# Thread: Riemann Hypothesis

1. I was wondering, if the Riemann Hypothesis is solved, would it have any major implications in mathematics, or would it be more or less merely an impressive accomplishment with little relative importance to extant mathematics (e.g., Fermat's Last Theorem).

2. Originally Posted by Ellatha
I was wondering, if the Riemann Hypothesis is solved, would it have any major implications in mathematics, or would it be more or less merely an impressive accomplishment with little relative importance to extant mathematics (e.g., Fermat's Last Theorem).
What is important is not the result itself, though people concerned with coding theory are very interested in that.

What is important are the new ideas that will likely be needed to construct a proof. That is the case with Fermat's Last Theorem which, contrary to your statement, generated a plethora of new ideas not the east of which included the proof of the Taniyama–Shimura conjecture.

Just like the proof of Fermat's Last Theorem, it would be hugely important.

3. If I'm not mistaken, many interesting and relatively important conjectures are made built off assuming a certain truth value of the RH. In my readings I'm sure I've ran into more than a couple topics in number theory mentioning the Riemann Hypothesis. So I'm guessing it's quite important...

4. Originally Posted by epidecus
If I'm not mistaken, many interesting and relatively important conjectures are made built off assuming a certain truth value of the RH. In my readings I'm sure I've ran into more than a couple topics in number theory mentioning the Riemann Hypothesis. So I'm guessing it's quite important...
There are certainly conjectures that are dependent on the Riemann Hypothesis, but they are of limited importance unless and until it is established or refuted.

It is generally presented as a conjecture in number theory, but it involves quite a bit of complex analysis and the general feeling is that any eventual proof will be very much analytic. Titschmarch wrote a classic text on the Riemann Zeta function, which is still one of the best treatments of that subject and Titschmarsh was a classical analyst. Simply to define the zeta function globally requires the notion of analytic continuation, a standard topic in complex analysis.

It is generally regarded as the most difficult and deep outstanding problem in mathematics and is the only one of the current Millenium Problems that was one of the original Hilbert problems. Fermat's Last Theorem and the Poincare conjecture, solved in the last several years are monumental results and represent extremely difficult proofs, but they pale in comparison to the difficulty that seems to come with the Riemann Hypothesis.

Some partial results have been proved, but by and large it has resisted all attacks.

Anyone who solves this problem is pretty much guaranteed a storied position in the history of mathematics.

5. Why is it difficult to solve, what is the problem with proving it? It seems to be that all the necessary information is already present to correctly test it.

6. Originally Posted by Ascended
Why is it difficult to solve, what is the problem with proving it? It seems to be that all the necessary information is already present to correctly test it.
Then do it.

7. Originally Posted by DrRocket
Then do it.
But why? Why would I want to sit checking whether non trivial zeros from 10^22 to say 10^1000 actually sit on the critical line just to prove something than most people are just accepting is probarbly true anyway. Just a waste of computing power if you ask me.

8. Work on the basis it is true and well if you do come up with a problem then hey you've solved it.

9. Originally Posted by Ascended
But why? Why would I want to sit checking whether non trivial zeros from 10^22 to say 10^1000 actually sit on the critical line just to prove something than most people are just accepting is probarbly true anyway. Just a waste of computing power if you ask me.
I have no idea why you would waste your time doing such a thing. It would not prove the Riemann Hypothesis, moreover the non-trivial zeros are not real numbers anyway so I can tell you with absolute certainty that there are no zeros between 10^22 and 10^1000 without having to check anything. The trivial zeros are the even negative integers and there are no other real zeros.

It has been shown that most (in a probabilistic sense) of the non-trivial zeros have real part 1/2.

The requirement is to show that ALL of the non-trivial real number have real part 1/2 and no amount of computer checking could do that.

It is becoming clear that don't even know what the Riemann Hypopthesis is. Do some reading.

http://www.claymath.org/millennium/R...is/riemann.pdf

10. Originally Posted by DrRocket
There are certainly conjectures that are dependent on the Riemann Hypothesis, but they are of limited importance unless and until it is established or refuted.

It is generally presented as a conjecture in number theory, but it involves quite a bit of complex analysis and the general feeling is that any eventual proof will be very much analytic. Titschmarch wrote a classic text on the Riemann Zeta function, which is still one of the best treatments of that subject and Titschmarsh was a classical analyst. Simply to define the zeta function globally requires the notion of analytic continuation, a standard topic in complex analysis.

It is generally regarded as the most difficult and deep outstanding problem in mathematics and is the only one of the current Millenium Problems that was one of the original Hilbert problems. Fermat's Last Theorem and the Poincare conjecture, solved in the last several years are monumental results and represent extremely difficult proofs, but they pale in comparison to the difficulty that seems to come with the Riemann Hypothesis.

Some partial results have been proved, but by and large it has resisted all attacks.

Anyone who solves this problem is pretty much guaranteed a storied position in the history of mathematics.
You're saying that the proof for the Riemann Hypothesis will far exceed the complexity of that of Fermat's Last Theorem? That's difficult to imagine.

