## Riemann Hypothesis and Coin flipping

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### Riemann Hypothesis and Coin flipping

Lut u(n) be the mobius function and let M(x)=sum(n less than x) u(n). It can be shown that the claim that M(x)=O(x^(1/2 +e)) for every e greater than 0 is equivalent to the Riemann Hypothesis.

See wiki under "Growth rate of Mobius function" for more details

http://en.wikipedia.org/wiki/Riemann_hypothesis

Now it can be shown that if K(x) represents the sum of heads minus the sum of tails obtained from a coin that is flipped x number of times, then we also have K(x)=O(O(x^(1/2 +e)) for every e greater than 0. Unfortunately, the proof for this case requires us assuming that the coin flips are random. This raises the following interesting questions.

1. Is there a fundamental difference between the Riemann Hypothesis, and the cion flipping problem?

2. Do we have to prove some sort of randomness condition for the primes numbers in order to prove the Riemann Hypothesis?

3. Is such a proof even possible?

jshort
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jshort,

Check out the Mertens and Möbius functions. The former is built upon the latter. Its structure is surprising and almost remeniscent of random behavior, but completely random it is not. I just find it mesmerizing.

Möbius function

http://en.wikipedia.org/wiki/M%C3%B6bius_function

Mertens function

http://en.wikipedia.org/wiki/Mertens_function

The Mertens functions is closely related to the world famous Riemann zeta function.

http://en.wikipedia.org/wiki/Riemann_zeta_function

This latter function is conjectured to have a certain property,

http://en.wikipedia.org/wiki/Riemann_hypothesis

but while few people doubt the veracity of the statement, nobody in over a century has been able to find a proof.
Giacomo

Georg Riemann found a vital clue. He discovered that the secrets of the primes are locked inside something called the zeta function. And, Riemann worked out that if the zeros really do lie on the critical line, then the primes stray from the 1/ln(x) distribution exactly as much as a bunch of coin tosses stray from the 50:50 distribution law. This is a startling conclusion. The primes aren't just unpredictable, they really do behave as if each prime number is picked at random, with the probability 1/ln(x) -- almost as if they were chosen with a weighted coin. So to some extent the primes are tamed, because we can make statistical predictions about them, just as we can about coin tosses.

More to follow...
Giacomo

Giacomo wrote: So to some extent the primes are tamed, because we can make statistical predictions about them, just as we can about coin tosses.

Wouldn't this mean that the primes are untamed because of this. After all, the tossing of a coin is random.

With regards to "true randomness", even though the primes may not have this property, they are still chaotic enough that I believe one can make conclusions about them by conclusions made about coin flipping (we just won't be able to prove these conclusions).

jshort
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Posts: 1637
Joined: 08 Oct 2006

Many of the famous conjectures about the distribution of prime numbers basically assume that the primes behave like a random sequence with 1/log n distribution, with one restriction: no prime is divisible by any other prime (random numbers do not satisfy this restriction).
One such conjecture has already been discussed here, the Bateman-Horn conjecture. There are many others, e.g. the conjectures about the distribution of spacing between zeros of the Riemann zeta function follow from such behavior of the primes.
Phalcon

jshort,

jshort wrote:Wouldn't this mean that the primes are untamed because of this. After all, the tossing of a coin is random..

The primes are not random. If you ask for the first 100 primes, you always get the same answer, but if you ask for the first 100 coin flips, you get different answers every time.

jshort wrote:With regards to "true randomness", even though the primes may not have this property, they are still chaotic enough that I believe one can make conclusions about them by conclusions made about coin flipping (we just won't be able to prove these conclusions).

Also, the error term for primes represents a difference between two deterministic functions; it's vague because we don't know a lot about one of the functions, not because it's random.

As of now, the majority of mathematicians I've talked to believe that it's just a coincidence that the uncertainty for prime numbers matches the variability for coin tosses.

There are a number of intriguing but unexplained analogies between the behavior of the Riemann Zeta Function and some probabilistic problems, although no one seems optimistic that these can be used to prove RH.
Giacomo

Giacomo wrote:The primes are not random. If you ask for the first 100 primes, you always get the same answer, but if you ask for the first 100 coin flips, you get different answers every time.

