# Hints that a dynamical system may lie behind the distribution of primes

## Main Question or Discussion Point

http://secamlocal.ex.ac.uk/~mwatkins/zeta/NTfourier.htm" [Broken]

This is along the lines of what I have suspected about the primes that there is something there that is far deeper and has a real impact on both math in general and physical reality.

Last edited by a moderator:

Related Linear and Abstract Algebra News on Phys.org
- There is an even more "amazing" fact, it's proved that there is a "Harmonic model of gas" so its total partition funciton Z is equal to the riemann Zeta function..(If Bosons) and $$\zeta(2s) / \zeta(s)$$ (If Fermions).. the frecuencies of every particle (infinitely many of them) is $$\hbar \omega (k) = log(p_k )$$ k=1,2,3,4,5,... (primes) this is called the "Riemann Gas"...

- There is an even more "amazing" fact, it's proved that there is a "Harmonic model of gas" so its total partition funciton Z is equal to the riemann Zeta function..(If Bosons) and $$\zeta(2s) / \zeta(s)$$ (If Fermions).. the frecuencies of every particle (infinitely many of them) is $$\hbar \omega (k) = log(p_k )$$ k=1,2,3,4,5,... (primes) this is called the "Riemann Gas"...
I can't find much on that but would love to learn more do you have links to some references?  Unfortunately "Playdo" i myself am stuck in this problem..  you could try to learn something about "Statistical Physics" (involving partition function) at Wikipedia    Unfortunately "Playdo" i myself am stuck in this problem..  you could try to learn something about "Statistical Physics" (involving partition function) at Wikipedia  So do you usually make statments of fact about things you cannot completely prove? I mean it is one thing to be armchair and point to someone elses clearly written work, but to simply say I think this is true but can't prove it. At least make an argument showing why you think it might be true or even what you really mean.

Office_Shredder
Staff Emeritus
Gold Member
I think he means that he knows it's true, but doesn't know how to prove it. Example, it's a question on a homework assignment

I could be wrong though

HallsofIvy
Using Solid state (i recommend you "Ashcorft & Mermin : SOlid State Physics) using the definition of Partition function and specific Heat.. i've been able to recover the Integral equation involving $$\pi (e^{t})$$ (precisely the inverse of the k-th frequency) $$\omega (k) = log(p_k)$$ ,unfortunately this does not simplify the problem.. what i have asked is if there would be a method knowing the "Entropy" , "gibbs function" or similar ,which can be calculated knowing the partition function, and from this to get the density of states (in 1-D is just the inverse of the derivative $$\frac{d\omega}{dk}$$ multiplied by a constant, if we were able to calculate the "speed of sound " for the lattice or density of states we could calculate every prime..at the moment the only chance would be to use X-rays (if we had a portion of the Riemann gas of course) to calculate the frequencies... of course this is impossible since Riemann gas does not exist 