Riemann hypothesis

Check it out at Mathworld, here:

http://mathworld.wolfram.com/RiemannHypothesis.html

Best regards,

/Clay

Check out the book by Devlin on the Millennium Problems

http://www.amazon.com/Millennium-Problems-Greatest-Unsolved-Mathematical/dp/0465017290

You can see the 7 Millennium problems here:

(click on 'Awards' at top)

http://www.claymath.org/index.php

(no relation, sigh)

First of the prizes has been awarded but the
mathematician refuses to appear.

Best regards,

/Clay

I don't know of any ebooks, but I recently read Du Satoy's "Music of the Primes" and it gives a good layman's overview of Riemann's zeta function and the Riemann Hypothesis. It also gives an explicit formula that will generate all the primes.

The hypothesis is pretty straightforward to state: the nontrivial zeros of the zeta function all have a real part of 1/2. But no one has proved this yet in the last 150 years or so.

Hi,
There still have been a problem in understanding prime # distribution. And the reason is because prime # at some intervals behaves under a certain rule, but then behaves at some other intervals in a random manner. So in general the question is: Is there a general rule that governs prime # distribution? Many great thinkers and mathematicians tried their best to solve this problem, starting from Euclid passing through Euler, Gauss, and many more. But it turned out to be that the Remann hypothesis to be the most accurate result to an error of root n.

f(x) = int dx/ln(x) = n + O(root[n(ln(n))^3])
the limit of integration 2 to pn
where pn = nln(n)

So if anyone can prove the above hypothesis, would earn a prize of $1 million.

One consequence of the Riemann hypothesis has to do with the distribution of prime numbers.

a) Riemann used this hypothesis to formulate a way to calculate the number of primes less than a specific number.

b) One interpretation of this was that the real parts of the Riemann zeta function, at its non-trivial zeros, indicate the error in the prediction of a prime number. To put this into other words, the error term in the prime number theorem can be inferred from the positions of the zeros.

c) This further leads to the speculation that the average gap between the nth prime and the (n+1)th prime is given by log(n).


Getting from the original hypothesis to a, and from a to b, and from b to c are each the subject of entire volumes. If you are going to take on any work concerning the Riemann Hypothesis, I wish you the very best luck. If you prove it you will be a millionaire, and even if you only come up with something publishable you will at least be a great mathematician.