I disagree with this logic. First of all you have to view the primes as a single event of coin tossing. Second, the only reason you say that the first 100 primes are deterministic is because you've already calculated them. We could say the same thing regarding the coin tosses if we had gone in and already flipped the 100 coins. The probability functions only change from 50:50 to 0:1 only after you have gone in and observed whether or not the number in question is prime, or whether the coin toss was a head or tail.

In my opinion there is no fundamental difference between primes and the randomness associated with coin tossing.

Giacomo wrote:Also, the error term for primes represents a difference between two deterministic functions; it's vague because we don't know a lot about one of the functions, not because it's random.

How could you possibly know this?

Giacomo wrote:There are a number of intriguing but unexplained analogies between the behavior of the Riemann Zeta Function and some probabilistic problems, although no one seems optimistic that these can be used to prove RH.

In my opinion, if the primes can be treatedly randomly, then this could imply that the Riemann Hypothesis has no proof based on some structure of the primes, and that it is just some probabalistic accident that the hypthesis is true.

jshort
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Joined: 08 Oct 2006

jshort,

The prime-counting function pi(n) is deterministic. We don't have a formula for it, but we do have a method to determine its value for any n. The key difference between the prime-counting problem and the coin-tossing problem is that, if we are thinking about probabilities, we would run the coin-tossing experiment many times, and the probability functions tell us something about the average behavior of these many experiments. If you calculate pi(n) many times for the same n you always get the same answer, unlike the coin-tossing experiment where usually each experiment gets different results.

Maybe it would be more useful to think about 'statistics' than 'probability'. Statistics helps us deal with the unknown, even if it is deterministic.
Giacomo

I scowered over wikipedia articles and other google-browsed links, but found a hard time finding:

What exactly will it mean when/if the Riemann hypothesis is proved? I heard things about ties to quantum physics, and distribution of the primes. What will it mean for us all?
smokeybob

smokeybob,

The failure of the Riemann hypothesis (RH) would create havoc in the distribution of prime numbers. And the havoc would spread further. This is why we, mathematicians, long to prove the hypothesis.

Early in the 20th century, mathematicians made a daring conjecture: that the Riemann zeros could correspond to the energy levels of a quantum mechanical system.

Quantum mechanics deals with the behaviour of tiny particles such as electrons. Crucially, its equations work with complex numbers, but the energy of a physical system is always measured by a real number. So energy levels form an infinite set of numbers lying along the real axis of the complex plane--a straight line.

This sounds like Riemann's zeros. The line of zeros is vertical, rather than horizontal, but it is a simple bit of maths to rotate it and put it on top of the real line. If the zeros then match up with the energy levels of a quantum system, the Riemann hypothesis is proved.

For decades, this idea was only wishful thinking.

Then in 1972 came a hint that it could work. Hugh Montgomery, at the University of Michigan, had found a formula for the spacings between Riemann zeros. Visiting the Institute for Advanced Study at Princeton, he ran into physicist Freeman Dyson at afternoon tea, and mentioned his formula. Dyson recognized it immediately. It was identical to a formula that gives the spacings between energy levels in a category of quantum systems—quantum chaotic systems, to be precise.

Chaos theory applies to physical systems so sensitive to their starting conditions that they are impossible to predict. Almost all complicated systems are chaotic.

The quantum versions of these systems have a jumble of energy levels, scattered apparently at random but in fact spaced according to Montgomery's formula. Quantum chaotic systems include atoms bigger than hydrogen, large atomic nuclei, all molecules, and electrons trapped in the microscopic arenas called quantum dots. Could the Riemann zeros fit one of these quantum chaotic systems?

In the late 1980s, Odlyzko picked an assortment of systems, and compared their energy levels with the Riemann zeros. He found that when he averaged out over many different chaotic systems, the energy level spacings fitted the Riemann spacings with stunning precision.

