Can Your Insights on Primes Unlock the Riemann Hypothesis?

[shmoe]

Ok, I see that. There is something special about 6 to me, but that's just because of the way I've approached all this. As long as the modulus is a multiple of 6 the deal is the same.

The modulus doesn't need to be a multiple of 6 either!

I'm not sure about this. I think it can be shown that the converse must be true infinitely often - although I cannot actually show it yet. At least, I'm, not simply suggesting that I cannot find a reason that these numbers must have such a divisor. Rather, I'm suggesting there are reasons why infinitely often they will not have such divisors.

That was just how your statements (3) and (4) looked to me.

It may very well be that N+/-1 are primes infinitely often. It may also be the case that the N+1's are prime infinitely often and the N-1's are prime infinitely often yet there are only finitely many twin primes to be had this way (like what we can say about 30n+11 and 30n+13 for example).

Would it be true to say that given an infinite number of spins a roulette wheel must output zero infinitely many times? Or, can an infinite number of spins produce a finite number of zeros? The latter seems counterintuitive to me.

I'm ok with that. But what about where there are an infinite number of trials? Do we just take the limit as the outcome?

You can't spin a roulette wheel infinitely many times . Any (reasonable) model for the space of infinitely many roulette spins will have the probability of the set of events with only a finite number of 0's as being zero. This doesn't mean these events don't exist (at least in an abstract sense, none of the events exist in the real world). We can get into a more detailed discussion of what we mean when we talk about probability, but this is getting a little off topic. You of course know the primes aren't really random, this is really an imprecise way of thinking about densities of sequences which I'll talk about more below.

I've probably missed your point, but as far I can tell this doesn't seem to address my question. If, as I think you said, the probability of N+/-1 being a twin prime necessarily goes to zero as P increases, then doesn't this mean that there are not infinitely many twin primes? (Although, I suppose there could be an infinity of twin primes not at N+/-1. Is this what you meant by saying it was not a complete argument?) On the other hand, if this probability does not go to zero (or never reaches its limit) then wouldn't this mean there is no highest twin prime?

By 'not a complete argument', I mean it makes some leaps that haven't been justified, and is possibly not true. These kinds of probabilistic arguments, like Hardy&Littlewoods for twin primes, are not proofs. However, they can give results that look suprisingly accurate. The conjectured density of twin primes has held up to the actual data quite remarkably. In the case of N+/-1, there is much less data to go on, but tests up to something like P<120,000 do support the suggested result (N(P) gets unworkably huge fast, so the data is more limited), and the assumptions that can't be proven rigorously are still very 'reasonable'.

Let me put what the 'probability goes to zero' means in more precise terms. Let's just consider the sequence of perfect squares, 1, 4, 9, 16, 25, ... Let S(x) be the number of perfect squares less than or equal to x, so S(10)=3, S(19)=4, and so on. We actually have S(x)=[sqrt(x)], where [] is the floor function, i.e. S(19)=[sqrt(19)]=[4.358...]=4.

The ratio of squares to the total number of integers less than x is S(x)/x, considering x an integer for simplicity. If we randomly select an integer from 1, 2, 3, ..., x, the probability we get a perfect square is simply S(x)/x. Now S(x)/x=[sqrt(x)]/x~1/sqrt(x) (when y is very large, [y] is pretty close to y), so we can in a sense say the probability an integer less than x is a perfect square is 1/sqrt(x). Since 1/sqrt(x)->0 as x->infinity, S(x)/x->0 as x-> infinity as well. Hence we are justified in saying "the probability an integer less than x is a perfect square goes to zero as x goes to infinity", the true meaning behind this probability statement is in behavior of S(x)/x.

Now, even though the density of the squares goes to zero (or their probability if you prefer), there are still infinitely many of them. So even if we could prove the density of the twin primes goes to zero, it would not tell us that there are finitely many of them- you can have an infinite sequence whose density tails off and approaches zero.Derbyshire's book was trying to explain some very complicated maths without using much maths. A healthy knowledge of complex analysis is really needed to appreciate and understand the Zeta function. Something like Hardy and Wright's Number theory text covers elementary number theory. 'elementary' in this sense means no use of anything outside a calculus of one variables course, and in reality you don't need a background beyond high school math to understand most of it. Same goes for most number theory texts with 'elementary' in the title. If you haven't already, you should give some a shot.

