Weierstrass theorems and primes.

[tpm]

Does 'Weirstrass theorem' allow the existence of an entire function so:

$$f(z)= g(z) \prod _p(1- \frac{x}{p^{k}})$$

so for every prime p then f(p)=0 , and k>1 and integer??

the main question is to see if a function can have all the primes as its real roots

 

[matt grime]

Of course a function can have the primes, and only the primes, as its only roots, although you clearly want the function to be analytic, don't you, eljose?