKM

You're very kind!!

I just edited my question while you were typing this reply. Ha ha! Oops.
I'll try to work through your reply very thorough above, a bit (with my limited experience), but do check out my edited question, if you don't mind. It will tell you where I got confused more ... in regular language. I do pick up some things from what you've written and from the equations sometimes, too, since you're interspersing explanatory prose, too.

I always wondered, by the way, what is the T ------ is that T for Trivial? I assume not, but am not sure.

Thanks! I'm not an idiot; I am expanding on my readings about it.

And OH SO YOU USE TRIVIALS AS WELL ... is that towards counting the Gamma function? I.e., what part of the process do they come in? They are not byproducts, but USED for something?

Kraflyn

Hello.

Ha ha, go get it, tiger!

You can handle it Yourself, CKM, i believe in You. This is the easy part of proving something about zeta. Try it!

Let me tell You something about first steps into zeta instead.

Everything begins with:

Fundamental theorem of arithmetic: For every [itex]n \in \bf{N}[/itex] there exists a unique sequence of natural numbers [itex]h(p)\in \bf{N}_0[/itex] such that
[itex]n=p_1^{h(p_1)} p_2^{h(p_2)} ... =\prod_p p^{h(p)}[/itex]
Factorization is unique for every [itex]n[/itex] and product runs over all primes [itex]p[/itex].

One can easily prove this.

Let us continue by defining
[itex]\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}[/itex]
One can also rewrite this as another definition of zeta,
[itex]\zeta(s)=\prod_p \frac{1}{1-{1}/{p^s}}[/itex]
Last two equations can be converted one into another just by applying fundamental theorem of arithmetic. So this is all very basic.

OK. Consider definition [itex]\zeta(s)=\prod_p \frac{1}{1-{1}/{p^s}}[/itex]. None of factors vanishes nor is unbounded if [itex]\Re s >1[/itex]. This means zeta has no singularities nor zeros in region [itex]\Re s >1[/itex]. So, You see, we can get basic informations on zeta rather easily.

In order to proceed, we need complex analysis: we would next deploy Gamma function! However, this will do for now. The point is: first steps into zeta are easy and fun!

Cheers.

Kraflyn

Hi.

If I got You correctly, You ask what is T: do You mean what is [itex]\Gamma[/itex]? [itex]\Gamma[/itex] is capital greek letter Gamma. Function [itex]\Gamma(s)[/itex] is the factorial Gamma function:

[itex]\Gamma(n)=(n-1)!=(n-1)(n-2)(n-3)\dots3\times 2 \times 1[/itex]

It is defined for all complexes [itex]s[/itex] except negative integers and zero, where it becomes unbounded.

I seriously hope I got You right, because otherwise... I'm doomed c:

Cheers.