I'll talk about number theory all day, I'm not bored yet.

 

[Canute]

You can't spin a roulette wheel infinitely many times .

Quite right. Just as well given that I once was half-addicted to playing it. But could we not say that for a non-halting roulette wheel the quantity of zeros will be greater than any finite quantity of zeros?

Let me put what the 'probability goes to zero' means in more precise terms. Let's just consider the sequence of perfect squares, 1, 4, 9, 16, 25, ... Let S(x) be the number of perfect squares less than or equal to x, so S(10)=3, S(19)=4, and so on. We actually have S(x)=[sqrt(x)], where [] is the floor function, i.e. S(19)=[sqrt(19)]=[4.358...]=4.

The ratio of squares to the total number of integers less than x is S(x)/x, considering x an integer for simplicity. If we randomly select an integer from 1, 2, 3, ..., x, the probability we get a perfect square is simply S(x)/x.

I'm fine up to here.

Now S(x)/x=[sqrt(x)]/x~1/sqrt(x) (when y is very large, [y] is pretty close to y), so we can in a sense say the probability an integer less than x is a perfect square is 1/sqrt(x).

I see that S(x)/x=sqrt(x)/x, but do not understand ~1/sqrt(x). Also, where did y suddenly appear from, and what do the square brackets around [y] signify? I'd like to fully understand the point you're making here.

Since 1/sqrt(x)->0 as x->infinity, S(x)/x->0 as x-> infinity as well. Hence we are justified in saying "the probability an integer less than x is a perfect square goes to zero as x goes to infinity", the true meaning behind this probability statement is in behavior of S(x)/x.

Lets' put this on hold until I've conquered the previous para.

Now, even though the density of the squares goes to zero (or their probability if you prefer), there are still infinitely many of them. So even if we could prove the density of the twin primes goes to zero, it would not tell us that there are finitely many of them- you can have an infinite sequence whose density tails off and approaches zero.

Yes, I see that. My question was badly phrased. If, as I think you said, the probability of N+/-1 being a twin prime necessarily goes to zero as P increases, then there seem to be two possibilities. First, the probability actually falls to zero, in which case there are not infinitely many twin primes at N+/-1, or, second, the probability does not actually fall to zero, in which case there are infinitely many of them. Is this not so?

Something like Hardy and Wright's Number theory text covers elementary number theory. 'elementary' in this sense means no use of anything outside a calculus of one variables course, and in reality you don't need a background beyond high school math to understand most of it. Same goes for most number theory texts with 'elementary' in the title. If you haven't already, you should give some a shot.

Are you seriously suggesting that I would understand any of H & L's book? I find that hard to believe. Even your definition of 'elementary' is beyond me. I have no idea what is either inside or outside a calculus of one variables course, whatever that is. It seem a little unlikely I'll get anywhere with H & L.

 

[shmoe]

Quite right. Just as well given that I once was half-addicted to playing it. But could we not say that for a non-halting roulette wheel the quantity of zeros will be greater than any finite quantity of zeros?

If you had some roulette wheel that will spin forever, even thousands of times a second, it's still possible that if you check in on it in 20 years that you haven't spun a zero yet. You might start to wonder if your wheel isn't rigged, but there's no way to be absolutely sure.

We can make a decent analogy with coin flipping. An infinite sequence of flips can be modeled by a sequence of 0's and 1's. If you think about the binary expansion of real numbers on the interval [0,1] these are exactly the same as the coin flips. The set of numbers in [0,1] that have only finitely many zeros is a countable set, the rest are uncountable. This is the sense that getting only a finite number of heads in an infinite sequence of coin flips is extremely rare, it's like a countable set vs an uncountable set.

I see that S(x)/x=sqrt(x)/x, but do not understand ~1/sqrt(x). Also, where did y suddenly appear from, and what do the square brackets around [y] signify? I'd like to fully understand the point you're making here.

