Offizieller Riemann-Hypothese Faden - Page 3

jasong · 2005-09-10, 23:25

Quote:

Originally Posted by cheesehead

Impact of obscure math on society. Hmmm ...

Well, there are the earlier examples of practical applications of special relativity (a consideration in designing CRT TV electron guns, and then there's the bomb thing), Boolean algebra (computers), and the difficulty of factoring products of large primes (crypto). But how that applies to your quest for sources, I don't know.

Judging from what everyone on here says about R.D. Silverman, he might be a good source.

Numbers · 2005-09-11, 15:36

Although I do not profess to be an expert in this � or anything else for that matter � I have just read a book about the Riemann Hypothesis, from which it seems that there might be one way that this might impact on society. The result of a chance meeting between Hugh Montgomery and John Dyson at Princeton in 1972 was a thing known as the Montgomery-Odlyzko Law which states that: The distribution of the spacings between successive non-trivial zeros of the Riemann zeta function (suitably normalised) is statistically identical with the distribution of eigenvalue spacings in a GUE operator.

IOW, if certain quantum-dynamical systems are modelled in a Gaussian-random Hermitian matrix, the eigenvalues of the matrix bear a statistical similarity to the non-trivial zeroes of the Riemann zeta function. If anyone can ever make anything of this it �might� (big might, and this is my guess, it's not in the book) lead to some new form of atomic or nuclear energy, hopefully one that does not produce huge amounts of toxic waste.

The only other way I can think of that this might impact on society is if someone ever proves it. Then, for one day, newspaper sales might go up 5%.

The book, by the way, was Prime Obsession by John Derbyshire.

Primeinator · 2005-09-16, 21:46

Okay thanks. R.D. Silverman, care to make a post?

philmoore · 2005-09-16, 22:56

I'll respond, since this was a topic that interested me when I was in graduate school. I haven't kept up with it, but you might look at a couple of articles by Michael Berry:

Berry, M V, 1986, ‘Riemann's zeta function: a model for quantum chaos?’ in Quantum chaos and statistical nuclear physics, eds. T H Seligman and
H Nishioka, Springer Lecture Notes in Physics No. 263, 1-17.

and

Berry, M V, 1987, Proc.Roy.Soc A413, 183-198, ‘Quantum chaology’, (The Bakerian Lecture).

The second article may be less technical. You can download them from Berry's website:
http://www.phy.bris.ac.uk/research/t...lications.html

He suggests that zeta functions in general may offer a paradigm for understanding quantum chaos. Nowadays, many people have become interested in the problem of nudging chaotic systems into a certain behavior, and this may eventually have practical applications.

As for the PvsNP problem, I don't know much, but Richard Feynman wrote a couple of seminal papers in the early 1980's about problems which were assumed to scale so badly with size as to be intractable on conventional computers, and he argued that quantum computers offered the only possibility of tackling such problems.

Primeinator · 2005-09-17, 00:08

Thanks Philmoore, bookmaring the site as I type :)

ColdFury · 2005-09-17, 01:30

Quote:

As for the PvsNP problem, I don't know much, but Richard Feynman wrote a couple of seminal papers in the early 1980's about problems which were assumed to scale so badly with size as to be intractable on conventional computers, and he argued that quantum computers offered the only possibility of tackling such problems.

It has been since proven that Quantum Computers are not an NP oracle. IE, NP problems scale just as poorly on Quantum Computers as classical ones.

philmoore · 2005-09-17, 04:00

Another interesting website on quantum chaos, including the connection to number theory and zeta functions, is Matthew Watkins site:

http://www.maths.ex.ac.uk/~mwatkins/zeta/physics.htm

The quantum mechanics link, in particular, seems to have some interesting links to articles about the Riemann zeta function.

Primeinator · 2005-09-17, 14:44

Thanks, I'll bookmark that one as well.

Citrix · 2005-10-15, 01:23

The infinite sum of 1/n^s, with s > 1, is equal to the infinite product of

1/[1 - p^(-s)] where p ranges over the prime numbers.

I have understood the problem so far, but what is s? I suppose it is the zeta function? How to calculate s?

May be we can have a wiki for RH. So people won't ask the same question over and over again. Any good explanations?

Citrix

jinydu · 2005-10-15, 05:15

In this context, s is the independent variable.

Numbers · 2005-10-15, 07:01

Jinydu is absolutely right, as usual.

