1. ## Riemann Hypothesis

Today I recieved an e-mail from my mathematics advisor (not sure how to call him). He is an expert in complex analysis.

I heard today that there will soon be an announcement of a proof of the Riemann Hypothesis!
You might be excited. But I am not. . It is a wonderful problem, I wish to solve it, not anybody else.

2. haven't there been many proofs proposed?

3. Originally Posted by tukeywilliams
haven't there been many proofs proposed?
Yes. And all of them false. Hopefully this will be false too.

Interesting fact: Many people think that the Riemann Hypothesis is a number theory problem, but in fact it is really a problem from complex analysis!

4. Originally Posted by ThePerfectHacker
Interesting fact: Many people think that the Riemann Hypothesis is a number theory problem, but in fact it is really a problem from complex analysis!
one of the candidates for the dean of our math department described it as a "number theory" problem ... maybe he mumbled complex analysis, i don't remember ...

5. Originally Posted by Jhevon
one of the candidates for the dean of our math department described it as a "number theory" problem ... maybe he mumbled complex analysis, i don't remember ...
It has connections to number theory but it is certainly not a number theory problem. "If the zeta function is analytically extended to $\mathbb{C}\setminus \{0\}$ then the (non-trivial) zeros have real part equal to 1/2". That is exactly what the hypothesis says. Look at the words: zeta function, analytic continuation, $\mathbb{C}$, real part. All of those are words from complex analysis, not from number theory.

6. Originally Posted by ThePerfectHacker
It has connections to number theory but it is certainly not a number theory problem. "If the zeta function is analytically extended to $\mathbb{C}\setminus \{0\}$ then the (non-trivial) zeros have real part equal to 1/2". That is exactly what the hypothesis says. Look at the words: zeta function, analytic continuation, $\mathbb{C}$, real part. All of those are words from complex analysis, not from number theory.
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...

7. 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)
You mean Taniyama-Shimura conjecture (now called Taniyama-Shimura-Wiles theorem ). There is a lot of work on a concept introduced by Lejuenne Dirichlet called L-series. In fact, one of the millenium problems, the Birch-Swinnerton-Dyer conjecture is based a little bit on the Dirichlet L-series. The problem is some of the results are conjectural because they need the Riemann hypothesis, but I am no expert on this so I am not entirely sure.

i have to read up on how Riemann came up with the hypothesis...
How? All the history I know is that Riemann wrote a short paper "On the number of primes less than a given magnitude", one of his few he ever wrote, but he had to introduce his new zeta function. And then he investigated its zeros. If I remember, Riemann was able to solve for some of the zeros himself, those were found scrippled around in his notes somewhere (I cannot confirm the origin of this last sentence, it needs reference).

8. Ramanujan tried to prove the Riemann Hypothesis right?

9. Originally Posted by tukeywilliams
Ramanujan tried to prove the Riemann Hypothesis right?
I do not know. I would guess that no. Because Ramanujan was not very knowledgable. And when I say knowledgable I do not mean he was stupid, I mean he did not know very advanced concepts in math. This is coming from Hardy, who was the advisor of Ramanajun, he explains that Ramanajun was unfamilar with many advanced concepts in complex analysis. Since the Riemann hypothesis is very complicated problem I assume Ramanajun could not understand it to the level that he could solve it. However, Hardy did a breakthrough work by proving there are infinitely many zeros on the critical line.

10. Just recently... de Branges from Purdue claimed he had a proof. But, of course, he was wrong and later apologized. Another statement of the hypothesis though, is this:

$\sigma(n) \leq H_n + ln(H_n)e^{H_n}, \ for \ all \ n \ \in \ \mathbb{N}$

$\sigma(n)$ is the sum of the positive divisors of n.
$H_n$ is the nth harmonic number.

It's an elementary equivalent to the Riemann Hypothesis.

I don't know nearly enough math to begin solving it, but I am quite fascinated by the problem.

11. Here's an interesting article.

K2Crypt

12. Originally Posted by ThePerfectHacker
You might be excited. But I am not. . It is a wonderful problem, I wish to solve it, not anybody else.
If I don't prove that it's impossible to prove the Riemann Hypothesis (like I can), why not ;p

13. The nice thing about this problem is that it is not as difficult to understand as the other millenium problems.

14. If they proved the riemann hypothesis, wouldn't they also be able to prove loads of number theory conjectures (or disprove them)?

15. Originally Posted by Skinner
If they proved the riemann hypothesis, wouldn't they also be able to prove loads of number theory conjectures (or disprove them)?
You should read the previous posts (#7).

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