11. Originally Posted by Ascended
But why? Why would I want to sit checking whether non trivial zeros from 10^22 to say 10^1000 actually sit on the critical line just to prove something than most people are just accepting is probably true anyway. Just a waste of computing power if you ask me.
Are you saying that a simple test-in-check can validate a conjecture? I don't think that's good enough in any serious mathematical context. If the first few (whatever power of 10) values are in line with the conjecture, then that only gives a sense that it might be true. Really, I don't see how this merits anything since we should only hold to the standard of rigorous proof, which is entirely different than checking the values of some sample.

I guess it may be acceptable considering a finite set of values. For sake of example, let's pretend there are only 10^21 primes. A simple computation run should verify any relatively trivial conjecture about the primes. The Goldbach conjecture would simply be a known property of those primes it concerns. But since there's really an infinitude of primes, we must depend on generalized proofs if we want to be logical and rigorous.

Back to the RH, it seems like there are infinitely many nontrivial zeroes in the zeta function, but then again I'm not certain of anything

12. Originally Posted by Ellatha
You're saying that the proof for the Riemann Hypothesis will far exceed the complexity of that of Fermat's Last Theorem? That's difficult to imagine.
Sincd no one has a proof or even a promising approach to a proof, no one can possibly make such a possitive assertion without appearing foolish. I certainly did not.

What I said is that the Riemann Hypothesis is considered by many mathematicians to be the most deep and difficcult open problem in mathematics. I said that other difficult problems were settled with proofs that are extremely complex and difficult to follow for all but a few specialists. One might therefore expect that a proof of the Riemann hypothesis would require new ideas and very difficult proof techniques, but to state that as an absolute fact would be to display monumental ingorance of mathematics--which you have just done.

Lomonosov's theorem took the experts by surprise. It produced the best result to date on the invariant subspace problem and is not particularly difficult. So sometimes hard problems can be solved in a clever manner that is not too difficult for a good mathematician to follow. But Lomonosov's theorem is unusual in that regard.

Please cease trying to put words into my mouth. You lack both the knowledge and intelligence to do any such thing.

13. Originally Posted by epidecus
Back to the RH, it seems like there are infinitely many nontrivial zeroes in the zeta function, but then again I'm not certain of anything
There are infinitely many nontrivial zeros on the line Re(z)=1/2 (the "critical line"). That is not particularly obvious. It is a theorem of Hardy and Littlewood.

14. Originally Posted by DrRocket
Sincd no one has a proof or even a promising approach to a proof, no one can possibly make such a possitive assertion without appearing foolish. I certainly did not.

What I said is that the Riemann Hypothesis is considered by many mathematicians to be the most deep and difficcult open problem in mathematics. I said that other difficult problems were settled with proofs that are extremely complex and difficult to follow for all but a few specialists. One might therefore expect that a proof of the Riemann hypothesis would require new ideas and very difficult proof techniques, but to state that as an absolute fact would be to display monumental ingorance of mathematics--which you have just done.

Lomonosov's theorem took the experts by surprise. It produced the best result to date on the invariant subspace problem and is not particularly difficult. So sometimes hard problems can be solved in a clever manner that is not too difficult for a good mathematician to follow. But Lomonosov's theorem is unusual in that regard.

Please cease trying to put words into my mouth. You lack both the knowledge and intelligence to do any such thing.
And why exactly are you so angry?

15. Originally Posted by Ellatha
And why exactly are you so angry?
I am not angry. I am a bit bemused by your pseudo-intellectual bullshit and attempt to sound authoritative, or even informed, when it is clear that you are clueless.

On the other hand, it is necessary to set things straight for those who might be reading some of this and actually trying to learn something.

There is no meaningful way to rank proofs according to difficulty or complexity, though a competent mathematician can make gross judgments. The Wiles's proof of the Taniyama-Shimura conjecture (which Ribet showed implied Fermat's Last Theorem) and Pereleman's proof of the Poincare Conjecture are pretty universally recognized as being extremely difficult. But which is the more difficult depends on the background of the person attempting to read them. So statements to the effect that a proof would "far exceed the complexity of that of Fermat's Last Theorem" due to Wiles or the proof of the Poincare Conjeccture due to Perleman are rather inane as there is no scale by which to measure complexity or difficulty. I don't even know what it would mean to "far exceed" the complexity of those proofs.

What I do know is that any valid proof will be a very great achievement indeed.

Given the long-standing prominence of the Riemann Hypothesis and the fact that a great many extremely talented mathematicians have attempted to prove, thus far without success, one would expect that any eventual proof will involve important new ideas, and be a difficult proof by most standards. If it was easy someone would have proved it by now. But no one knows for sure, and it is even possible that it might be shown to be undecidable, though I think that would surprise everyone.

16. You're "bemused by my attempt at trying to sound authoritative?" All I did was ask a simple question because I thought you said something you apparently didn't. I really don't know what your problem is but it is clear it is impossible for me to deal with you on any level despite trying to avoid conflict with you the entire time I've posted on this forum. Anyway good luck and goodbye.

