Why is RH so difficult to prove? - Page 2

R.D. Silverman · 2008-02-22, 17:16

Quote:

Originally Posted by Damian

Thanks all for your replys.
This may be a bit of topic but I was thinking which is the simplest known proof that $\pi$ is irrational.
Based on the assumption that $\zeta(2) = \fra{\pi^2}{6}$ is irrational, wouldn't it imply that $\pi$ is also irrational?
.

Where does one get such an assumption? It requires proof.
However, a proof that pi^2/6 is irrational does lead to an immediate
proof that pi is irrational.

And your use of the word 'simplest' is vague. AFAIK, a consensus
(among mathematicians) 'simplest' proof is the one by Niven. See

http://www.lrz-muenchen.de/~hr/numb/pi-irr.html

Damian · 2008-02-22, 18:02

[QUOTE=R.D. Silverman;126485]

Quote:

Originally Posted by Damian

Thanks for your reply.
I know what you mean, and I'm not asking for a proof of RH.
But I think each person can have it's own opinion as to why RH hasn't been proved.
QUOTE]

No. Every person is NOT entitled to such an opinion.
In particular, you are not so entitled.

Every *informed* person is entitled to an opinion. Your knowledge
of mathematics has not even reached the point where you have
mastered high school level math.

Thank you for your kind reply. I know I'm not entitled to an opinion, but I think you are. If you think you are entitled to an opinion I would be interested on hearing about it, even if I won't understand it because it is beyound my high school level math. It's for a local newspaper I want to write your opinion on the RH, and everyone's elses.

Damian · 2008-02-22, 18:07

Quote:

Originally Posted by R.D. Silverman

Where does one get such an assumption? It requires proof.
However, a proof that pi^2/6 is irrational does lead to an immediate
proof that pi is irrational.

Thanks for your reply. As far as I'm conserned that $\zeta(2)$ is irrational is already proven, so it wouldn't be necesary to assume it.
What I don't know is if that proof uses the fact that $\pi$ is irrational, or doesn't.

Orgasmic Troll · 2008-02-23, 05:19

Quote:

Originally Posted by Damian

Thanks for your reply. As far as I'm conserned that $\zeta(2)$ is irrational is already proven, so it wouldn't be necesary to assume it.
What I don't know is if that proof uses the fact that $\pi$ is irrational, or doesn't.

the fact that pi is irrational is already proven as well, therefore by your reasoning, I submit an even simpler proof of the irrationality of pi:

pi is irrational
QED

Damian · 2008-02-23, 06:49

Quote:

Originally Posted by Orgasmic Troll

the fact that pi is irrational is already proven as well, therefore by your reasoning, I submit an even simpler proof of the irrationality of pi:

pi is irrational
QED

Thanks for your reply.
Sure, that is the simplest demonstration that pi is irrational. But I was searching for simple proofs that doesn't use the fact that pi is irrational to deduce it.

retina · 2008-02-23, 07:49

Quote:

Originally Posted by Damian

But I was searching for simple proofs that doesn't use the fact that pi is irrational to deduce it.

Damian · 2008-02-23, 13:53

Quote:

Originally Posted by retina

google:

Thanks for your replay.
You are incredibly right. I copy/pasted into google "But I was searching for simple proofs that doesn't use the fact that pi is irrational to deduce it.", and pressed the "I'll be lucky" button. On the page it founds
(http://mathforum.org/kb/message.jspa?messageID=3853756), there is an incredibly simple proof that $\pi$ is irrational.
It simply says:

Quote:

In 1761 Johann Heinrich Lambert continued the continued
fraction investigations of Euler and proved that e^x and
tan(x) (radian measure) were each irrational for every
nonzero rational number x. Thus, Lambert proved
Pi is irrational. [Consider tan(Pi/4).]

I'll write it down here again using tex.

1) If $x$ is rational, $tan(x)$ must be irrational
2) It follows that if $tan(x)$ is rational, $x$ must be irrational.
3) Since $tan(\pi/4)=1$ , $\pi/4$ must be irrational; therefore, $\pi$ must be irrational.

Thanks.

jasong · 2008-02-24, 03:38

Does retina deserve credit for those search terms, or just encouraging you to use Google? The irony involved is amazing.

/me resolves to look up irony to make sure I'm not committing the same error as a certain singer.

/me looked it up, and is now more confused than ever. That word has always confused the heck out of me. Must be something about the way my brain works that makes that word impossible to fully comprehend.

m_f_h · 2008-02-25, 12:42

Quote:

Originally Posted by Damian

Thanks for your replay.
1) If $x$ is rational, $tan(x)$ must be irrational
2) It follows that if $tan(x)$ is rational, $x$ must be irrational.
3) Since $tan(\pi/4)=1$ , $\pi/4$ must be irrational; therefore, $\pi$ must be irrational.
Thanks.

2 useless comments regarding (1):
(a) This is not true for x=0.
(b) IMHO, this statement is roughly equivalent to the statement that \pi is irrational, so the proof of the latter is just hidden in the proof of the former.

Quote:

Originally Posted by R.D. Silverman

No. Every person is NOT entitled to such an opinion. In particular, you are not so entitled.

:-D ! (ROTFL etc.)
I love it!

Orgasmic Troll · 2008-02-25, 18:44

Quote:

Originally Posted by Damian

Thanks for your reply.
Sure, that is the simplest demonstration that pi is irrational. But I was searching for simple proofs that doesn't use the fact that pi is irrational to deduce it.

<sigh>

You clearly don't know what a proof is. For one thing, the "proof" I gave you was to illustrate how silly the notion of using that $\zeta(2) = \frac{\pi^2}{6}$ is irrational to prove that pi is irrational. Why do you get to use the fact that $\zeta(2)$ is irrational? Just because some guy said it? What if I tell you the guy was wrong? Can you prove me wrong? If you can't, then you don't really have any business using that fact in your proof.

R.D. Silverman · 2008-02-25, 19:18

Quote:

Originally Posted by Damian

But we know for example that Zeta(2) = Pi^2/6 (Euler proof of the Basel problem), and for negative integers (the bernoulli numbers) but what about numbers that are not on the real line?
Not knowing the exact values of Zeta mean we can only aproximate them. And I think it is difficult to get a proof of something that can only be approximated.

What does it mean to "know" an exact value??? Certainly you can
not mean an exact decimal value, for such is impossible.
The exact value of Zeta(3) is Zeta(3). What more is needed?
Or do you mean "know" in the sense of "represent in terms of a
finite number of known constants"? If so, then we can only ever "know"
the exact value of a very small number of numbers. Indeed. It is
easily proven that the set of such numbers is countable.

If I tell you that the answer to a problem is sqrt(2), is that an "exact"
answer? If so, why isn't giving an answer as zeta(3) an "exact" answer?
Both values are the results of evaluating well-known functions at
integers.

2008-02-24, 03:38	#19
jasong "Jason Goatcher" Mar 2005 3507₁₀ Posts	Does retina deserve credit for those search terms, or just encouraging you to use Google? The irony involved is amazing. /me resolves to look up irony to make sure I'm not committing the same error as a certain singer. /me looked it up, and is now more confused than ever. That word has always confused the heck out of me. Must be something about the way my brain works that makes that word impossible to fully comprehend. Last fiddled with by jasong on 2008-02-24 at 03:41

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