Offizieller Riemann-Hypothese Faden

ThomRuley · 2003-07-02, 18:17

I was looking for information about Mersenne primes, and I came across something about the Reimann hypothesis. Could someone explain what that is? Also, I read that if this were proven, it would speed up our searching a lot. How? I assume that other people on this list know more about this than I do.

cheesehead · 2003-07-03, 00:28

The Riemann Hypothesis is not something as simple as most other theorems and stuff discussed in this forum. (But there is a $1 million prize offered for proving it, so maybe that'll give you incentive to wade through the following details.)

The Rieman Hypothesis involves a function of a complex (like a + bi, where i = the square root of -1) variable. The function, called the zeta function, is the sum of an infinite series. The terms of the infinite series are reciprocals of integers raised to a power which is a complex number (the argument of the zeta function).

For certain values of the complex variable, the infinite sum equals 0. Those values are called the "zeros" of the zeta function (like zeros (AKA roots) of a polynomial -- the zeros of the polynomial x^2 - 4 are +2 and -2 because those are the values of x for which x^2 - 4 = 0).

The Riemann Hypothesis is that all of the zeros, except for certain "trivial" zeros, of the zeta function have a real part = 1/2. That is, all the nontrivial zeros of the zeta function are of the form 1/2 + bi, where b is some real number and i is the square root of -1.

Somehow (and I don't know the details), this assertion is linked to basic properties of prime numbers.

The following references may be throwing you into deep water, but at least take a look at the text comments ...

Within Eric Weisstein's MathWorld is
Riemann Hypothesis at
http://mathworld.wolfram.com/RiemannHypothesis.html

Among Chris Caldwell's extensive "Prime Pages" is
The Riemann Hypothesis at
http://www.utm.edu/research/primes/notes/rh.html

philmoore · 2003-09-23, 21:55

I've been away on vacation, but when I started to read this thread, I laughed so hard I think I broke a rib!

In doing contour integrals in complex function theory, one often has to define what he/she means by "THE square root of x", and the answer often depends on the particular problem under consideration. However, in algebra, we often agree to the convention that THE square root of a negative number is the imaginary number bi where b is positive. However, i is just a symbol, and your algebra should still be consistent if you substituted -i everywhere in place of i, but then you would have to change your convention. The one thing you do lose is that the law: "The square root of ab equals the square root of a times the square root of b", although true for positive numbers a and b, is no longer true in general.

So, on to Riemann. I picked up a fascinating book called "Riemann's Zeta Function" by H. M. Edwards, published in 1974 but reissued in 2001 as a Dover paperback, a good deal at $14.95. It is fairly heavy reading and requires some background in complex variables, but it does a great job at placing Riemann's original paper (included in translation as an appendix) in its historical context. Riemann's only published paper on the zeta function dates from 1859, but most of his assertions in that paper were only proved in the time between 1893 and 1905, while his conjecture about all "non-trivial" zeros having real part equal to 1/2 is still unsettled. I'm currently reading about early (pre-computer) calculations which showed that the first hundred or so zeros all satisfy the Riemann hypothesis. Of course, now that has been extended by computer to the first 1,500,000,000 zeros. In spite of this overwhelming numerical evidence, Edwards seems to say that there is no obvious reason why the RH should be true!

GP2 · 2003-09-23, 23:20

Quote:

Originally posted by philmoore
I'm currently reading about early (pre-computer) calculations which showed that the first hundred or so zeros all satisfy the Riemann hypothesis. Of course, now that has been extended by computer to the first 1,500,000,000 zeros. In spite of this overwhelming numerical evidence, Edwards seems to say that there is no obvious reason why the RH should be true!

Actually, at latest count, 515 billion zeros have been calculated by the Zetagrid distributed-computing project: http://www.zetagrid.net/servlet/service/statistic

and over a billion more every day.

There are a number of mathematical conjectures which are known to have counterexamples at extremely high numerical values. An example relevant to prime numbers: pi(x) > Li(x) for some unknown very large x.

http://www.utm.edu/research/primes/howmany.shtml#hist

jinydu · 2004-04-13, 02:49

Quote:

Originally Posted by littleone

I was recently assigned a math project that involved writing a paper about a math subject that we don't know about. I choose to do a paper on the riemann hypotesis. Now, I have no idea what it is or why it is so important. If anyone can help me to understand it I would be much obliged.
Thank You,
Littleone

Ok, I'll try to explain it as well as I can.

Consider the infinite sum: 1/1 + 1/2 + 1/3 + 1/4 + ...

Obviously, as you add more and more terms, the sum gets larger and larger. However, the sum eventually exceeds any number you choose, no matter how large (i.e. 1000, 1,000,000, M40, etc.). We say that the sum diverges.

Now consider the infinite sum: 1/(1^2) + 1/(2^2) + 1/(3^2) + 1/(4^2) + ...

Clearly, this sum also gets larger and larger as you add more and more terms. Unlike the previous case, the sum eventually converges to a particular number: ([pi^2]/6).

We can define a function called the zeta function:

Zeta(x) = 1/(1^x) + 1/(2^x) + 1/(3^x) + 1/(4^x) + ...

It can be proven that this function is defined only for x > 1.

However, using a technique called analytic continuation, mathematicians have defined with another function (called the Riemann zeta function). The Riemann zeta function is more complicated than the zeta function, but it has the big advantage of being defined for all complex numbers except 1, instead of just all real numbers greater than 1. In addition, the Riemann zeta function gives the same value as the original Zeta function for all real numbers greater than 1. (Note: the input variable for the Riemann zeta function is usually written as s).

There are some values of s for which the Riemann zeta function gives the output 0. These values are known as zeroes. It has been known for a long time that s = -2, -4, -6, -8, -10, ... -2n (where n is a positive integer), ... are all zeroes. In 1859, Riemann conjectured that the only other zeroes for the Riemann zeta function all lie on a critical line with the equation Re(s) = 1/2. This is known as the Riemann Hypothesis.

To this day, the Riemann Hypothesis has resisted all attempts to prove it. Many consider it the greatest unsolved problem in mathematics today. In 2000, the Clay Mathematics Institute announced it would reward $1M US for a proof of the Riemann Hypothesis (although it did not announce an award for a disproof).

So far, computers have been used to test the Riemann Hypothesis for "small" zeroes (small in the sense of being close to the real line). A distributed computing project called ZetaGrid (http://www.zetagrid.net/zeta/index.html) uses volunteer CPU power (similar to GIMPS) to look for zeroes. As of the latest news ZetaGrid has looked through the first 721 billion potential zeroes of the Riemann zeta function without finding any. Most mathematicians believe that the Riemann Hypothesis is true. However, it should be remembered that infinity is a LONG way away. There have been several examples of mathematical conjectures that have been disproved at very large numbers. There could be a zero at 722 billion that will be found tomorrow. Or there could be a zero at 1 trillion that will be found next year. Or maybe, there is a zero bigger than M40 that is too large to be found by any computer, present or future. Thus, the problem remains open.

R.D. Silverman · 2004-04-13, 16:49

Quote:

Originally Posted by biwema

How does the Zetagrid Project verify the Riemann hypothesis? I doubt, that they just check the line Re(x)=0.5. In that case they would find a lot of zeroes, but no counterexample.
Where else do they search then? Is the whole gauss plane searches, or just a strip near the known zeroes, where it is more probable that new zeroes appear? Finding Zeroes in a 2dimensional area might be quite tricky.

One computes a contour integral along the perimeter of a box surrounding the
critical line. The value of the integral gives the number of zeros within
the box. Independently one computes zeros on the critical line itself up to
a specified height. When the two counts match you know that you have found all the zeros.

