How do non-trivial zeros of the Riemann Zeta Function occur?

[Jameson]

I've tried my best to understand the Riemann Zeta Function on my own, but I appeal to the knowledge of you guys to help me understand more.

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

$$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$$

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

 

[matt grime]

You need something called analytic continuation.

Recall that log(1+z) has a power series for |z|<1, something like z-z^2/2+z^3/3-..., right? Well, just because that power series isn't valid for any z with |z|>1 doesn't mean that log(1+z) isn't a perfectly good function of any complex number (except z=-1).

Suppose now I giev youi the series 1+x+z^2+z^3+... this only converges for |z|<1 as well, but can you think of a way to define a function that agrees with this for |z|<1 but is valid for more z?

With the zeta function we start simply with a series and then try to see if we can extend it to an analyitc function, and it turns out we can to all of the complex plane except for some (can't remember how many points) points in the ciomplex plane.


The idea is that the existence of a Taylor series about some point is purely local datum, and if we can patch together lots of local data then we can create a global function.

Here, local means defined in some smaler area around points, the small area depending on the point, and global means defined on all of the complex numbers (possbly with some poles)


De Branges is the mathematician you are referring to at Perdue, and he perhaps falls under tha title "boy who cried wolf".

 

[shmoe]

de Branges 'apology' can be safely ignored. The method he hoped to use has been shown to not work by Conrey and Li.

You're right that the Dirichlet series only converges when the real part of s is greater than 1. However it can be analytically continued to the rest of the complex plane (with a simple pole at s=1) in exactly one way. That it can actually be continued is non-trivial and hugely important, that this continuation is unique comes from some complex analysis.

These sort of questions have come up many times here. try doing a search for "Zeta" or "Riemann" or both, e.g.

https://www.physicsforums.com/showthread.php?t=66708&highlight=riemann explains analytic continuation a little bit

https://www.physicsforums.com/showthread.php?t=78799&highlight=riemann
mentions something about how to find the trivial zeros

https://www.physicsforums.com/showthread.php?t=68125&highlight=riemann
expalins a little on how non-trivial zeros can be located.

Look about, come back with any more questions you have.

 

[Jameson]

Thank you both for your posts.

shmoe - I've looked at the different posts and have some questions. I'll ask one just for now.

The Zeta function can be continued analytically by the functional equation:

$$\zeta(s)=2^{s}\pi^{s-1}\sin(\pi s/2)\Gamma(1-s)\zeta(1-s)$$

I'm familiar with the Gamma Function, but it's still hard for me to see the connection with the Zeta function.

You said there are many proofs of the analytic continuation. Could you please point one out or briefly explain if possible? I can see how it works, but really want to understand it. Thanks again.

 

[Hurkyl]

The gamma function is just one of those functions that pops up everywhere. Try http://mathworld.wolfram.com/RiemannZetaFunction.html for more info on the zeta function.

 

[Jameson]

Thanks Hurkyl. I'm actually pooling most of my information from the mathworld site. It's thourough, but hard to take in all at once. At least for myself.

Mathworld mentions the analytic continuation, but I only see it for s>1, not for s<1. Nor do I see the continuation I copied in my previous post.

 

[fourier jr]

i read on wikipedia that the zeroes of the zeta function & the prime numbers satisfy some duality property (the "explicit formulae") which show that the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes. could someone explain more?

 

[fourier jr]

never mind... a google search found me this page
http://www.maths.ex.ac.uk/~mwatkins/zeta/NTfourier.htm

looks like the riemann hypothesis, primes, etc all have weird physical & geometric implications. i wouldn't have thought that about what seems to be a number theory problem. cool

 

[shmoe]

You said there are many proofs of the analytic continuation. Could you please point one out or briefly explain if possible? I can see how it works, but really want to understand it. Thanks again.

You can find several in Titchmarsh's "Theory of the Riemann Zeta Function", a couple in Ivic's "Riemann Zeta Function", and of course a couple in Riemann's original "On the Number of Primes less than a given Magnitude" (a translation of this can be found somewhere online, or see Edward's book for a translation and more details). Also many analytic number theory texts will have a proof or two e.g Iwaniec and Kowalski. Riemann's original will be the most bare bones (and a strongly recommended read if you're interesting in this sort of thing), Edward's an excellent guide to accompany it. Check with a source (or two) and come back with any questions.


As to the harmonic nature of the zero's, I think it's easiest to see in the explicit formula:

$$\psi(x)=x-\sum_\rho\frac{x^\rho}{\rho}-\frac{\zeta'(0)}{\zeta(0)}-\frac{1}{2}\log(1-x^{-2})$$

where the sum is over the non trivial zeros of zeta $$\rho$$, taken in order of increasing magnitude,and $$\psi(x)$$ is counting prime powers with logarithmic wieght, e.g. $$\psi(7)=\log(2)+\log(3)+\log(2)+\log(5)+\log(2)+\log(7)$$ (the extra log(2) is 'counting' 4 with wieght log(2)). This $$\psi$$ may look nastier than the usual $$\pi(x)$$, but it turns out to be the more natural thing to work with. It still has jumps at primes ( and relatively smaller ones at prime powers) and it's not too difficult to go back and forth between results for pi and psi.

The important bit to notice in this explicit formula is the sum over the zeros. If you concentrate on any specific zero, the corresponding term is oscillating as x grows as $$\rho$$ is a complex number (the term is also real when paired with it's conjugate counterpart), so this infinite sum is really the sum of an infinite number of waves, each corresponding to a non-trivial zero. The other thing to notice is as the magnitude of these zeros grows, the amplitude of these waves, the magnitude of $$x^{\rho}/\rho$$ becomes smaller and smaller. It's kindof interesting to get a a few hundred zeros (say from odlyzko's website) and muck about with some computations involving the explicit formula in a truncated form. I want to say there's a java applet somewhere that let's you play with this (try google).

Another thing to notice is that the prime number theorem is essentially that $$\psi(x)~x$$ so everything else in the explicit formula above is the "error term" (of course work was involved in showing this 'error term' is actually smaller than the main term). Another factor in the magnitude of each wave is the real part of the zero (if $$\rho=\sigma+it$$ then $$|x^\rho|=x^\sigma$$). Since the zeros come in 4's symmetric about s=1/2 (if z is a zero, so is 1-z, and the conjugates of z and 1-z are zeros as well, there's only 2 when z is on the critical line not 4), the best you can hope for to get a small amplitude here is having the real part equal to 1/2 (since a zero with real part 1/4 means you'd have one with real part 3/4). Hence the error term is smallest if we knew the Riemann Hypothesis were true and all zeros in this sum had real part 1/2. Using asymptotics for the number of zeros up to a given magnitude (the "Riemann-von Mangoldt" formula), you can make the connection between the location of the zeros and the magnitude of the error term explicit.

edit- http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html has the java applet (and more) I mentioned above at http://www.math.ubc.ca/~pugh/Psi/