Explain Riemann Hypothesis: Imaginary Numbers & Interval

[Weave]

Can someone please explain to what exactly the Riemann Hypothesis is?
My friend said it is something to do with imaginary numbers and how they behave in a certain interval- just wondering.

 

[yenchin]

It's quite difficult to explain the Riemann Hypothesis is a short forum post though. Perhaps you should tell us how much math background you have so that we can explain accordingly.

If you have the time, whether you are a complete layman or you know a lot of math, you may enjoy reading the book Prime Obssession (. The author said that if you don't understand Riemann Hypothesis after reading the book, you probably never will.

 

[mathwonk]

it invovles a connection between zeroes of a certain holomorphic function and the distribution of prime numbers.

riemann knew that a complex ["meromorphic"] function was essentially determined by the location of its zeroes and "poles" (places where it equals infinity).

so if there is a holomorphic function which is determined by the distribution of the primes, then the zeroes and poles of that holomorphic function will say something about the distribution of primes.

euler wrote down a beautiful infinite product relating primes and ordinary integers, based on the unqiueness of prime factorization, as follows:

the formal product of the factors 1/(1-1/p) is the product of the geometric series (1 + 1/p + 1/p^2 +1/p^3 +...) for all primes, equals the series 1+ 1/2 + 1/3 + 1/4 + 1/5 +...
of reciprocals of all integers, by unique prime factorization,

Although this does not converge, the corresponding product, over all primes, of the factors (1 - 1/p^s), where s>1 does converge.

Riemann then defined a complex function zeta(s) to be this product over all primes,

and equal to the infinite sum of 1/n^s over all integers.

although this representation only converges for |s| > 1, riemann shiowed it actually extended to be meromorphic in the entire plane.

since it is determiend entirely, by eulers product, by the distribution of primes, its zeroes and poles should hold crucial information on that distribution.

the poles are at the negative integers i believe, but the zeroes are harder to find.

riemann showed they were clustered near the line where re(z) = 1/2, and deduced a very precise formula for the number of primes less than a given integer, improving an estimate of Gauss.

If the zeroes were actually all on the line re(z) = 1/2 one would get an even better estimate for the distribution of primes.

The search for this holy grail of number theory continues to this day.

does this help?

 

[mathwonk]

i am not a number theorist by the way so this should be taken with a grain of salt, as comments by an interested amateur.

i learned what i said here however by actually reading riemann, which i recommend to you over all other sources.

 

[Weave]

Thanks mathwonk. Reading rieman? It might be too advanced for me, I'm only in Calc by the way, lol. I was just interested because I heard its one of the biggest things people are trying to prove right now. I also hear a lot of peoples research depend on it.

 

[mathwonk]

heed my words, the great people are easier to understand than their imitators. i.e. it is easier to understand something explained by someone who actually understands what they are talking about than someone who does not.

indeed you may not understand it, but then again you may understand something. and anything you get is worth its weight in gold.

reading the masters is a good habit to form. at some point people who do this will find themselves way beyond others.

 

[Weave]

That can be true..do you have any recommendations on books for Rieman and any other math/physics books as well?

 

[yenchin]

If you want to read Riemann's paper, you can get it for free on the internet:
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/

I usually find it helpful to start with a modern text, and then going back to read the masters' only after you have at least some understanding of the material. But of course the choice is yours.

 

[waht]

Where are the zeros of zeta of s?
G.F.B. Riemann has made a good guess;
They're all on the critical line, saith he,
And their density's one over 2 pi log t.

This statement of Riemann's has been like a trigger,
And many good men, with vim and with vigour,
Have attempted to find, with mathematical rigour,
What happens to zeta as mod t gets bigger.

The efforts of Landau and Bohr and Cramer,
Littlewood, Hardy and Titchmarsh are there,
In spite of their effort and skill and finesse,
In locating the zeros there's been little success.

In 1914 G.H. Hardy did find,
An infinite number do lay on the line,
His theorem, however, won't rule out the case,
There might be a zero at some other place.

Oh, where are the zeros of zeta of s?
We must know exactly, we cannot just guess.
In order to strengthen the prime number theorem,
The integral's contour must never go near 'em.

Let P be the function p minus Li,
The order of P is not known for x high,
If square root of x times log x we could show,
Then Riemann's conjecture would surely be so.

Related to this is another enigma,
Concerning the Lindelöf function mu sigma.
Which measures the growth in the critical strip,
On the number of zeros it gives us a grip.

But nobody knows how this function behaves,
Convexity tells us it can have no waves,
Lindelöf said that the shape of its graph,
Is constant when sigma is more than one-half.

There's a moral to draw from this sad tale of woe,
which every young genius among you should know:
If you tackle a problem and seem to get stuck,
Use R.M.T., and you'll have better luck.



Words by Tom Apostol (revised slightly by cph).

 

[mathwonk]

well i looked at some more modern works but they were not nearly as clear to me as riemann.