Note that if there is a zero off the line it will come in pairs because of the
functional equation.

All non-trivial zeros must be in the critical strip and there are theorems which
say that zeros can not be found 'too close' to real(s) = 1 [i.e. there are
known zero-free regions].

philmoore · 2004-04-16, 17:20

To littleone:

You ask why the Riemann Hypothesis is important. My answer is that, if it were proven, it would show that the distribution of the prime numbers behaves "randomly". Of course, at another level, we might say that the primes are not random, and that Eratosthenes' sieve gives a deterministic algorithm for finding all the primes. The prime number theorem says that the "probability" of a large integer x being prime is about 1/ln x. (I put "probability" in quotes because x is either prime or not, so the true probability is either 1 or 0, but we mean probability in the sense of average over a large number of possible x values.) Consider a random sequence of zeros and ones where the probability is 1/ln n that the n-th element of the sequence is 1. The Riemann Hypothesis says that the error term in the pi(x) prime counting function is essentially the same as one of these random sequences.

If you come up with more questions, feel free to ask them here.

R.D. Silverman · 2004-06-09, 17:16

Loius DeBranges has struck again.

He has just claimed a proof of RH. As I understand it, a formal
manuscript is not yet available...

akruppa · 2004-06-09, 17:46

Interesting times for number theory... AKS, arithmetic progressions, twin primes - and now the grand prize?

On his home page is a PDF titled Apology for the proof of the Riemann hypothesis but it seems (from first glance) to be more of a history of the subject. Looks interesting, though, and at least in part readable for non-mathematicans.

Alex

ewmayer · 2004-06-09, 18:50

Quote:

Originally Posted by Bob Silverman

Loius DeBranges has struck again.

Can you fill those of us not familiar with the man in on the previous history here?

philmoore · 2004-06-09, 20:27

Louis De Branges is best known for proving the Bieberbach Conjecture in 1984. He has been working on the Riemann Hypothesis now for many years, and has announced new proofs of it on a number of occasions. The current version from his web-site is dated May 24, 2004 and appears to be an updated version of one he posted last August. On the web-page Alex cited, look at the paper "Riemann zeta functions (in pdf format)." It is 124 pages long. I saw objections a few years ago from Conrey and Li, with arguments of Sarnak, claiming that there were problems with De Branges' approach, but I don't know if those objections apply to his current manuscript.

2003-07-02, 18:17	#1
ThomRuley May 2003 123₁₆ Posts	Offizieller Riemann-Hypothese Faden I was looking for information about Mersenne primes, and I came across something about the Reimann hypothesis. Could someone explain what that is? Also, I read that if this were proven, it would speed up our searching a lot. How? I assume that other people on this list know more about this than I do.

2004-06-09, 17:16	#8
R.D. Silverman "Bob Silverman" Nov 2003 North of Boston 16524₈ Posts	DeBranges and RH Loius DeBranges has struck again. He has just claimed a proof of RH. As I understand it, a formal manuscript is not yet available...

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2003-07-03, 00:28	#2
cheesehead "Richard B. Woods" Aug 2002 Wisconsin USA 1E0C₁₆ Posts	The Riemann Hypothesis is not something as simple as most other theorems and stuff discussed in this forum. (But there is a $1 million prize offered for proving it, so maybe that'll give you incentive to wade through the following details.) The Rieman Hypothesis involves a function of a complex (like a + bi, where i = the square root of -1) variable. The function, called the zeta function, is the sum of an infinite series. The terms of the infinite series are reciprocals of integers raised to a power which is a complex number (the argument of the zeta function). For certain values of the complex variable, the infinite sum equals 0. Those values are called the "zeros" of the zeta function (like zeros (AKA roots) of a polynomial -- the zeros of the polynomial x^2 - 4 are +2 and -2 because those are the values of x for which x^2 - 4 = 0). The Riemann Hypothesis is that all of the zeros, except for certain "trivial" zeros, of the zeta function have a real part = 1/2. That is, all the nontrivial zeros of the zeta function are of the form 1/2 + bi, where b is some real number and i is the square root of -1. Somehow (and I don't know the details), this assertion is linked to basic properties of prime numbers. The following references may be throwing you into deep water, but at least take a look at the text comments ... Within Eric Weisstein's MathWorld is Riemann Hypothesis at http://mathworld.wolfram.com/RiemannHypothesis.html Among Chris Caldwell's extensive "Prime Pages" is The Riemann Hypothesis at http://www.utm.edu/research/primes/notes/rh.html

