I do not think that is the actual problem. Maybe some modified problem. There is a Riemann Hypothesis (which is proved) for function fields. Maybe this is some modifed problem?
There are parts that look similar, yes, but I doubt there's enough of a connection to conclude there's a proof of the RH.
Most likely it's just something to grab passing mathematicians' attention - as it seems to have done
I'd say if they were more confident of having a proof, or of even providing a stepping-stone, that they would hold out in the hope of cracking it - considering the prize that's at stake...
No one's going to remember the penultimate step in cracking the hypothesis...
All numerical evidence to date supports the hypothesis, but its actual truth is unknown.
For information on the -function and the Riemann Hypothesis, you should read:
T. M. Apostol (1976) Introduction to Analytic Number Theory
H. A. Priestley (2003) Introduction to Complex Analysis
For other popular accounts (Like du Sautoy's), read:
J. Derbyshire (2003) Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
K. Sabbagh (2003) The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
K. Devlin (1988) Mathematics: The New Golden Age
K. Devlin (2002) The Millenium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time
There's more of course... The first two are good intros to analysis and the zeta function, the last five are good popular books that aren't too mathematically rigorous.
Here is the arxiv.org link
[0807.0090] A proof of the Riemann hypothesis
When I found out about the Prize Problems, the one that appealed to me the most was the Riemann Hypothesis. I wanted to solve that one. However, when I heard TPH wanted to solve that, I decided to pick something else [since it had to do with number theory, and I have no clue what it is...yet...]. So I have decided to take on the Navier-Stokes Problem.
...Do you think that the Riemann Hypothesis has been "officially" solved??
Like I said the original RH is not number theory it is complex analysis.
I hope you solve that one. It has to do with differencial equations, stuff that you like.So I have decided to take on the Navier-Stokes Problem.
No. The paper is either a RH formulation in number fields not the original problem. And RH in number fields is solvable. If the person claims it is the original problem then it is probably wrong. Just read what Wikipedia says about RH, if it does not say anything about anyone writing a paper which has a solution then it means it is not solved....Do you think that the Riemann Hypothesis has been "officially" solved??
I forgot that you mentioned that earlier...
Thanks! If this problem hasn't been solved, I wish you luck in beating others to it. The Navier-Stokes Problem is a pretty interesting problem...and the whole thing doesn't need to be solved in order to win the prize...I think. They request that only one of the four statements [they propose] needs to be proven. See here.I hope you solve that one. It has to do with differencial equations, stuff that you like.
I read this weeks ago and I agree with what you said. I also heard that it takes a couple years for the Institute to examine the submitted proof before the problem has been declared as solved or not solved...No. The paper is either a RH formulation in number fields not the original problem. And RH in number fields is solvable. If the person claims it is the original problem then it is probably wrong. Just read what Wikipedia says about RH, if it does not say anything about anyone writing a paper which has a solution then it means it is not solved.
Terence Tao believes that the proof is not valid!
Dr. Koornwinder feels Professor Zhang's paper is also not correct:
http://staff.science.uva.nl/~thk/art/comment/ZhangRuimingComment.pdf
I've recently been reading 'The Music of the Primes' by Marcus du Sautoy. Being still at school, I can't really fully understand the whole problem, let alone solve it, but I have had a couple of thoughts:
1. Has anyone ever looked at non-primes, because if some of them are on the line it may not be as useful.
2. Has anyone thought of looking at the area (or volume) under the graph, for example between neighbouring zeroes?
I apologise if these have been thought of already, but if not, I hope that they provide some headway into the problem, or some enjoyment in trying them out.
1. What?
2. This is actually kind of interesting, here's why:
Given , order the zeros of corresponding to increasing ordinates, i.e. for , .
Define and .
Right now it's known that and , however it's conjectured that and .
If this conjecture was true, then could lead you to believe the integral along the critical line converges, but might lead you to think otherwise.
I'll tinker with this question for a bit and see if I can come up with actual results.