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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Distribution of Primes

**Physics Forums | Science Articles, Homework Help, Discussion**