Sine Waves and Primes: Exploring the Connection Through Music

[Canute]

Hi folks. I'm back to make a fool of myself again. It seems that I have a proof that the 'music' of the primes is caused by the interaction of a collection of sine waves. However, I have no idea whether this is old hat, trivial or interesting. Can somebody here tell me?

PS. I don't know why this question appears twice. The other can be deleted but it won't let me do it.

 

[shmoe]

The "music of the primes" saying comes from the prime distribution being expressed as a sum of waves whose frequencies depend on the zeros of zeta (and other terms). Roughly, there's 'music' if these waves all have amplitudes growing at the same rate- equivalent to the riemann hypothesis (a zero off the critical line would correspond to a 'nasty' wave).

I guess I don't understand what it is you think you've proven because your claim that the 'music' is cause by an interaction of sine waves seems to be exactly the thing that caused the word 'music' to be introduced in the first place.

 

[Canute]

Thanks. I understand the point about the zeta function (very roughly). What I'm suggesting is not that the distribution of primes can be analysed in terms of sine waves, but rather that the occurence of primes can be shown to be caused by them. This is a slightly different claim. It may already be obvious to everyone, but this is what I don't know.

To put this another way - There is a simple function that outputs sine waves and in this way generates a complete list of primes up to any n. Of course, chances are this is a trivial result, it would be very surprising if it wasn't, but I'm having trouble confirming that it is.

 

[matt grime]

Any such function with the properties you imply it has, if it were to exist, would be a fundamental break through, assuming that 'simple' really were 'simple'.

 

[shmoe]

What I'm suggesting is not that the distribution of primes can be analysed in terms of sine waves, but rather that the occurence of primes can be shown to be caused by them.

While I would argue there's no difference in these claims, you could say this is already included in the explicit formula for the usual psi(x) or (pi(x)). Roughly it gives psi(x) as a sum of waves (infinitely many of them) plus a couple terms that are continuous. psi(x) is discontinuous at primes (and prime powers), so if you like, the discontinuities are coming from the interaction of the 'waves'. Whether you want to say these waves determine the primes or these waves are determined by the primes is up to you (I'm more inclined to view them as the same thing, primes and zeros of zeta are interrelated).

Of course, chances are this is a trivial result, it would be very surprising if it wasn't, but I'm having trouble confirming that it is.

It's rather difficult to judge if something is new, usefull, or otherwise interesting without actually having details on what it is. Based on your claim of a simple function that can be used to generate primes up to n, my best suggestion is to compare it with the usual prime generating functions (see http://mathworld.wolfram.com/PrimeFormulas.html for example) and perhaps how efficient your method is compared to the usual methods for generating lists of primes like the sieve of erathosthenes and improvements such as the wheel sieve.

 

[Canute]

Yes, I take the point about not giving enough details. It's unfair I know, but I want to be a bit cagey until I've got some idea of what it is that I've stumbled across.

I realize that psi(x) can be given by the sum of a collection of waves, if that's the right way to put it. But do we know why this works?

Although I've read du Sautoy and Derbyshire, and some bits and pieces online, I have never seen the mathematical mechanism that detirmines the pattern of the primes mentioned. Not being a mathematician I don't know whether this is because it is too trivial to mention or whether it is not known.

Matt, your comment suggests it is not known, but is this what you meant? (Yes, it is simple by the way, one shortish sentence even in plain English).

What I am able to do is generate a list of all the multiples of the primes that occur at 6n+/-1, with no starting list of primes and without any redundancy (without having to calculate any other numbers). This then leaves the primes as the blanks. This is not particularly useful, although it is a much more efficient method than that of Aristosthenes. But while it may not be all that useful for calculation purposes it does completely explain why the primes are distributed as they are. It occurred to me that being able to explain this might have some impact on the provability of Riemann hypothesis, but I have absolutely no idea whether this is true.

Pardon the waffle. I'll try to come back with something more clear and specific.

 

[shmoe]

I realize that psi(x) can be given by the sum of a collection of waves, if that's the right way to put it. But do we know why this works?

Sure, the explicit formula was proven long ago by von Mangoldt.

