Problem Number 8

[Leonardo]

Hi folks

 

I'm not a mathematician but I'd like to ask some simpleminded question about Riemann's hypothesis.

 

It seems to me that the only way the hypothesis can be proved is by showing that the mathematical mechanism that generates the primes is exactly the same mechanism that generates the non-trivial zeros. Am I right about this?

 

If I am then, bearing in mind the origins of the zeta function, would I also be right in thinking that a proof would require showing that the primes are generated or determined by an interaction of interacting sine waves akin to the process that generates the harmonics of a vibrating string? Thanks.

[shmoe]

Well the zeros and primes are already known to be intimately linked via the Riemann-von Mangoldt explicit formula (stated by Riemann, proven by von Mangoldt). It gives the prime counting function (the number of primes up to x) in terms of an infinite sum indexed by the nontrivial zeros of zeta. Each term in this sum is a kind of wave like I think you're after.

 

If you've managed to somehow prove a zero free region for zeta (in the critical strip) you can use this explicit formla to translate to a bound for the error term in the prime number theorem, and vice versa (!). So theoretically you could prove the Riemann hypothesis by proving the error term in the PNT is "small enough". However this would be unlikely- historically the strongest error terms have come directly from the strongest zero free regions, not the other way around. It would also be very odd if that ended up happening, after all the zeta function is studied to give insight into primes.

[Leonardo]

Thanks. I can't say I understand too much of that, but I get the general idea. The question about waves came from reading that the zeros are in some way like tuning forks, and that the interaction of their different frequencies in some way defines the primes. Is this roughly correct?

 

In an in principle sort of way I'm trying to establish the relationship between the mechanism that generates the primes and the mechanism that generates the zeroes. Specifically, I'm wondering what the wave-related mathematics of Riemann's zeta function, the mechanism that generates the zeros, tells us about the nature of mechanism generating the primes.

 

I'm also interested in what it would mean to prove Riemann's hypothesis. As far as I can see to do this we'd have to be able to predict the positions of the zeros to infinity. I have no idea what doing this would entail. However, because of the close link between zeros and primes I'm wondering whether being able to predict the position of the primes to infinity isn't almost as good as a proof of the hypothesis, as long as it could be shown that the mathematical mechanism generating the zeros and the primes are equivalent. Does this make sense? If it's all nonsense feel free to say so.

[shmoe]

"The question about waves came from reading that the zeros are in some way like tuning forks, and that the interaction of their different frequencies in some way defines the primes. Is this roughly correct?"

 

Yes. You can think of the frequency being determined by the imaginary part of the zero, and the amplitude being determined by the real part of the zero. The furthur the zero is from the critical line (real part equals 1/2) the larger the amplitude, and in some sense the worse behaved the corresponding "wave" is (it's like you have one really loud tuning fork), and therefore the more erratic the distribution of the primes (worse error term in the prime number theorem).

 

A good exercise is to take the Riemann-von Mangoldt explicit formula and the first few hundred zeros (ordered by increasing imaginary part, zeros can be found on Andrew Odlyzko's website for example) and see how close the formula comes to giving the prime counting function with the truncated sum over your finite number of zeros (vary the number of zeros as well). You can also plot the "wave" each zero corresponds too.

 

To prove RH, you don't have to know exactly the locations of the zeros, 'just' that their real part is 1/2. It's not necessary to predict their imaginary parts (though quite alot is known about this already, like asymptotics for the number of zeros less than a certain height). I'm not sure exactly what you have in mind for "predict(ing) the position of the primes to infinity", but a way to quantify this would be bounds on the error term in the prime number theorem. In this interpretation, you could prove RH by knowing enough about the position of the primes (like I said before though, this is not a promising method to prove RH).

 

If you want to quantify all of this handwaving about how zeros and primes are related, you should look up the explicit formula. There's plenty of online resources to be found, and plenty of good texts (e.g. Edwards, Ivic).

[Leonardo]

That's very helpful, thanks. Just one more thing. You say it would be possible to prove RH by knowing enough about the position of primes. What would one have have to know about them? If one came up with a function that produced the primes would this allow RH to be proved? Or, would one then have to go on show that this function was equivalent to to R's zeta function?

 

I take your point about 'handwaving'. Unfortunately the explicit formulae are over my head mathematically, which is why I have to ask what are probably daft questions. I haven't yet found a discussion of Riemann's zeros sufficiently simple that I can follow it. Pathetic really.

[shmoe]

The prime number theorem asserts that [math]\pi(x)\approx Li(x)[/math], where [math]Li(x)=\int_{2}^{x}\frac{dt}{\log t}[/math]. The question of just how good this approximation is is directly related to the location of the zeros- roughly the smaller the error the closer RH is to being true. If you can show that [math]\pi(x)=Li(x)+O(\sqrt{x}\log x)[/math] by some method then you'll have proven the Riemann hypothesis (conversely RH implies this error term).

 

There are elementary proofs of the prime number theorem (e.g. Selberg and Erdos), "elementary" meaning it avoids analytic methods and appealing to results about Zeta, not that they are simple. The error terms that these proofs are able to show will in turn say something about the zeros of zeta. However, they imply much weaker results about the zeros than we can achieve by attacking them directly with analytic techniques.

 

You also might want to look up the Merten's function. It's pretty simple to understand and has a nice related equivalence with RH. (though this also seems like a hopeless way to tackle RH)

 

Pathetic really.

 

Nahh, it's a very, very complicated beast. If you haven't already, you might want to check out some of the "popsci" books on Zeta. My usual reccomendation is to start with Derbyshire's "Prime Obsession" and work your way up to Edward's textbook (which also contains a translation of Riemann's original paper).

 

Check out http://www.math.ubc.ca/~pugh/Psi/ it's a nice version of what I called a "good exercise" in my last post. There's some other nice applets on this site as well. It has the explicit formula at the top (not for pi(x) but a related function) that you should be able to understand the bits of if you understand complex exponentials (which aren't that difficult) and the fact that the zeros come in conjugate pairs.

[Leonardo]

Thanks for all that. I'll have a look around some of your references.