That's still not enough. To prove the Riemann hypothesis, researchers must pinpoint a specific quantum system whose energy levels correspond exactly to the zeros, and prove that they do so all the way to infinity. Which, of all the different systems, is the right one?

Berry and his colleague Jonathan Keating have made one suggestion. In a chaotic system, an object usually moves unpredictably, but sometimes its path will cycle back on itself in a "periodic orbit". Berry and Keating think that the right quantum system will have an infinite collection of periodic orbits, one for each prime number.

Nicholas Katz and Peter Sarnak predicted that the system should have a special kind of symmetry called symplectic symmetry.

Both of these clues should help quantum chaologists zero in on the one system that will prove the Riemann hypothesis.

Some mathematicians are working on the following approach :

To create a system that already includes the prime numbers. To understand how, you have to imagine a quantum system not as a particle bouncing around an atom, say, but as a geometrical space. It sounds odd, but it represents one of the weird things about quantum systems: they can be two or more things at once.

To understand this approach think of the Schrödinger's cat space, where any quantum object can find itself in a "superposition" of different states.
Giacomo

And,

Gauss, when he was a young man, stated that the number of primes less than x is asymptotic to x/log(x).

Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function.

Riemann checked the first few zeros of the zeta function by hand. They satisfy his hypothesis.

By now over 1.5 billion zeros have been checked by computer. Very strong experimental evidence.

But in mathematics we require a proof. A PROOF gives certainty, and it gives understanding: it helps us understand why a result is true.
Giacomo

And,

Prime numbers do matter a great deal, especially when you’re dealing with numbers larger than the number of atoms in the universe.

Whenever you make a purchase online and there’s a little padlock in the corner of your browser, that encryption is based on the fact that huge prime numbers are hard to pick out, and large non-primes are hard to divide up into primes. This encryption which keeps the majority of internet information safe, is called RSA encryption.

It’s quite possible that once the Riemann Hypothesis is worked out we’ll have to find some new type of encryption.
Giacomo

It’s quite possible that once the Riemann Hypothesis is worked out we’ll have to find some new type of encryption.

This was suggested in an episode of "Numb3rs", it's the first time I hear this unfounded assertion from a professional mathematician.
Phalcon

it's the first time I hear this unfounded assertion from a professional mathematician.

I merely wrote that it is possible. That's all. It is merely a conjecture.

Time will tell. When that day comes, we all will be able to see it, to prove it or disprove it.
Giacomo

Okay, so lets say someone finds a quantum mechanical system who's energy levels correspond directly to the zeros of the Zeta function. That means that we've found... the best quantum mechanical system ever?
smokeybob

smokeybob,

We need to exercise caution, here. Yes, caution, my friend ! Because we have invented after all some bizarrely twisted geometries that are based on the primes.

These mathematicians decided to build a quantum state space out of the prime numbers.

[ Note that the primes are a bunch of isolated numbers, nothing like the smooth expanses of space in which we can measure things like angles and lengths.]

To put all the primes in a soup-mix, an infinite-dimensional space was constructed. In the first dimension, measurements are made with 2-D geometry, in the second dimension with 3-D geometry, and so on to include all the prime numbers.

They managed to prove that their prime-based quantum system has energy levels corresponding to all the Riemann zeros that lie on the critical line.

This will be good if we can make one last step, which is to prove that there aren't any extra zeros hanging around, unaccounted for by their energy levels.

They managed to replace the Riemann Hypothesis with an equally difficult question.

So, caution!

We need some major new idea here. We need to find a clever way to make it in the lab.

Then, we will be able to get the Riemann zeros out just by observing its spectrum.

The idea is to use the mathematics based on the Riemann zeta function to predict the behavior of chaotic systems. Most models of quantum chaos are complicated and difficult to calculate. The Riemann zeros, by comparison, are easy to compute.

If we prove the Riemann hypothesis using a quantum system, the link will be firmly established.

Using the mathematics of the zeta function, scientists will be able to predict the scattering of very high energy levels in atoms, molecules and nuclei, and the fluctuations in the resistance of quantum dots in a magnetic field.

