Can a Generalized Function for f(n) Reveal Insights into Prime Distribution?

[Edwin]

Define the following funtion f(n) = the finite product of sin(pi*C/n) from n = 2 to n, where C is any integer >= 2. As it turns out, for each integer C, the product terminates to zero at n the smallest prime factor of C.


For example, suppose you consider C = 3

f(2) = sin(pi*3/2) = 1
f(3) = sin(pi*3/2)*sin(pi*3/3) = 0
f(4) = sin(pi*3/2)*sin(pi*3/3)*sin(pi*3/4) = 0
.
.
.
f(n) = 0 for any integer greater than 2

If you define F as the family of functions of the form above for all positive integers C >= 2, then the set of all those functions will have positive values for integers n in the closed interval [2, "smallest prime factor of C"), and will be zero for all other integers "greater than the smallest prime factor of C." The question I have is as follows:

n! (n factorial) is defined for all integer n greater than or equal to 0. However, the gamma function is considered the generalized factorial function, because it reduces to n factorial for positive integers. I was wondering if it is possible to construct a gamma-like function for f(n), that is, a “generalized function for f(n)” with C being a free parameter?

Inquisitively,

Edwin

p.s. would such a function have any connection to the Riemann Zeta Function?

 

[eljose]

A beatiful example of prime distribution is given by the identity:

\zeta (s) = \prod _ p (1-p^{-s})^{-1}

From the language of Physics this is equal to the fact that if we consider a NOn-interacting Phonon Gas (Phonon== Photons) in which every single particle is subjected to an Harmonic potential Cw(k)x^2

In that case... the "frecuencies" are w(k)=\hbar log (p_k ) where p(k) is the k-th prime...


This would be wonderful if the "riemann gas" Really existed... HOwever we can,t use X-ray scattering or other experimental tool to get the "frecuencies" and hence the primes..sigh!

 

[Edwin]

One of the things I am interested in is looking for connections between prime numbers, or primality, and physical phenomenon. Along those lines, if one could find a connection between prime numbers and something like Bessel's functions, might that enable one to connect prime numbers to physical phenomenon in a meaningful way?

Inquisitively,

Edwin

 

[Hurkyl]

One of the things I am interested in is looking for connections between prime numbers, or primality, and physical phenomenon.

Magicidada. Though I don't know if that's the sort of thing you're looking for.

Anyways, if you're looking for physical things, this thread might be better off in one of the physics forums. Lemme know if you want me to move it.

 

[Edwin]

Hurkyl,

Thanks! That is a good idea; please do move this thread to one of the physics threads. It would be very interesting to hear what perspectives and insights physicists and engineers might have into possible connections between physical phenomenon and prime numbers. I remember reading somewhere about a connection found between quantum physics (pertaining to nuclei of large atoms on the periodic table), prime numbers, and the Riemann Zeta function. Are you familiar with any work like this?

Inquisitively,

Edwin

 

[eljose]

Hello edwin..as you pointed before there is a connection,...it's clearly explained at the webpage (Wikipedia):

http://en.wikipedia.org/wiki/Hilbert-Pólya_conjecture

With several examples, I'm a physicist but also interested in Number theory and prime distribution...

 

[BoTemp]

Edwin, the most obvious generalization to me would be allowing n to take any value greater than 2, and formulating the product from the set S = {n, n-1, n-2, ..., n - floor(n) + 2}. f(n,C) will have zeros, whenever s contained in S divides C. You could've used the phrase "smallest factor" instead of "smallest prime factor", they'll always be the same.

Also, don't know if you care, but this function will not be uniformly positive. sin(3pi/2) = -1.

As far as I know there aren't any known phenomena that depend explicitly on prime numbers, unless you count those which are related to the zeta function. That pops up in physics occasionally. I've seen zeta(4) for blackbody radiation spectrum, and zeta(3/2) for BE condensate critical temperature, but that's all I know.

 

[HallsofIvy]

Just out of curiosity, what in the world does (phonons== photons) mean?
I know what phonons are and I know what photons are but I just can't make sense out of that!

 

[lokofer]

Just out of curiosity, what in the world does (phonons== photons) mean?
I know what phonons are and I know what photons are but I just can't make sense out of that!

"Photon" is a Quantum of light, "Phonon" is defined in a similar manner but It's a Quantum of "sound" (not exactly) they are produced by the lattice when vibrating under an "Harmonic" or similar potential...

- Dispersion relation \omega (k) = log(p_k ) "k" is equivalent to the classical momentum although \hbar k = p= \frac{2 \pi \hbar }{\lambda} where "Lambda" is the wave function of the system... a "peculiarty" of Riemann Gas (if existed) is that a "big prime" means short wavelenght, and for short wavelenghts is where "Quantum Effects" appear, so we could say that .."The Quantum Physics of the riemann Gas is what dictates the behavior of the k-th prime for k-->oo ", in other words, prime distribution is given by a Quantum effect...since for k=0 then \omega (0) =0 we're dealing with "acoustic phonons"...there're lots and lots of method to obtain these dispersion relations...unfortunately they're all experimental, so you need some "portion" of the gas to apply scattering...but Has anyone ever seen a bottle saying "Riemann gas" ..very dangerous not inhalate...:bigrin: :bigrin: