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

  • Thread starter Playdo
  • Start date
  • #1
88
0

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:

Answers and Replies

  • #2
131
0
- 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 [tex] \zeta(2s) / \zeta(s) [/tex] (If Fermions).. the frecuencies of every particle (infinitely many of them) is [tex] \hbar \omega (k) = log(p_k ) [/tex] k=1,2,3,4,5,... (primes) this is called the "Riemann Gas"...
 
  • #3
88
0
Karlisbad said:
- 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 [tex] \zeta(2s) / \zeta(s) [/tex] (If Fermions).. the frecuencies of every particle (infinitely many of them) is [tex] \hbar \omega (k) = log(p_k ) [/tex] 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?
 
  • #4
131
0
:cry: :cry: Unfortunately "Playdo" i myself am stuck in this problem..:frown: :frown: you could try to learn something about "Statistical Physics" (involving partition function) at Wikipedia:confused: :confused:
 
  • #5
88
0
Karlisbad said:
:cry: :cry: Unfortunately "Playdo" i myself am stuck in this problem..:frown: :frown: you could try to learn something about "Statistical Physics" (involving partition function) at Wikipedia:confused: :confused:
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.
 
  • #6
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,750
99
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
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
41,833
955
And everyone knows that a statement on a homework assignment can't possibly be incorrect!
 
  • #8
131
0
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 [tex] \pi (e^{t}) [/tex] (precisely the inverse of the k-th frequency) [tex] \omega (k) = log(p_k) [/tex] ,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 [tex] \frac{d\omega}{dk} [/tex] 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...:redface: of course this is impossible since Riemann gas does not exist :frown:
 

Related Threads on Hints that a dynamical system may lie behind the distribution of primes

  • Last Post
Replies
8
Views
3K
Replies
1
Views
4K
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
6
Views
5K
  • Last Post
2
Replies
37
Views
8K
  • Last Post
Replies
2
Views
2K
Replies
20
Views
5K
Replies
1
Views
2K
  • Last Post
Replies
1
Views
3K
Top