The same applies to any situation where waves bounce around chaotically, such as light waves and sound waves.

We tackle these problems without knowing the truth of the Riemann Hypothesis. And, once we prove it, then it should lead to an efficient way of deciding whether a given large number is prime. No existing algorithms designed to do this are guaranteed to terminate in a finite number of steps.

However, proving the Riemann hypothesis won't be the end of the story. There will be harder questions to tackle. The fun never stops with math !

For example,

- The question of randomness and order with prime numbers.
- Once we prove the Riemann hypothesis using a quantum system, what else can we discover when we dig deeper ?

If you believe that we live in a mathematical universe or that mathematics is the key to the universe, you might as well contemplate on the prime numbers and ask what else can we discover, what else can the prime numbers reveal to us ?
Giacomo

Is it a reasonable question to ask, when you think the Reimann hypothesis will be solved?

I remember a while ago some mathematicians saying something about proving/discovering a property about L-class functions, of which the Reimann Zeta function is a member of, and how it would help lead to a solution to this age-old problem. Is it meaningful to say, that given what we know now and progress towards the solution that in a 95% confidence interval a solution will be obtained in the next 20 years?
smokeybob

I don't know when exactly it will be solved.

I'm still working on it like many other mathematicians.

The Riemann zeta-function itself is now reasonably well understood, its L-function relatives are not. L-functions are analytic continuations(1) of the more general Dirichlet series:

L(s) = 1 + a2/2s + a3/3s + a4/4s + ...

where ai are complex numbers.

Just like Riemann's function, L-functions can be represented by infinite products involving the primes. They also satisfy particular functional equations(2).

functional equations http://en.wikipedia.org/wiki/Functional_equation

(1) Analytic continuation : http://en.wikipedia.org/wiki/Analytic_continuation

(2) Functional equations : http://en.wikipedia.org/wiki/Functional_equation

Many problems in Number Theory are connected to these functions (L-functions).

Here's our strategy:

Think of Riemann Hypothesis as the proverbial needle in the haystack, then to prove L-functions claim which is an even bigger claim would be to dump more hay on your original haystack in order to find the needle. In everyday life situations this will not make sense, but in mathematics this makes good sense.

more to follow....
Giacomo

smokeybob

Understanding L-functions

L-functions are categorized in the first place by degree. The degree one L-functions are the Riemann zeta function and Dirichlet L-functions. Degree two L-functions conjecturally all arise from primitive, cuspidal modular forms, both the holomorphic and non-holomorphic (Maass) forms. For degree higher than two, examples include convolutions of degree two L-functions; there are specific examples of higher degree L-functions that do not arise in this way, for example Siegel modular forms that are not lifts.
Giacomo

L-functions

A new mathematical object, an elusive cousin of the Riemann zeta-function.

L-functions underpin much of twentieth century number theory. They feature in the proof of Fermat's last theorem, as well as playing a part in the recent classification of congruent numbers, a problem first posed one thousand years ago.

The Riemann zeta-function started all L-functions, goes back to Leonhard Euler and Bernhard Riemann, and contains deep information regarding the distribution of prime numbers. Many mathematicians believe that other L-functions also contain invaluable insights into number theory. The problem is that few of them are explicitly known.

To understand L-functions, let's firstly consider the Riemann zeta-function. In the eighteenth century, the legendary mathematician Leonhard Euler considered the infinite series

L(s) = 1/1s + 1/2s + 1/3s + 1/4s + ...

where s is a real number. On the face of it, this series does not seem to have much to do with prime numbers. However, Euler showed that this series is equal to the infinite product

zeta(s) = 2s/(2s - 1) * 3s/(3s - 1) * 5s/(5s - 1) * 7s/(7s - 1) ...

which contains one factor for each of the primes 2, 3, 5, 7, etc.

The value of the series — or the lack of one — depends on the value of s. When s is less than or equal to 1, it is possible to make the sum ever larger simply by adding more terms — that is, the series does not converge, it diverges. For example, for s = -2, the series is

L(-2) = 1 + 4 + 9 + 16 + ...