S(x) isn't exactly sqrt(x), sqrt(x) may not even be an integer, we actually have S(x)=[sqrt(x)]. This [] function is just telling you to round down to the nearest integer, so S(10)=[sqrt(10)]=[3.162...]=3 for example.

However when x is big, so is sqrt(x) and you will have [sqrt(x)] very close to sqrt(x) relative to the size of sqrt(x). For example, sqrt(12345)=111.10... but we actually have S(12345)=[sqrt(12345)]=111, the percent difference is very small and gets smaller as x grows.

If you're happy with saying S(x)~sqrt(x) when x is large, that's enough. So S(x)/x~sqrt(x)/x=1/sqrt(x).

The key point from this if you take x large enough, the proportion of squares in the set {1, 2, ..., x} is as small as we like.

Yes, I see that. My question was badly phrased. If, as I think you said, the probability of N+/-1 being a twin prime necessarily goes to zero as P increases, then there seem to be two possibilities. First, the probability actually falls to zero, in which case there are not infinitely many twin primes at N+/-1, or, second, the probability does not actually fall to zero, in which case there are infinitely many of them. Is this not so?

The probability can go to zero while you still have infinitely many of them. Think about the sequence of perfect squares for the moment.

Are you seriously suggesting that I would understand any of H & L's book? I find that hard to believe. Even your definition of 'elementary' is beyond me. I have no idea what is either inside or outside a calculus of one variables course, whatever that is. It seem a little unlikely I'll get anywhere with H & L.

You never know until you try. Most elementary number theory books have no calculus in them at all. Hardy and Wright do use the big O notation and the like, so you will have plenty of asymptotic statements, so some idea of limits will be desirable. Hardy's book admitedly isn't the easiest intro, but it has some great stuff in it. You might find a book like Silverman's "A Friendly Introduction to Number Theory" to be a good place to start, but there are lots of options- check nearby libraries.

 

[Canute]

If you had some roulette wheel that will spin forever, even thousands of times a second, it's still possible that if you check in on it in 20 years that you haven't spun a zero yet. You might start to wonder if your wheel isn't rigged, but there's no way to be absolutely sure.

I'm sure you're right about this, but I don't get it. 20 years is a finite time. If after 500 million years there are no zeros I'd be sure the wheel is rigged, and even that is the blink of an eye in eternity.

... but we actually have S(12345)=[sqrt(12345)]=111, the percent difference is very small and gets smaller as x grows.

Got that.

If you're happy with saying S(x)~sqrt(x) when x is large, that's enough. So S(x)/x~sqrt(x)/x=1/sqrt(x).

Ok. Does ~ mean equivalent to, proportional to, roughly the same as, or something like that?

The key point from this if you take x large enough, the proportion of squares in the set {1, 2, ..., x} is as small as we like.

Is it? I can't see how it could become as small as zero.

The probability can go to zero while you still have infinitely many of them. Think about the sequence of perfect squares for the moment.

I'm confused about this point. I see that the proportion of squares below x approaches zero as x increases, but I can also see that it never reaches zero. Likewise, the probability of some n being prime falls to zero as n increases, but we know it never actually reaches zero.

But whether it does or does not reach zero my question still seems relevant. If you are saying that N+/-1 eventually falls to zero as P increases then there are not infinitely many twin primes of the form N+/-1. If it does not then there are. Is there something wrong with this reasoning?

 

[shmoe]

I think I see the problem. Whenever I've said "goes to zero" it doesn't necessarily mean whatever I was talking about is ever actually equal to zero, rather it gets arbitrarily close as in the sense of a limit.

\lim_{x\rightarrow\infty}f(x)=0

means that for any \epsilon&gt;0 we can find a N so that if x\geq N then |f(x)-0|&lt;\epsilon. "as small as we like" is in the choice of \epsilon, "as x->infinity" is in the lower bound of big N making sure x is 'large enough'.

S(x)/x is always greater than zero, but gets as close to zero as we like. By taking large enough x values, we can make S(x)/x very close to zero:

S(10^10)/10^10=0.00001
S(10^20)/10^20=0.0000000001

But it will never reach zero. I think you understood this and it was just a terminology gap.

Ok. Does ~ mean equivalent to, proportional to, roughly the same as, or something like that?

Asymptotic. Writing f(x)~g(x) as x goes to infinity means that f(x)/g(x) approaches 1, as in a limit. This means the relative error between the two get's very small (the percent difference) but the absolute error may still be very 'large' (|f(x)-g(x)| is the absolute error). You can think of it as "close for large values of x".

 

[Canute]

I think I see the problem. Whenever I've said "goes to zero" it doesn't necessarily mean whatever I was talking about is ever actually equal to zero, rather it gets arbitrarily close as in the sense of a limit... But it will never reach zero. I think you understood this and it was just a terminology gap.

Yes, I think it was. This leaves my question.

If the probability of N+/-1 being a twin prime falls (all the way) to zero as P increases then there are not infinitely many twin primes of the form N+/-1. If it does not then there are. Is this reasoning flawed? Or, if the probability does not fall (all the way) to zero then does it just mean there may be infinitely many?

 

[shmoe]

Yes, I think it was. This leaves my question.

If the probability of N+/-1 being a twin prime falls (all the way) to zero as P increases then there are not infinitely many twin primes of the form N+/-1. If it does not then there are. Is this reasoning flawed? Or, if the probability does not fall (all the way) to zero then does it just mean there may be infinitely many?

If the density of the sequence goes to zero, then there may or may not be infinitely many elements. The squares example (or the primes themselves) have density going to zero, yet there are infinitely many of them. Any finite set will have it's density going to zero. If you have at least one element, then the density never actually equals zero. For example the probability of selecting an even prime from {1, 2, ..., x} is 1/x for all x. No matter how big x is, it's never zero.

If it does not go to zero, then there must be infinitely many of them.

 

[Canute]

If the density of the sequence goes to zero, then there may or may not be infinitely many elements.

Hmm. I thought perhaps that it was possible to show that the probability of N+/-1 being (both) prime never becomes zero, and that this would mean there are infinitely many twin primes, but it looks like this approach will not work.

I think I need a bit of time to digest what I've learned here. It has all been very helpful, and thanks for taking the time to explain so much. Can we leave it here for the moment, as far as twin primes go? I'll come back with some more questions when I've given more thought to what you've said so far.

Would you be able to clarify for me the relationship between the non-trivial zeros of RH's function and the distribution of primes? Or do I need to know more mathematics before you can do that? Also, am I right to suppose that RH must be proved by reference only to the zeta function itself, or would the proof have to depend in some way on the actual behaviour of the primes?

 

[shmoe]

Hmm. I thought perhaps that it was possible to show that the probability of N+/-1 being (both) prime never becomes zero, and that this would mean there are infinitely many twin primes, but it looks like this approach will not work.

It looks to me like the density will be going to zero, but the heuristic argument could be off (though I would doubt it to be that off). You might want to search around to see how far people have gone with an exhaustive effort on primorial primes (the N+/-1) and factorial primes (these a n!+/-1). The P<12000 was a few years old I think. More data would be good to compare the heuristic against.

Would you be able to clarify for me the relationship between the non-trivial zeros of RH's function and the distribution of primes? Or do I need to know more mathematics before you can do that? Also, am I right to suppose that RH must be proved by reference only to the zeta function itself, or would the proof have to depend in some way on the actual behaviour of the primes?

See https://www.physicsforums.com/showthread.php?t=88468 for now, the last post explains the "explicit formula" (<-words to google) a little bit. That gives it in the simplest version for you, in terms of the function that counts primes as well as prime powers with a logarithmic weight rather than the usual pi(x). The counting function is more complicated, but the right hand side is much simpler that what you would have seen in Derbyshire's book. Sadly the link I give at the bottom doesn't seem to work right now, I'll hopefully be able to find a new one.

https://www.physicsforums.com/showthread.php?t=73459 explains a little bit about the equivalence of the error term in the prime number theorem vs locations of zeros.

Also search around this forum, this has been talked about more than just those posts. I'm taking the lazy route of links now, but I'll be able to expand more later.