This is not particularly useful, although it is a much more efficient method than that of Aristosthenes. But while it may not be all that useful for calculation purposes it does completely explain why the primes are distributed as they are.

So it's much more efficient then Eratosthenes, yet not usefull for calculation purposes? This doesn't make sense, people generate lists of primes all the time for various reasons a faster method would be of great interest. What is the run time of your algorithm?

 

[Canute]

Sure, the explicit formula was proven long ago by von Mangoldt.

Would you have time to tell me a bit about this proof?

So it's much more efficient then Eratosthenes, yet not usefull for calculation purposes? This doesn't make sense, people generate lists of primes all the time for various reasons a faster method would be of great interest. What is the run time of your algorithm?

What I'm saying, I suppose, is that knowing why the primes behave as they do, and even having a simple algorithm for generating them, is not the same thing as being able to find them quickly. There are other methods that have been honed over centuries.

Ps. Pardon the idiotic spelling of Eratosthenes.

 

[shmoe]

Would you have time to tell me a bit about this proof?

Roughly you can express psi(x) as a contour integral involving zeta(s), actually there's a zeta'(s)/zeta(s) term in it. You can then evaluate this integral using Cauchy's residue theorem, you have poles at all the zeros of zeta(s) which is where the sum over the zeros comes from.

You can find full details in most analytic number theory texts as well as details in Riemanns original paper. You might want to check out Edward's book on Zeta, or Ingham's classic book on the distribution of primes, or really countless other references (I can give more if you like). Edward's is probably the most accessible, but it's still a textbook, and you'll need to know some complex analysis. This website looks like a pretty good rundown too:

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

What I'm saying, I suppose, is that knowing why the primes behave as they do, and even having a simple algorithm for generating them, is not the same thing as being able to find them quickly. There are other methods that have been honed over centuries.

So yours isn't more efficient then Erathosthenes? By "efficient" I mean the number of steps it would take to find the primes up to a given N, this isn't saying anything about how simple or complicated an algorithm is. It looks like your 'simple' doesn't imply 'fast'. If Erathosthenes doesn't fall under 'simple' I really don't know what would.

 

[matt grime]

What I meant was that any genuinely simple function that returned pi(x) would be a genuine breakthrough, where simple means is a nice expression involving elementary functions in few variables; the function f(n)=p_n, where p_n is the n'th prime is in one sense simple, yet is not 'simple' in another. You're now talking about algorithms for computing values, 'simple' algorithms abound, or rather, simplistic algorithms abound.

Just state what you're doing, rather than being vague, and explain why it is a 'simple' function that generates lists of primes. A function that generates lists of primes doesn't to me mean one that is algorithmic like a sieve, but one that takes an input value and performs a small number of algebraic operations (small can be quite big actually, and we allow for truncation of infinite series providing this works, since we only want integer values.).

 

[DaTario]

I have arrived at a function f_n(x) such that the solutions of f_n(x) = 0 are all the prime number from 2 to n. No explicit list of primes is included in the functions structure.

Somebody has told me that there are several functions like this one and their usefulness vary a lot, being just a small number of some use.

I have then published this function but in the humble form of a question in a math magazine.

best Regards

DaTario

 

[Canute]

I think I need to go away and think a few things through. I can see now that unless I provide all the details my questions are a rather meaningless. Thanks for the help, and sorry for being so vague. I'll ponder some more and return when I've got my act together. Bye for now.

 

[eljose]

for the primes if you define the formula:

$$G(x)=\sum_{p}x$$ then if you define $$I(x)=G(x)-G(x-1)$$ it presents two "magical" features:

a)all the primes $$p_n$$ are fixed point of it in the form that if p is prime then $$I(p)=p$$ for every prime.

b)if n is composite then $$I(x)=0$$ root of it.

If PNT is correct then $$G(x)\sim{li(x^2)}$$

this is obtained from the properties of the integral of Logarithmic integral "li".

 

[JeremyEbert]

I think I need to go away and think a few things through. I can see now that unless I provide all the details my questions are a rather meaningless. Thanks for the help, and sorry for being so vague. I'll ponder some more and return when I've got my act together. Bye for now.

Sine wave relation to primes:
http://tubeglow.com/test/Fourier.html