However, if s is greater than 1, the series converges to a finite value. Taking s = 2, for example, gives

L(+2) = 1 + 1/4 + 1/9 + 1/16 + ...

which sums to pi2/6 once infinitely many terms have been added. Therefore, as a function of the variable s, Euler's series is only valid for values of s that are greater than 1 — for all other values of s, the series adds to infinity.

Riemann developed a method of extending Euler's series to a function that is valid for all values of s. He found a function that agrees with Euler's series for values of s that are greater than 1, but also gives a finite value for all other values of s, including complex values. This analytic continuation of Euler's series is now known as the Riemann zeta-function.

Riemann showed that this continuous function of a complex variable had deep connections to prime numbers, which are not only real numbers, but also discrete. In particular, he found that the way the primes are distributed along the number line is related to the values of s for which his zeta-function is zero. He also conjectured for which values of s this happens, but he could not prove it — this is the famous Riemann hypothesis, one of the most important open problems in mathematics.

While the Riemann zeta-function itself is now reasonably well understood, its L-function relatives are not. L-functions are analytic continuations of the more general Dirichlet series, as I've posted before.
Giacomo

I found that very helpful. And thank you!

So do these L-class functions have a particular maximum order? Or are they just way harder "polynomials", where the order goes up to infinity. But we find some interesting properties about, zeroth, first and second order polynomials and real-life things - is this similar to L-class functions, but on steroids?
smokeybob

smokeybob wrote:I found that very helpful. And thank you!

So do these L-class functions have a particular maximum order? Or are they just way harder "polynomials", where the order goes up to infinity. But we find some interesting properties about, zeroth, first and second order polynomials and real-life things - is this similar to L-class functions, but on steroids?

You're welcome.

So far we have discovered the third degree transcendental ones. There are two types of L-functions: algebraic and transcendental, and these are classified according to their degree.

Well, they are crucial and powerful functions. We only have scratched the surface. There is a lot of work ahead of us. The work is divided into theoretical, algorithmic, experimental and data gathering. It will keep us busy for a long time. It's a major battle. All branches of number theory are touched by these functions and modular forms. This is very very tough. This is Ph.D. level, it is my level. I need more time to deepen my understanding of these functions.

Some mathematicians are not convinced that these functions will bring us closer to Riemann-hypothesis. Some were even opposed to the idea of using them in number theory. But, here we are, we discuss them, we use them. They are taken seriously, but the sceptics are still out there. That's fine. We just need proof.
Giacomo

I just finished reading a fantastic book on the Riemann Hypothesis that touched on a lot of the topics covered in this post. Namely, where it comes from, how it relates to functions like the Prime counting function, the PNT, Li, Moebius' function, and quantum mechanics. It was a really awesome book, and as some of the reviews on Amazon stated, it really is tough to put down.

It's called: Prime Obsession, by John Derbyshire - and Google books will let you read the first 3 (of about 21) chapters to get a taste of what the content is like. I'd say its a book for all audiences, with those versed in mathematics having a much quicker read, and an enjoyable time seeing what came out of some of these amazing mathematicians' heads so long ago.
smokeybob

### Re: Riemann Hypothesis and Coin flipping

Let u(n) be the mobius function and let M(x)=sum(n less than x) u(n). It can be shown that the claim that M(x)=O(x^(1/2 +e)) for every e greater than 0 is equivalent to the Riemann Hypothesis.

Now it can be shown that if K(x) represents the sum of heads minus the sum of tails obtained from a coin that is flipped x number of times, then we also have K(x)=O(O(x^(1/2 +e)) for every e greater than 0. Unfortunately, the proof for this case requires us assuming that the coin flips are random. This raises the following interesting questions.

1. Is there a fundamental difference between the Riemann Hypothesis, and the cion flipping problem?

2. Do we have to prove some sort of randomness condition for the primes numbers in order to prove the Riemann Hypothesis?