RH could theoretically be proved with no reference to the zeta function itself. The theory of the zeta function does rely on the euler product, partly a consequence of unique factorization, so most of it can't be considered independant of the primes. However, this is really the only location it turns up, and any zeta results you can prove by first proving something about the prime distribution are much weaker than you get with the complex theory itself.

 

[mathwonk]

this reminds me of that show "who wants to be a billionaire?", put on by the same people who made candid camera i think.

 

[Nimz]

Cool stuff here.

Regarding the probability going to zero, I have two examples that illustrate opposite extremes. The first is the set {1, 2, 3}. If you look at the probability that a natural number below x is in that set, the probability will get as close to zero as you could desire, but it would still never actually reach zero, despite being a finite set.

The other example is the probability that a real number will be a natural number {1, 2, 3, ...}. Here the probability is exactly zero that a given number will be a natural number, regardless of the interval. However, there are infinitely many natural numbers. This is due to the fact that real numbers form an uncountable infinite set, whereas natural numbers form a countable infinite set.

These examples are intentionally ludicrous - to demonstrate that you cannot say anything about the size of a set based on probabilities.

What caught my eye more than anything else, though, was the mention of Fourier with the primes. I once looked at the sum over primes of (1/p)(sin p). It probably would not be very pleasant to listen to, though, with all the discontinuities.

 

[Canute]

this reminds me of that show "who wants to be a billionaire?", put on by the same people who made candid camera i think.

Yeah, I thought mention of $s would liven things up. I wish I hadn't done it now.

 

[Canute]

Thanks for the links. I should have searched the threads before posting anything. I came across this.

"The first instance of this to be observed involved the Selberg trace formula (discovered in the 1950's) which concerns the geodesic flow on a Riemann surface, relating its periodic orbits and its energy levels, i.e. eigenvalues of the Laplace-Beltrami operator. Here the orbits correspond to the primes and the energy levels to the Riemann zeta zeros. The latter correspondence lends credence to the spectral interpretation of the Riemann zeta function, and the overall situation suggests the existence of some kind of mysterious dynamical system underlying (or "lurking behind" as N. Snaith put it in her Ph.D. thesis) the distribution of prime numbers."

This is the sort of comment that confuses me. What is mysterious about the dynamical system lurking behind the distribution of primes? It doesn't seem mysterious to me, so what am I overlooking?

"The wider phenomenon of correspondence between the explicit formulae of number theory (of which the Riemann-Weil formula is just one, important, special case) and dynamical trace formulae points to some fundamental issue of duality which is currently a great mystery, and may turn out to be hugely significant in our understanding of both mathematical and physical reality."

What is this issue of duality? Or is it inexplicable at my level of mathematics?

 

[shmoe]

This is the sort of comment that confuses me. What is mysterious about the dynamical system lurking behind the distribution of primes? It doesn't seem mysterious to me, so what am I overlooking?

I don't think you have the same definition of a dynamical system. I don't know this kind of stuff well enough to offer anything resembling a plain english explanation, so this will be very vague.There is some hope that the zeta function will be attatched to some kind of dynamical system with some nice properties. If we knew what this dynamical system was (we don't know, though there are many clues to it's behavior), the hope is a proof of the riemann hypothesis will follow. I said it would be vague.

What is this issue of duality? Or is it inexplicable at my level of mathematics?

The explicit formula for the prime counting function is an example of this kind of duality. We have two objects, the primes and the zeros of zeta, and we have a connection between sums over these objects. This allows you to translate results about one of these objects to the other. For example you can go back and forth between smaller error term for the prime number theorem and larger zero free regions for zeta. Historically we 've always used an improved zero free region to prove a sharper error term, though the reverse is theoretically possible.

 

[Canute]

But the quote says a dynamical system underlies the distribution of prime numbers, not the zeta function.

 

[shmoe]

But the quote says a dynamical system underlies the distribution of prime numbers, not the zeta function.

See the 'duality' bit again, these things can't be seperated. If something is connected to zetas zeros, then it's connected to the primes. Here the connection between the primes and this hypothetical dynamical system lies through the zeta function- it will be more 'natural' to connect this system (which we don't even know exists) with the zeta function than directly to the primes.

 

[Canute]

That may be so, but it doesn't seem to alter the original statement except to say that this supposed dynamical system underlies both the distribution of primes and the zeta zeros. It still says that such a system might underly the primes, and presumably would have done so long before the zeta function was invented. I still find the statement odd.

Do mathematicians mean the same by 'dynamical system' as physicists?

 

[shmoe]

Yes, same as physicists.

Maybe I should ask you, if you don't find it mysterious, please explain it to us!

 

[Canute]

Well, I'm sure this will be a misunderstanding on my part and this system is mysterious after all, but this is what I meant.

The distribution of primes is caused by the distribution of multiples of primes. (I.e if a number n at 6n+/-1 is ~prime then it is because it is a multiple of a prime at or below sqrtn). Only two in every six multiples of a prime p have any effect on the distribution of primes above 6p. These multiples can be predicted. (The quantity of relevant multiples of a prime p in a range R of numbers is R/3p. E.g. the quantity of multiples of 101 occurring at 6n+/-1 between 606 and 1212 is 2). In calculating pi(x) the complication is that these multiples cannot simply be summed. A correction term is required (because many non-primes at 6n+/-1 are multiples of more than two primes). This correction term is complex, but I can't see what's mysterious about the mechanism.

 

[shmoe]

That is not a 'dynamical system'. What you've just described is how a basic sieve works. The correction term can be worked out (I mentioned 'inclusion-exclusion' before) and allow you to find pi(x) given say a list of primes less than sqrt(x), this has been known all the back to Legendre, sieves themselves back to Eratosthenes. This is definitely not what Snaith is referring to as mysterious.

I do have some idea how the zeros are supposed to be related to some unkown Hermitian operator (i.e. the ramndom matrix theory stuff), but I don't think I have a chance at properly explaining how this operator will be connected to a dynamical system. There's the paper by Berry and Keating "The Riemann-Zeros and Eigenvalue Asymptotics", SIAM Review, vol 41, no. 2, pp 236-266 that goes into some detail on what this dynamical system will probably look like, I've been meaning to give it a thorough read but haven't got around to it yet.

 

[Canute]

As usual I'm not understanding something here. The original comment I quoted speaks of the primes, not the zeta zeros. I realize that what I described does not constitute a dynamic system, nevertheless it is the rule or mechanism that determines the position of primes. If this mechanism is not the system spoken of, then how can there be another one? Any dynamic system would have to produce the same outputs.

Does the quoted remark mean that this mechanism can be modeled as a dynamic system?

 

[shmoe]

Sure, it's a rule that determines where the primes are, but remember one of the goals of number theorists is to improve the error term in the prime number theorem. Sieves and other elementary methods (essentially meaning no complex analysis) have produced some results, but nothing like we can do with the zeta function as far as the error term goes.

Yes, zeros and primes are the same thing in a sense, so if you find out what's up with one of them you know about the other. In this sense this dynamical system would tell you about the primes, and can be thought of the thing controlling them (though you could probably think of the primes as controlling the dynamical system), so you could think of it as 'the same thing' as what you've described in a way.

However, if you ran into this mythical dynamical system in a back alley it would probably be impossible to draw a connection between it and primes without the zeta function in hand to translate back and forth. This is the sense that I mean the dynamical system is more naturally attatched to zeta and not the primes.

 

[Canute]

Hmm. I still can't understand why this system is said to be mysterious, but never mind.

I think I need to stop here and go away to think some things through, now you've explained some of the mathematics to me. Thanks for all your help and patience. I'll put the rest of my thoughts in better order and may be back to ask you some more questions. I can at least now see the direction I need to head in.

Many thanks
Canute

 

[shmoe]

Hmm. I still can't understand why this system is said to be mysterious, but never mind.

They don't even know what the dynamical system they are looking for is, let alone if it even exists, that is the mystery. If they could find this system like they hope, the riemann hypothesis would be solved, that is how transparent this hypothetical dynamical system is expected to make things.

Many thanks
Canute

You are welcome.