Riemann Hypothesis: Explaining \zeta(s) & Diagrams

[steven187]

hello all

after doing a bit of research on the riemann hypothesis I came along this paragraph, in which I don't understand, especially the first sentence , how would one be able to show that?

It can be shown that $$\zeta (s) = 0$$ when s is a negative even integer. The famous Riemann hypothesis concerns the locations of the remaining, ‘nontrivial’ zeros of $$\zeta(s)$$. It asserts that they all have real part equal to 1/2.

It would be great if someone could elaborate on what this paragraph actually means? thanxs
also is there anyway I could get a diagram in which would show some kind of pattern to make me understand this? Iv been trying to come up with one on mathematica but I end up getting weird looking things,

steven

 

[HallsofIvy]

I'm not sure I can say it in any simpler terms. It can be shown that for the "Riemann zeta function", $$\zeta(x)$$, $$\zeta(-2)= 0$$, $$\zeta(-4)= 0$$, $$\zeta(-6)= 0$$, and that, in general $$\zeta(-2n)= 0$$ for any positive integer n (so that -2n is a negative even integer).

There exist other solutions to $$\zeta(s)= 0$$ but all know solutions either are negative even integer or complex numbers of the form (1/2)+ iy- that is, the real parts are 1/2. The "Riemann hypothesis" is that all non-real ("nontrivial") solution have real part equal to 1/2.

 

[steven187]

well I am going to explain where I am getting confused
$\zeta(-2)=\sum_{n=1}^{\infty}1/n^{-2}=\sum_{n=1}^{\infty}n^2=1^2+2^2+3^2+...=\infty$
now let's look at it from another perspective
[itex]\zeta(-2)=\prod_{p}(1-1/p^{-2})^{-1}=\prod_{p} (\frac{1}{1-p^2}) =(\frac{1}{1-2^2})(\frac{1}{1-3^2})(\frac{1}{1-5^2})...(\frac{1}{1-p^2})[/tex]
[tex]=(\frac{-1}{3})(\frac{-1}{8})(\frac{-1}{24})...(\frac{1}{1-p^2})\longrightarrow 0 [/itex]
well that's where I am getting confused , well in terms of non trivial solutions, how could you show that $$\zeta(\frac{1}{2}+ti)=0$$ for any value of t?, well does it equal to 0 for any value of t? or for some values of t? if the zeros at t are discrete then is there anything special about the pattern it gives? and would it be true to say that there are infinitely many zeros of the riemann zeta function - which all lie on the critical line except for the trivial solutions? also attached is what i produced while playing around with mathematica, but I don't know how to relate it back to the riemann zeta function? if anybody understaands the diagram could you please explain? thanxs

Steven

 

[shmoe]

You've seen that $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ when Re(s)>1. You've probably also seen Euler's product form, $\zeta(s)=\prod_{p\text{ prime}}(1-p^{-s})^{-1}$, also valid where Re(s)>1. Using this product form it can be shown that zeta has no zeros in the right half plane Re(s)>1.

Now to define zeta when Re(s)<=1 we need to use analytic continuiation. There are many different proofs that such a continuation exists to the entire complex plane (with a pole at s=1), if you've got some complex analysis behind you, you should look one up (any text on zeta will have one). It can then be shown that zeta satisfies this glorious Functional Equation:

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

This is really one of the best things since sliced bread. The Dirichlet series used to define zeta is pretty well behaved (except the unfortunate problem of only converging where Re(s)>1). If we want to know what happens at $$\zeta(s)$$ and Re(s)<0, the functional equation let's us translate this to a question about some well understood junk $$2^{s}\pi^{s-1}\sin(\pi s/2)\Gamma(1-s)$$, and $$\zeta(1-s)$$, where Re(1-s)>1 and we can use our nice Dirichlet series (or Euler product version) to handle this.

For example to find the zeros of zeta with Re(s)<0 we look for zeros and poles on the right hand side of the functional equation (poles of one factor may cancel zeros of another). We know $$2^{s}\pi^{s-1}\Gamma(1-s)\zeta(1-s)$$ is never zero here (Euler product handles the zeta term) and have no poles here either. So when Re(s)<0, $$\zeta(s)=0$$ exactly when $$\sin(\pi s/2)=0$$, which is the negative even integers, -2, -4, -6, etc.

So we've managed to find all zeros of zeta outside the region 0<=Re(s)<=1.These were (relatively) easy to find, and are called the Trivial zeros. It's also known that there are no zeros with Re(s)=0 or 1, which leaves 0<Re(s)<1 as the place left to examine, this is known as the critical strip and is where the interesting stuff happens. It's known there are infinitely many zeros here, in fact it's known there are infinitely many on the critical line Re(s)=1/2. There are methods of actually locating all the zeros in the critical strip with imaginary part less than some finite bound. Billions of zeros in the critical strip have been found this way, and so far all have been on the critical line. The conjecture that all these zeros are in fact on the critical line is the Riemann hypothesis.

ps. I typed this before seeing your second post, I'll respond to that in a moment.

 

[shmoe]

Ah, an important thing you've missed is that the series used to define zeta does not converge if the real part of s is less than 1 (same for the product form) and thus only defines a function on this right half plane. The analytic continuation is the key if you want to extend this function to all complex numbers. Have you seen much complex analysis?

As for $$\zeta(1/2+it)$$ equalling zero, yes it does and infinitely often (this is the critical line I mentioned above). See https://www.physicsforums.com/showthread.php?t=68125&highlight=zeta for my attempt to simplify how to find zeros and approximate them numerically. We would definitely love to say more than we currently know about the distribution of the t values of the zeros, the distribution appears to have some nice properties.

Your plot appears to be the the absolute value of $$\zeta(1/2+it)$$ for t from -30 to 30. Not sure what to say about it, but notice the symmetry about zero (have you seen the reflection principle?) and where it touches down to zero, when t is plus or minus, 14.13..., 21.02..., 25.01... These are the first few zeros of zeta.

 

[snoble]

Can I suggest checking out the mathworld page http://mathworld.wolfram.com/RiemannZetaFunction.html. Since the zeta function has become so popular in popular mathematics mathworld seems to have gone out of its way to write a very thorough discussion including a discussion of analytic continuations and lots of diagrams to assist understanding.

 

[saltydog]

Two more suggestions:

The book "Prime Obsession"

A nice article in American Scientist called "The Spectrum of Riemannium". Very interesting:

http://www.americanscientist.org/template/AssetDetail/assetid/21879

 

[mathwonk]

there is a nice book by harold edwards on this problem, including a translation into english of the original short paper by riemann.

 

[shmoe]

Edward's book is "Riemann's Zeta Function", not that it would have been hard to guess. Two really great features are the translation mathwonk mentioned and that it's a cheapie Dover book. I especially recommend it to anyone who plans to read Derbyshire's "Prime Obsession", as both follow the same classical lines and Edwards fills in the details that a motivated reader will find lacking in Derbyshire's popsci book (Derbyshire seems to have used Edwards as his main source). Derbyshire has some nice history in it that you don't find in the usual texts, so I found it a worthwhile read as well.

There's lots of info available on the web as well. A nice summary of the state of things by Conrey appeared in Notices a couple years back http://www.ams.org/notices/200303/fea-conrey-web.pdf you have to register to read it (it's free and it's the AMS, so you can probably trust them)

 

[mathwonk]

the advantage of edwards is the actual paper of riemann. i.e. the most important source to read is riemann himself. as edwards says so eloquently in his preface.

 

[shmoe]

The excellent quote from Edward's preface "If you read this book without reading the primary sources you are like a man who carries a sack lunch to a banquet."