2003-09-23, 21:55	#3
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 2⁵×5×7 Posts	I've been away on vacation, but when I started to read this thread, I laughed so hard I think I broke a rib! In doing contour integrals in complex function theory, one often has to define what he/she means by "THE square root of x", and the answer often depends on the particular problem under consideration. However, in algebra, we often agree to the convention that THE square root of a negative number is the imaginary number bi where b is positive. However, i is just a symbol, and your algebra should still be consistent if you substituted -i everywhere in place of i, but then you would have to change your convention. The one thing you do lose is that the law: "The square root of ab equals the square root of a times the square root of b", although true for positive numbers a and b, is no longer true in general. So, on to Riemann. I picked up a fascinating book called "Riemann's Zeta Function" by H. M. Edwards, published in 1974 but reissued in 2001 as a Dover paperback, a good deal at $14.95. It is fairly heavy reading and requires some background in complex variables, but it does a great job at placing Riemann's original paper (included in translation as an appendix) in its historical context. Riemann's only published paper on the zeta function dates from 1859, but most of his assertions in that paper were only proved in the time between 1893 and 1905, while his conjecture about all "non-trivial" zeros having real part equal to 1/2 is still unsettled. I'm currently reading about early (pre-computer) calculations which showed that the first hundred or so zeros all satisfy the Riemann hypothesis. Of course, now that has been extended by computer to the first 1,500,000,000 zeros. In spite of this overwhelming numerical evidence, Edwards seems to say that there is no obvious reason why the RH should be true!

2004-04-16, 17:20	#7
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 10001100000₂ Posts	To littleone: You ask why the Riemann Hypothesis is important. My answer is that, if it were proven, it would show that the distribution of the prime numbers behaves "randomly". Of course, at another level, we might say that the primes are not random, and that Eratosthenes' sieve gives a deterministic algorithm for finding all the primes. The prime number theorem says that the "probability" of a large integer x being prime is about 1/ln x. (I put "probability" in quotes because x is either prime or not, so the true probability is either 1 or 0, but we mean probability in the sense of average over a large number of possible x values.) Consider a random sequence of zeros and ones where the probability is 1/ln n that the n-th element of the sequence is 1. The Riemann Hypothesis says that the error term in the pi(x) prime counting function is essentially the same as one of these random sequences. If you come up with more questions, feel free to ask them here.

2004-06-09, 17:46	#9
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	Interesting times for number theory... AKS, arithmetic progressions, twin primes - and now the grand prize? On his home page is a PDF titled Apology for the proof of the Riemann hypothesis but it seems (from first glance) to be more of a history of the subject. Looks interesting, though, and at least in part readable for non-mathematicans. Alex

2004-06-09, 20:27	#11
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 2⁵×5×7 Posts	Louis De Branges is best known for proving the Bieberbach Conjecture in 1984. He has been working on the Riemann Hypothesis now for many years, and has announced new proofs of it on a number of occasions. The current version from his web-site is dated May 24, 2004 and appears to be an updated version of one he posted last August. On the web-page Alex cited, look at the paper "Riemann zeta functions (in pdf format)." It is 124 pages long. I saw objections a few years ago from Conrey and Li, with arguments of Sarnak, claiming that there were problems with De Branges' approach, but I don't know if those objections apply to his current manuscript.