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Sine Waves and Primes 
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#1
Feb1506, 12:42 PM

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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. 


#2
Feb1506, 04:31 PM

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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. 


#3
Feb1606, 05:17 AM

P: 1,499

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. 


#4
Feb1606, 06:56 AM

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Sine Waves and Primes
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'.



#5
Feb1606, 10:43 AM

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#6
Feb1706, 06:28 AM

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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 realise 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 occured 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. 


#7
Feb1706, 08:14 AM

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#8
Feb1706, 10:13 AM

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Ps. Pardon the idiotic spelling of Eratosthenes. 


#9
Feb1706, 11:18 AM

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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 


#10
Feb1706, 12:01 PM

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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.). 


#11
Feb1706, 11:30 PM

P: 555

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 


#12
Feb1806, 06:11 AM

P: 1,499

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.



#13
Apr906, 08:51 AM

P: 501

for the primes if you define the formula:
[tex] G(x)=\sum_{p}x [/tex] then if you define [tex] I(x)=G(x)G(x1) [/tex] it presents two "magical" features: a)all the primes [tex] p_n [/tex] are fixed point of it in the form that if p is prime then [tex] I(p)=p [/tex] for every prime. b)if n is composite then [tex] I(x)=0 [/tex] root of it. If PNT is correct then [tex] G(x)\sim{li(x^2)} [/tex] this is obtained from the properties of the integral of Logarithmic integral "li". 


#14
Mar2211, 12:54 PM

P: 205




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