# Thread: Riemann Hypothesis

1. apparently it has been proved: here

2. 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?

3. The RH has been proven?. Wow, that is a big deal. First I have heard of it. I am sure it'll have to be scrutinized ad nauseum before it is accepted as a valid proof.

4. 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...

5. Originally Posted by Jhevon
yes. i know the theorem. just curious, what doors would it open if the theorem is actually proven? are there theorems in existence that are like "if the Riemann Hypothesis were true, then ..." like there were with the Takamuri conjecture? (i'm almost certain i spelt that wrong)

i have to read up on how Riemann came up with the hypothesis...
There are lots of theories that begin by assuming the Riemann Hypothesis is true. I think it's Marcus du Sautoy's book "The Music of the Primes" that gives an estimate.

6. Originally Posted by echoes
There are lots of theories that begin by assuming the Riemann Hypothesis is true. I think it's Marcus du Sautoy's book "The Music of the Primes" that gives an estimate.
All numerical evidence to date supports the hypothesis, but its actual truth is unknown.

For information on the $\zeta$-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.

7. 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??

8. Originally Posted by Chris L T521
[since it had to do with number theory, and I have no clue what it is...yet...].
Like I said the original RH is not number theory it is complex analysis.
So I have decided to take on the Navier-Stokes Problem.
I hope you solve that one. It has to do with differencial equations, stuff that you like.
...Do you think that the Riemann Hypothesis has been "officially" 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.

9. Originally Posted by ThePerfectHacker
Like I said the original RH is not number theory it is complex analysis.
I forgot that you mentioned that earlier...

I hope you solve that one. It has to do with differencial equations, stuff that you like.
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.

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.
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...

10. Originally Posted by MathGuru
Terence Tao believes that the proof is not valid!

11. Originally Posted by tukeywilliams
apparently it has been proved: here
Dr. Koornwinder feels Professor Zhang's paper is also not correct:

http://staff.science.uva.nl/~thk/art/comment/ZhangRuimingComment.pdf

12. 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.

13. For two, I think that the area under the line would diverge.
But who knows......(-:

14. Originally Posted by RobertK
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 $\gamma_n>0$, order the zeros of $\zeta(s)$ corresponding to increasing ordinates, i.e. for $\rho_n=\beta_n+i\gamma_n$, $\gamma_n\leq \gamma_{n+1}$.

Define $\mu = \liminf\frac{(\gamma_{n+1}-\gamma_n)\log\gamma_n}{2\pi}$ and $\lambda = \limsup\frac{(\gamma_{n+1}-\gamma_n)\log\gamma_n}{2\pi}$.

Right now it's known that $\mu<.5154$ and $\lambda>2.69$, however it's conjectured that $\mu = 0$ and $\lambda = \infty$.

If this conjecture was true, then $\mu = 0$ could lead you to believe the integral along the critical line converges, but $\lambda = \infty$ 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.

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