Oooh... you're getting into some serious math now.
The stable waves on a circle are precisely those waves which oscillate an integer number of times as they traverse. In other words, one for each integer. Further, every function on the circle can be expressed as a sum of the sine waves. That sum is called the spectral decomposition. (This is Fourier series.)
With clever choices of functions, you can get some profound results. For example, picking a saw-tooth wave and doing the spectral decomposition gives the identity
1 + 1/4 + 1/9 + 1/16 + ... = pi^2/6
And, by the way, the left hand side is the zeta function evaluated at 2.
And about functions that are zero at each composite... You may want to check out Dirichlet characters. They are periodic functions which behave nicely under multiplication. Whenever an integer and the period have a common factor, the character will be zero at that integer.
It's not going to be zero at all composites, but it's on the right track.
The stable waves on a circle are precisely those waves which oscillate an integer number of times as they traverse. In other words, one for each integer. Further, every function on the circle can be expressed as a sum of the sine waves. That sum is called the spectral decomposition. (This is Fourier series.)
With clever choices of functions, you can get some profound results. For example, picking a saw-tooth wave and doing the spectral decomposition gives the identity
1 + 1/4 + 1/9 + 1/16 + ... = pi^2/6
And, by the way, the left hand side is the zeta function evaluated at 2.
And about functions that are zero at each composite... You may want to check out Dirichlet characters. They are periodic functions which behave nicely under multiplication. Whenever an integer and the period have a common factor, the character will be zero at that integer.
It's not going to be zero at all composites, but it's on the right track.