A translation of Riemann's paper can be found online as well http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf

 

[steven187]

hello all

thanxs for the great info guys especially the references I guess I really need to buy some books on this stuff, this topic is so wide that i can't even imagine how wide it is, well I havnt actually studied complex analysis yet but hopefully soon, now you quoted that

The analytic continuation is the key if you want to extend this function to all complex numbers.

but if It can be shown that $$\zeta (s) = 0$$ for s being a negative even integer in this case s isn't a complex number? like I understand its complex roots but i don't understand how it could have negative even integer real roots?

what kind of properties does the distribution of t have?

also on my graph in mathematica how come when i take the absolute value sign of it doesn't plot?

now how does the riemann hypothesis relate to the distribution of prime numbers? i have a feeling that it is related to eulers product form but i don't know how? also if riemann hypothesised it why didnt he prove it? if he lived longer do you guys believe that he could have proved it? also why is it so important to the field of maths and how would it be useful in aplying it into the physical world? and do you believe that anybody would be able to solve it in the near future? if not what is stopping humans from solving it?

 

[shmoe]

but if It can be shown that $$\zeta (s) = 0$$ for s being a negative even integer in this case s isn't a complex number? like I understand its complex roots but i don't understand how it could have negative even integer real roots?

Think of the real line as lying in the complex plane. -2 is a complex number with 0 imaginary part if you like.

If you hope to understand the zeta function better, at least a basic grasp of complex analysis is necessary. The texts on zeta will take these things for granted.

what kind of properties does the distribution of t have?

The link gvien by saltydog is a nice stab at explaining the pair correlation conjecture.

also on my graph in mathematica how come when i take the absolute value sign of it doesn't plot?

You mean when you take the absolute value sign off? Without this absolute value sign, $$\zeta(1/2+it)$$ is roaming around the complex plane, it's not a real valued function so it probably won't work with the normal plotting routine (I'm not too familiar with mathematica though). Try plotting the same graph but take the real part or imaginary part instead of absolute value (sorry don't know the command). You should also be able to do a parametric plot or something to see it twirl around in the complex plane.

now how does the riemann hypothesis relate to the distribution of prime numbers? i have a feeling that it is related to eulers product form but i don't know how? also if riemann hypothesised it why didnt he prove it? if he lived longer do you guys believe that he could have proved it? also why is it so important to the field of maths and how would it be useful in aplying it into the physical world? and do you believe that anybody would be able to solve it in the near future? if not what is stopping humans from solving it?

The location of the zeros is directly related to the error term in the prime number theorem. Showing that there were no zeros on the line Re(s)=1 is how the prime number theorem was first proved. Essentially the closer we know all the zeros lie on the critical line, the better we know how the primes are distributed. Look for Riemann's explicit formula, it gives the prime counting function exactly as a certain sum over the non-trivial zeros of zeta.

Riemann briefly mentions his attempts to prove all the zeros were on the critical line, but resigns to put this task aside for another time. Who can say if he would have been able to solve it eventually? It was believed for years that he had incredible insight to make this conjecture, I mean we can now check for thousands of zeros in seconds on computers, he had no such luxury. It was later found that he had in fact calculated the locations of the first few and found them to be on the critical line, so some of the mysticism was lost. However his techniques for doing this are essentially what the modern ones are based on (Riemann-Siegel formula), so was still many years ahead of his time. Tough to say if it will be solved soon, there's lots of work going in but the progress sometimes seems dismal compared to the goal. Really big breakthroughs are rare.

 

[mathwonk]

actually, riemanns collected works, begins with his 39 page thesis, which is a complete treatment of a basic course in complex analysis. "After understanding that", one is presumably ready to read his paper which is only 8 pages long.

But I would recommend just starting with the prime number paper, if that is your goal, and referring to complex analysis references as needed.

 

[mathwonk]

well now i have actually read riemann's paper on prime numbers, and consulted h.m. edwards as well, and i recommend doing that to understand what riemann did. in particular basic compelx analysis is not sufficient to read riemann's paper, but some Fourier analysis is needed, such as Fourier inversion. edwards is very helpful.