3. Is such a proof even possible?

Great questions. The answers to all 3 questions: 1 no, 2, yes, 3, yes, as shown by this paper in 2008 arXiv:0810.0095 "Modeling the creative process of the mind by prime numbers and a simple proof of the Riemann Hypothesis"
The key is prove that prime numbers are inherently or by definition unpredictable. To do this, one can define primes as uniqueness as uniqueness is by definition unpredictable. One then needs to come up with an algorithm to create uniqueness and show that such an algorithm can create all the primes by merely creating uniqueness.
gnom15
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### Re: Riemann Hypothesis and Coin flipping

gnom15 » February 24th, 2019, 12:36 pm wrote: arXiv:0810.0095 "Modeling the creative process of the mind by prime numbers and a simple proof of the Riemann Hypothesis"

Haven't followed this thread. But isn't a claim of a simple proof of RH a strong indicator of crankitude? After all, the problem is still open.

gnom15 » February 24th, 2019, 12:36 pm wrote:The key is prove that prime numbers are inherently or by definition unpredictable.

Nor are primes random. You can generate the sequence of primes using a perfectly deterministic process, namely trial division on each of the positive integers in turn. It's a programming exercise suitable for a beginner. Anything that's the output of a program isn't random, by definition. Random sequences are ones that aren't the output of any program. Turing, Kolmorogov, Chaitin, and all that.

ps -- I looked up the paper. The abstract is extremely cranky. OP claims to have a Ph.D., exemplifying John Baez's observation in his famous crackpot index that cranks often claim to have been educated, "as if that were evidence of sanity."

https://arxiv.org/pdf/0810.0095.pdf

pps -- Oh I see, someone resurrected a 11 year old thread just to post the link to that article. I'll just leave this post here in case anyone's interested. I skimmed the entire paper. It's cranky.
someguy1
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### Re: Riemann Hypothesis and Coin flipping

ppps -- This was a hell of a great thread. Well worth being resurrected for any reason.
someguy1
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### Re: Riemann Hypothesis and Coin flipping

sounds like you know quite a bit about crank, then surely you know this, “A person with a new idea is a crank until the idea succeeds.” Mark Twain

Could you be more specific in your criticisms? Please name at least one logical flaw? That would be really helpful to all.

"The key is prove that prime numbers are inherently or by definition unpredictable. To do this, one can define primes as uniqueness as uniqueness is by definition unpredictable. One then needs to come up with an algorithm to create uniqueness and show that such an algorithm can create all the primes by merely creating uniqueness." Is there any specific flaw in this paragraph?
gnom15
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### Re: Riemann Hypothesis and Coin flipping

You said: "Nor are primes random. You can generate the sequence of primes using a perfectly deterministic process, namely trial division on each of the positive integers in turn. It's a programming exercise suitable for a beginner. Anything that's the output of a program isn't random, by definition. Random sequences are ones that aren't the output of any program. Turing, Kolmorogov, Chaitin, and all that."

Primes may be produced by a deterministic process but that does not mean primes can be predicted. So long primes cannot be predicted, they would appear random or pseudorandom. A proof for the non-predictability of primes should be extremely fundamental to number theory. However, experts are clueless in how to do this. For example, Terrence Tao wrote: "We have many ways of establishing that a pattern exists... but how does one demonstrate the absence of a pattern?" Uniqueness is by definition not corollary of a pattern and a proof of uniqueness is hence a proof of absence of a pattern. Math is the science of pattern. Should the concept of uniqueness or non-pattern, which is vital to being human, belong to math? Is it there already or not yet?
gnom15
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### Re: Riemann Hypothesis and Coin flipping

gnom15 » March 4th, 2019, 1:37 pm wrote:sounds like you know quite a bit about crank

I do. I posted the link to the article and gave my opinion. Others can read and decide for themselves. I'm not going to argue with you about it.

gnom15 » March 4th, 2019, 1:53 pm wrote:Primes may be produced by a deterministic process but that does not mean primes can be predicted.

Clever of you to focus on the math part, which isn't the cranky part. What's cranky is that the author (is it you?) thinks RH implies some cosmic truth about human consciousness. That's the cranky part.
someguy1
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