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January 18th, 2009, 04:01 PM   #1
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Riemann Hypothesis

Hi,

I've just joined in the hope that someone here will have enough time and patience to help a non-mathematician gain a better understand Riemann's hypothesis. I don't want to understand the mathematics as such, the Zeta function will always be incomprehensible hieroglyphics to me, but I do want to understand the principles at work. I want to understand what R's non-trivial zeroes actually are, their relationship to the primes, why they are sometimes likened to tuning forks, and a bit about the consequences of R's hypothesis being demonstrably true or false. I'm not being lazy. I've read the obvious books and trawled the Internet. But too much of it is over my head and what little remains does not address my questions. I have a fairly good understanding of the behaviour of the primes but little or no mathematics.

I could tentatively start be asking how the imaginary numbers fed into the Zeta function are chosen and why, and what distinguishes the numbers which transform into non-trivial zeroes from the rest. No doubt this question is perfectly idiotic, but I did say some patience would be required.

Thanks for any help.
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January 26th, 2009, 04:14 AM   #2
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Re: Riemann Hypothesis

Hmm. Obviously this was not a good question. I'll try another which is closely related.

A while ago I managed to prove that there is an infinite quantity of twin primes. Dr. Booker at Bristol Uni. agreed to have a look and congratulated me on constructing a correct heuristic proof, but then went on to say that the proof is not rigorous. I am still trying to find out why it is not rigorous. Indeed, I don't even understand how a proof can be correct and unrigorous. I would have expected it to be both or neither. My question is, then, why is my proof not rigorous.

The proof depends on showing that the behaviour of the products of the primes is such that for any two consecutive primes there is a twin prime between p squared and p1 squared. It is easy to show this for small primes, and easy to show that the quantity of TPs in this range increases with p. Yet I am told that such a proof cannot work because the interaction of prime products eventually becomes too complex to predict. I think this is nonsense, and that their behaviour is entirely predictable forever, in principle at least. What am I not understanding? Can someone explain to me how a proof can be correct but not rigorous.

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January 26th, 2009, 07:06 AM   #3
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Re: Riemann Hypothesis

Quote:
Originally Posted by Whoever
A while ago I managed to prove that there is an infinite quantity of twin primes. Dr. Booker at Bristol Uni. agreed to have a look and congratulated me on constructing a correct heuristic proof, but then went on to say that the proof is not rigorous. I am still trying to find out why it is not rigorous. Indeed, I don't even understand how a proof can be correct and unrigorous. I would have expected it to be both or neither. My question is, then, why is my proof not rigorous.
Of course without seeing the purported proof I can't say. But a heuristic is not the same as a proof. It's easy to demonstrate that the expected number of twin primes below n is n/(log n)^2.

Quote:
Originally Posted by Whoever
The proof depends on showing that the behaviour of the products of the primes is such that for any two consecutive primes there is a twin prime between p squared and p1 squared. It is easy to show this for small primes, and easy to show that the quantity of TPs in this range increases with p. Yet I am told that such a proof cannot work because the interaction of prime products eventually becomes too complex to predict. I think this is nonsense, and that their behaviour is entirely predictable forever, in principle at least. What am I not understanding? Can someone explain to me how a proof can be correct but not rigorous.
First, I believe the claim. You'd expect something like >> 16sqrt(n)/(log n)^2 twin primes in a gap that size, and small examples work.

But without seeing your proof, it's hard to say. You don't actually say anything about your method, only that it's "easy".
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January 28th, 2009, 07:14 AM   #4
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Re: Riemann Hypothesis

Quote:
Of course without seeing the purported proof I can't say. But a heuristic is not the same as a proof. It's easy to demonstrate that the expected number of twin primes below n is n/(log n)^2.
Yes, but is this not just a statictical rule, for which there may be many exceptions?

If I had an algorithm which generated only even numbers, could I not prove, by an analysis of the mechanics of this algorithm, that it can never produce an even number? Would this not count as a proof?

Quote:
You'd expect something like >> 16sqrt(n)/(log n)^2 twin primes in a gap that size, and small examples work.
Ok. We might expect this, but might we not be surprised? For my proof there would be no surprises. For any two consecutive primes there will always be a twin prime between p^2 and p1^2. (There may be a rare exception for very small primes, but none once the sequence properly gets going).

Quote:
But without seeing your proof, it's hard to say. You don't actually say anything about your method, only that it's "easy".
Basically, I show that for any two consecutive primes there can never be sufficient products of primes falling in the target range to prevent the appearance of at least one twin prime. It's really just simple wave mechanics. (I'm a musician and feel comfortable with waves). I feel like I'm saying that two waves of different frequencies must beat together in a certain way, as determined by the laws of mathematics and physics, only to be told that I might be wrong, and that if the waves are kept going for long enough they might eventually behave differently. This doesn't make much sense to me.

