I have been attempting to look into functions in which satisfy the following properties:

$ $ F(p) = 0; \text{for all prime numbers } p $ $ $ $ F(x) \ne 0; \text{for all non-prime natural numbers } x $ $

I have attempted searching through this site as well as Google, but have not come across any functions that have these exact properties. Although, I am not 100% sure as to what would be a good search criteria besides the name of this question.

Of course, a trivial construction would be something like:

$ $ F(x)=1-\text{is_prime}(x) $ $

Where is_prime is some function that maps primes to one and non-primes to zero.

Are there functions out there that satisfy these properties? If so what are some of these functions and what articles/papers can I read on them?