 

[mathwonk]

i have gained the following impression from my reading and would appreciate corrections from experts.

it seems that first euler noticed that, assuming the factorization of natural numbers into primes, that speaking purely formally, one has the sum of all the natural numbers n equal to the product of the infinitely many factors
(1+2 + 2^2 + 2^3 +...)(1+3+3^2+3^3+...)(...)(1+p+p^2+p^3+...)(...).


To try to render this meaningful, one can instead consider reciprocals, since now

(1+1/2+1/2^2+...) = 1/(1 - [1/2]),..., (1+1/p+1/p^2+...) = 1/[1 - 1/p], ...


so now at least all the factors are finite, and we may hope that the product of the factors
1/[1 - 1/p] for all primes p, equals the sum of the reciprocals of the natural numbers

1 + 1/2 + 1/3 + 1/4 +... however still both these infinite expressions diverge.


then euler thought to use exponents. indeed he knew the sum of the squares of the reciprocals of natural numbers equalled pi^2/6, etc...,

so he looked at the equation PRODUCT (over all primes p) of 1/[1 - (1/p)^s]

= SUM over all natural numbers n, of 1/n^s.

he considered it principally for integers s.

then perhaps legendre looked at this expression as a function of a real variable s, and noted it converges for any s > 1.


then riemann took it up and naturally for him asked for its fullest range of definition. thus he considered complex values of s, obtaining a complex meromorphic function called the zeta function.

i.e. for riemann a complex holomorphic function is determined globally by its values in any region, and moreover, his overriding point of view was that a meromorphic complex function is best understood by its zeroes and poles.

hence here is a function which is wholly determined by an expression involving only prime numbers, so its behavior reflects the nature, i.e. location, of the prime numbers, but which is best understood by considering its zeroes and poles (of which it has none for Re(s) > 1). thus the curious fact that s= 1 is a point where the original formula makes no sense, is replaced by the intersting fact that s=1 is the unique pole of the global extension of this function. hence the other crucial points must be the zeroes of the function, none of which are visible until Riemann's extension of the function.

now the problem riemann considered was that of determining roughly the number of prime numbers that are less than a given number x = pi(x).

according to edwards, euler's observations on the rate of divergence of the series 1+ 1/2 + 1/3 + 1/5 +...+1/p+... can be phrased as saying the density of primes is 1/log(x).

Then Gauss conjectured that the number pi(x) was well approximated by Li(x) = the integral from 0 to x of dt/ln(t), (up to a small constant).

now riemann's paper attempted to improve this approximation for pi(x) as follows:

Using his zeta function, and various manipulations of integral expressions for it, Riemann claimed to show that in fact Gauss's estimate Li(x) approximates more closely the number

Li(x) (roughly) = pi(x) + (1/2)pi(x^[1/2]) + (1/3)pi(x^[1/3]) + ...

i.e. not just the number of primes less than x but also 1/2 the number of squares of primes, plus 1/3 the number of cubes of primes, ...

hence he says that Li(x) overestimates pi(x) by a term of order of magnitude x^(1/2).

By solving the approximate equation above for pi(x) he obtains a formula approximating pi(x) (roughly) =
Li(x) - (1/2)Li(x^[1/2]) - (1/3)Li(x^[1/3]) - (1/5)Li(x^[1/5]) + (1/6)Li(x^[1/6]) - +...


where in this sum, only square - free integers appear, and the sign is determined by the number of prime factors, minus if an odd number, or plus if an even number.


Empirically this approximation is indeed superior to Gauss's, but Riemann's argument for the error expression between this formula and pi(x), i.e. for his claimed theoretical accuracy of his approximation, is apparently what needs the Riemann hypothesis on just where the zeroes of the zeta function are located for a rigorous proof.


I learned all this, such as it is, just today, from reading riemann and then edwards. I do not pretend to have understood either, but, if even roughly correct, this is a lot mroe than i knew before. the moral is that one learns some useful facts really quickly by reading riemann, more than in a lifetime of reading others.

 

[change257]

For steven187, the quick answer is that the analytic continuation of zeta is not accurately expressed by the well-know infinite sum for zeta. It just "ain't". I can explain SOME other things about primes and zeta, if you wish.