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

• Playdo
In summary, the author believes that there is something deeper to the primes than what is currently known, and that it has a real impact on math and physical reality. He provides references to a Wikipedia article on statistical physics and solid state physics, which he claims can help explain the prime phenomenon.
Playdo
http://secamlocal.ex.ac.uk/~mwatkins/zeta/NTfourier.htm"

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.

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- 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 statements 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.

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

And everyone knows that a statement on a homework assignment can't possibly be incorrect!

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

## 1. What is a dynamical system in relation to the distribution of primes?

A dynamical system is a mathematical model that describes the evolution of a system over time. In the context of the distribution of primes, it refers to a system that generates the sequence of prime numbers in a predictable and deterministic way.

## 2. How can the distribution of primes be explained by a dynamical system?

The distribution of primes can be thought of as a pattern that emerges from a complex and dynamic system. This system involves the interaction of various mathematical concepts, such as number theory, and the behavior of prime numbers can be described using dynamical models.

## 3. What are some hints that suggest a dynamical system may be behind the distribution of primes?

Some of the hints include the existence of patterns in the distribution of primes, the fact that prime numbers can be generated using simple algorithms, and the relationship between prime numbers and other mathematical concepts, such as the Riemann zeta function.

## 4. How can studying dynamical systems help us understand the distribution of primes?

Studying dynamical systems can provide insights into the underlying mechanisms that govern the behavior of prime numbers. By analyzing the patterns and behaviors of these systems, we can gain a deeper understanding of the distribution of primes and potentially uncover new connections and relationships.

## 5. What are some potential applications of using dynamical systems to study the distribution of primes?

One potential application is the development of more efficient algorithms for finding and generating prime numbers, which could have practical implications in fields such as cryptography. Additionally, understanding the behavior of prime numbers could also lead to advancements in other areas of mathematics, such as number theory and algebraic geometry.

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