17. Originally Posted by DrRocket
I have no idea why you would waste your time doing such a thing. It would not prove the Riemann Hypothesis, moreover the non-trivial zeros are not real numbers anyway so I can tell you with absolute certainty that there are no zeros between 10^22 and 10^1000 without having to check anything. The trivial zeros are the even negative integers and there are no other real zeros.

It has been shown that most (in a probabilistic sense) of the non-trivial zeros have real part 1/2.

The requirement is to show that ALL of the non-trivial real number have real part 1/2 and no amount of computer checking could do that.

It is becoming clear that don't even know what the Riemann Hypopthesis is. Do some reading.

http://www.claymath.org/millennium/R...is/riemann.pdf

Ok well thank you for the sensible answer, I wasn't actually being serious with my posts. Perhaps I need to work on my humour there lol.
Anyway yes I think it is quite clear here that I don't fully understand it by a long shot, even though I have read many different explanations, and in all honesty I would imagine there are far more intelligent people than I that don't fully understand it either.

I have read several descriptions of the Riemann Hypothesis that suggest it is about how all non trival zeros should lie on the critical line and that the real part should also always be 1/2. I've also come across the work of other mathematicians that suggest this has already been checked upto 10^22, further to this I came across another article suggesting this had now been checked upto 10^24. But also suggestions that because we are dealing with integers of integers this could require checking to possibly in excess of the 10^1000 mark.
Now I'm not claiming that this is all true because as previously stated I don't fully understand, just simply that this is what I have already read. If this isn't correct please feel free to show me what is in error. But also, and this is what I'm really trying to grasp here, is the issue with this particular problem that is making it so hard to solve. Perhaps this is hard to explain, perhaps it's just a simple thing I am missing here I really don't know. What I am asking is, for clarity and the purpose of helping every other person out there like me that isn't quite getting it can we have a clear logical explanation of what makes this problem so hard that it hasn't been solved in over a hundred years, cheers.

18. Originally Posted by Ascended

I have read several descriptions of the Riemann Hypothesis that suggest it is about how all non trival zeros should lie on the critical line and that the real part should also always be 1/2. I've also come across the work of other mathematicians that suggest this has already been checked upto 10^22, further to this I came across another article suggesting this had now been checked upto 10^24. But also suggestions that because we are dealing with integers of integers this could require checking to possibly in excess of the 10^1000 mark.
This makes no sense whatever.

The "critical line" IS the line in the complex plane where Re(z)=1/2. There is no "also" to it.

The non-trivial zeros of the Riemann zeta function are not integers, nor even real numbers. They are complex numbers with non-zero imaginary part. There is no such thing as an "integer of an integer".

Maybe before you tackle the Riemann hypothesis you ought to go learn what a complex number is. Try any book on "college algebra" (aka high school algebra).

19. Originally Posted by DrRocket

The "critical line" IS the line in the complex plane where Re(z)=1/2. There is no "also" to it.
Ok here it looks like you are actually accepting that according to RH non trivial zeros do sit on the critical line, cheers we can now put this to rest.

Originally Posted by DrRocket
The non-trivial zeros of the Riemann zeta function are not integers, nor even real numbers. They are complex numbers with non-zero imaginary part. There is no such thing as an "integer of an integer".
Here again this is helpful as I was a little baffled when I read that myself.

Originally Posted by DrRocket
Maybe before you tackle the Riemann hypothesis you ought to go learn what a complex number is. Try any book on "college algebra" (aka high school algebra).
Thank you for the recommendation here.

As for explanation of why the RH is quite so difficult I think the fact it wasn't forth coming clues us in here that you might be having a little difficulty with this, so please don't worry and thank you for the helpful answers you did manage to provide.

20. Originally Posted by DrRocket
There are infinitely many nontrivial zeros on the line Re(z)=1/2 (the "critical line"). That is not particularly obvious. It is a theorem of Hardy and Littlewood.
Much thanks. For no particular reason, I always find math involving complex plots or hypercomplex algebras interesting (beyond me yes, but cool nonetheless). I think watching videos of fractal structures generated in hypercomplex spaces (if that made any sense) was what really pulled me towards topics in higher math.

21. Revisiting an old thread.

I was under the impression that a mathematical statement whose truth value is unknown is said to be a conjecture, e.g. the Goldbach conjecture and many others. Once a valid and rigorous proof is established, it is then considered a theorem.

Why, then, is Riemann's famous problem known as a hypothesis? Is there a technicality behind it or is there nothing significant to it? In my mind, hypothesis is typically associated with the ordinary sciences where one inquires on observation and then follows a conclusion through experimentation. This isn't the case with the formal proofing nature of mathematics.

22. There are no hard a fast rules here, one authors lemma would be another's theorem would be another’s proposition while one author's conjecture would be another’s hypothesis. It really depends on the writing style.

23. Originally Posted by river_rat
There are no hard a fast rules here, one authors lemma would be another's theorem would be another’s proposition while one author's conjecture would be another’s hypothesis. It really depends on the writing style.
That's interesting. Thanks river_rat. (been off a little, sorry for the delay)

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