Because the language of maths was not quite so settled in 1859 as it is today, where any other mathematician would write f(x) as his function, Riemann wrote $\large \zeta(s)$ and everyone else has copied him. In studies of the zeta function, the argument is always given as s.

2005-10-15, 01:23	#31
Citrix Jun 2003 2·7·113 Posts	Riemann Hypothesis The infinite sum of 1/n^s, with s > 1, is equal to the infinite product of 1/[1 - p^(-s)] where p ranges over the prime numbers. I have understood the problem so far, but what is s? I suppose it is the zeta function? How to calculate s? May be we can have a wiki for RH. So people won't ask the same question over and over again. Any good explanations? Citrix

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2005-09-11, 15:36	#24
Numbers Jun 2005 Near Beetlegeuse 2²·97 Posts	Although I do not profess to be an expert in this � or anything else for that matter � I have just read a book about the Riemann Hypothesis, from which it seems that there might be one way that this might impact on society. The result of a chance meeting between Hugh Montgomery and John Dyson at Princeton in 1972 was a thing known as the Montgomery-Odlyzko Law which states that: The distribution of the spacings between successive non-trivial zeros of the Riemann zeta function (suitably normalised) is statistically identical with the distribution of eigenvalue spacings in a GUE operator. IOW, if certain quantum-dynamical systems are modelled in a Gaussian-random Hermitian matrix, the eigenvalues of the matrix bear a statistical similarity to the non-trivial zeroes of the Riemann zeta function. If anyone can ever make anything of this it �might� (big might, and this is my guess, it's not in the book) lead to some new form of atomic or nuclear energy, hopefully one that does not produce huge amounts of toxic waste. The only other way I can think of that this might impact on society is if someone ever proves it. Then, for one day, newspaper sales might go up 5%. The book, by the way, was Prime Obsession by John Derbyshire.

2005-09-16, 21:46	#25
Primeinator "Kyle" Feb 2005 Somewhere near M52.. 3×5×61 Posts	Okay thanks. R.D. Silverman, care to make a post?

2005-09-16, 22:56	#26
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 1119₁₀ Posts	I'll respond, since this was a topic that interested me when I was in graduate school. I haven't kept up with it, but you might look at a couple of articles by Michael Berry: Berry, M V, 1986, ‘Riemann's zeta function: a model for quantum chaos?’ in Quantum chaos and statistical nuclear physics, eds. T H Seligman and H Nishioka, Springer Lecture Notes in Physics No. 263, 1-17. and Berry, M V, 1987, Proc.Roy.Soc A413, 183-198, ‘Quantum chaology’, (The Bakerian Lecture). The second article may be less technical. You can download them from Berry's website: http://www.phy.bris.ac.uk/research/t...lications.html He suggests that zeta functions in general may offer a paradigm for understanding quantum chaos. Nowadays, many people have become interested in the problem of nudging chaotic systems into a certain behavior, and this may eventually have practical applications. As for the PvsNP problem, I don't know much, but Richard Feynman wrote a couple of seminal papers in the early 1980's about problems which were assumed to scale so badly with size as to be intractable on conventional computers, and he argued that quantum computers offered the only possibility of tackling such problems.

2005-09-17, 00:08	#27
Primeinator "Kyle" Feb 2005 Somewhere near M52.. 3·5·61 Posts	Thanks Philmoore, bookmaring the site as I type :)

2005-09-17, 04:00	#29
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 1119₁₀ Posts	Another interesting website on quantum chaos, including the connection to number theory and zeta functions, is Matthew Watkins site: http://www.maths.ex.ac.uk/~mwatkins/zeta/physics.htm The quantum mechanics link, in particular, seems to have some interesting links to articles about the Riemann zeta function.

2005-09-17, 14:44	#30
Primeinator "Kyle" Feb 2005 Somewhere near M52.. 1623₈ Posts	Thanks, I'll bookmark that one as well.

2005-10-15, 05:15	#32
jinydu Dec 2003 Hopefully Near M48 2×3×293 Posts	In this context, s is the independent variable.

2005-10-15, 07:01	#33
Numbers Jun 2005 Near Beetlegeuse 2²×97 Posts	Jinydu is absolutely right, as usual. Because the language of maths was not quite so settled in 1859 as it is today, where any other mathematician would write f(x) as his function, Riemann wrote $\large \zeta(s)$ and everyone else has copied him. In studies of the zeta function, the argument is always given as s.