Heisenburg Uncertainty Principle makes a prime number function impossible...

First you can set up something like the following...

The way it's arranged it makes it so it is 0 when a number in the range is composite. The integer peaks are at 101, 103, 107, and 109 which are the prime numbers between 100 and 110.
So that might make you wonder if there is some function that does a better job than this one, like it has peaks of 1 at all the prime numbers and is 0 elsewhere... In the frequency domain a function like that would look like:

This is how it would look with just frequency components corresponding to the first 5 primes, you can see the peak at 1/7 in order to make a signal that repeats every 7 numbers is already having to be compressed to fit in that space. The more frequency components you would add to correspond to larger primes the narrower the frequency band has to become to fit in that space.
  This is where the Heisenburg uncertainty principle comes in, it says the narrower these bands are in the frequency domain the less certain the function is in the time domain. Meaning there couldn't be a function with information about every prime in the frequency domain that gave any information at all in the time domain. Eventually the uncertainty results in so much error as to showing which number is prime that it becomes useless.
  So there can't really be a prime number function that does something like taking on a 1 value for primes and 0 otherwise.