Thanks for your help. Much appreciated.
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January 28th, 2009, 08:54 AM   #5
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Re: Riemann Hypothesis

Quote:
Originally Posted by Whoever
Quote:
Of course without seeing the purported proof I can't say. But a heuristic is not the same as a proof. It's easy to demonstrate that the expected number of twin primes below n is n/(log n)^2.
Yes, but is this not just a statictical rule, for which there may be many exceptions?
Indeed; that was his point: the expected number is n/(logn)^2, but that's just a heuristic which suggests that the number of twin primes is infinite.

Quote:
If I had an algorithm which generated only even numbers, could I not prove, by an analysis of the mechanics of this algorithm, that it can never produce an even number? Would this not count as a proof?
I really don't understand what you're trying to say, here.

Quote:
[snip]
[quote:r8en1efh]But without seeing your proof, it's hard to say. You don't actually say anything about your method, only that it's "easy".
Basically, I show that for any two consecutive primes there can never be sufficient products of primes falling in the target range to prevent the appearance of at least one twin prime. It's really just simple wave mechanics. (I'm a musician and feel comfortable with waves). I feel like I'm saying that two waves of different frequencies must beat together in a certain way, as determined by the laws of mathematics and physics, only to be told that I might be wrong, and that if the waves are kept going for long enough they might eventually behave differently. This doesn't make much sense to me.

Thanks for your help. Much appreciated.
Whoever[/quote:r8en1efh]
The problem with the wave analogy, is that waves follow a relatively fixed path. As far as we know, there's no function to generate all the primes, no less all the twin primes-- they aren't very predictable.

Getting back to your question, did he say it's a "good heuristic" or a "correct but non-rigorous proof"? A good heuristic just means it can make us reasonably certain the conjecture is true, but is not a proof. A "correct" proof that isn't rigorous is a proof that isn't quite right, but could easily be made rigorous, and would almost certainly be correct.
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January 28th, 2009, 09:04 AM   #6
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Re: Riemann Hypothesis

You asked "how a proof can be correct and unrigorous". I said that heuristics aren't the same as proofs, because the professor you quoted said you had a "correct heuristic proof". He didn't say that you had a proof (a heuristic proof is not a proof, just like a skew field is not a field), and you haven't shown one.

Quote:
Originally Posted by Whoever
Quote:
Of course without seeing the purported proof I can't say. But a heuristic is not the same as a proof. It's easy to demonstrate that the expected number of twin primes below n is n/(log n)^2.
Yes, but is this not just a statictical rule, for which there may be many exceptions?
Not really. If I said that there were between n/(log n)^3 and n/log n twin primes up to n (for n > 9, say), there are no exceptions -- but this hasn't been proven (your claim notwithstanding).

Quote:
Originally Posted by Whoever
For my proof there would be no surprises. For any two consecutive primes there will always be a twin prime between p^2 and p1^2. (There may be a rare exception for very small primes, but none once the sequence properly gets going).
I understand your claim, but you'll forgive (I hope) some skepticism. You haven't shown even a sketch of the proof, which has eluded Hardy, Ramanujan, Littlewood, de Polignac, Brun, Erdos, ...

Quote:
Originally Posted by Whoever
Basically, I show that for any two consecutive primes there can never be sufficient products of primes falling in the target range to prevent the appearance of at least one twin prime. It's really just simple wave mechanics. (I'm a musician and feel comfortable with waves). I feel like I'm saying that two waves of different frequencies must beat together in a certain way, as determined by the laws of mathematics and physics, only to be told that I might be wrong, and that if the waves are kept going for long enough they might eventually behave differently. This doesn't make much sense to me.
I get a little bit more worried every time you repeat the claim that it's simple. The conjecture has been open for what, 100 years? 150? If it were simple it would have been solved already.

Also, your description of the proof sounds like it proves something stronger than the existence of infinitely many twin primes, and I'm concerned that what it 'proves' is in fact false. Wouldn't that show that Omega(n) is bounded on average as n -> infty?
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January 28th, 2009, 09:24 AM   #7
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Re: Riemann Hypothesis

Quote:
Originally Posted by cknapp
The problem with the wave analogy, is that waves follow a relatively fixed path. As far as we know, there's no function to generate all the primes, no less all the twin primes-- they aren't very predictable.
Oh hush, you -- there's no particularly *simple* formula generating all primes, but there are plenty that generate all primes.

Quote:
Originally Posted by cknapp
Getting back to your question, did he say it's a "good heuristic" or a "correct but non-rigorous proof"? A good heuristic just means it can make us reasonably certain the conjecture is true, but is not a proof. A "correct" proof that isn't rigorous is a proof that isn't quite right, but could easily be made rigorous, and would almost certainly be correct.
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what he said
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January 29th, 2009, 04:56 AM   #8
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Re: Riemann Hypothesis

Quote:
Originally Posted by cknapp
Quote:
Originally Posted by Whoever
Quote:
Of course without seeing the purported proof I can't say. But a heuristic is not the same as a proof. It's easy to demonstrate that the expected number of twin primes below n is n/(log n)^2.
Yes, but is this not just a statictical rule, for which there may be many exceptions?
Indeed; that was his point: the expected number is n/(logn)^2, but that's just a heuristic which suggests that the number of twin primes is infinite.
Ok. But what I was getting at is this. A heuristic may show a trend or it may show a rule. Yes? To show a trend is not to prove much, but if a rule can be shown would this not count as a proof?

Quote:
[quote:3h88smcj]If I had an algorithm which generated only even numbers, could I not prove, by an analysis of the mechanics of this algorithm, that it can never produce an even number? Would this not count as a proof?
I really don't understand what you're trying to say, here.[/quote:3h88smcj]
I suppose I'm trying to understand the distinction between a mathematical proof and a mechanical proof, if I can put it that way.

Quote:
The problem with the wave analogy, is that waves follow a relatively fixed path. As far as we know, there's no function to generate all the primes, no less all the twin primes-- they aren't very predictable.
There is clearly a mechanism that generates the primes, and armed with an understanding of this the primes can be predicted. The only sense in which the primes are not predictable, it seems to me, is that it is difficult to find a manageable way of doing it. In principle they're as predictable as the integers. They are simply the gaps in the pattern of the products of the primes.

Quote:
Getting back to your question, did he say it's a "good heuristic" or a "correct but non-rigorous proof"? A good heuristic just means it can make us reasonably certain the conjecture is true, but is not a proof. A "correct" proof that isn't rigorous is a proof that isn't quite right, but could easily be made rigorous, and would almost certainly be correct.
Yes, that's exactly what I thought. But apparently I'm wrong. This is my problem. He said it is correct but non-rigorous, and this confuses me.

This is all very interesting. Thanks
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January 29th, 2009, 05:35 AM   #9
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Re: Riemann Hypothesis

Quote:
Originally Posted by CRGreathouse
You asked "how a proof can be correct and unrigorous". I said that heuristics aren't the same as proofs, because the professor you quoted said you had a "correct heuristic proof". He didn't say that you had a proof (a heuristic proof is not a proof, just like a skew field is not a field), and you haven't shown one.
This is a bit confusing. Perhaps I have not fully understood what a heuristic proof is. Could you say a bit about this?

Quote:
I understand your claim, but you'll forgive (I hope) some skepticism. You haven't shown even a sketch of the proof, which has eluded Hardy, Ramanujan, Littlewood, de Polignac, Brun, Erdos, ...
This is the problem. It makes no sense that I've succeeded where they failed. It would be unbelievable. Yet it seems to me I have.

I'm slightly reluctant to say much more about my proof since either it is flawed and I'm a fool, which is the most likely outcome, or it isn't and I don't want to give it away just yet. Before I post it I'd like to be sure I understand it properly myself. I suppose that's annoying.

Quote:
Quote:
Originally Posted by Whoever
Basically, I show that for any two consecutive primes there can never be sufficient products of primes falling in the target range to prevent the appearance of at least one twin prime. It's really just simple wave mechanics. (I'm a musician and feel comfortable with waves). I feel like I'm saying that two waves of different frequencies must beat together in a certain way, as determined by the laws of mathematics and physics, only to be told that I might be wrong, and that if the waves are kept going for long enough they might eventually behave differently. This doesn't make much sense to me.
I get a little bit more worried every time you repeat the claim that it's simple. The conjecture has been open for what, 100 years? 150? If it were simple it would have been solved already.
I'm not so sure. I think it's possible to be too clever sometimes. This is certainly true in philosophy. An innocent layman is likely to come at problems in an unusual way.

Quote:
Also, your description of the proof sounds like it proves something stronger than the existence of infinitely many twin primes, and I'm concerned that what it 'proves' is in fact false. Wouldn't that show that Omega(n) is bounded on average as n -> infty?
Um. Sorry. What is Omega?

I have wondered if I''ve proved something more, and have a suspicion it has a bearing on Riemann's hypothesis, if only I could understand it. But I daren't go there in a discussion with mathematicians.

I'll post a better summary of the proof when I've time.

Thanks
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January 29th, 2009, 05:40 AM   #10
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Re: Riemann Hypothesis

Quote:
[quote:34fggq7t]If I had an algorithm which generated only even numbers, could I not prove, by an analysis of the mechanics of this algorithm, that it can never produce an even number? Would this not count as a proof?
I really don't understand what you're trying to say, here.[/quote:34fggq7t]
Oh Hell. I meant to say that it could not produce an odd number. No wonder this question was incomprehensible.
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