I have this function:
$$f(x) = \prod_{p\text{ is prime}} \left(1 - \frac{x^2}{p^2}\right)$$
Now, this function can be said to be an infinite degree polynomial with zeros on each of the primes and their negatives and nowhere else.
Properties of $f(x)$ that I'm aware of:
- $f(x)$ is an even function.
- The zeroes of $f(x)$ are the set of prime numbers union the set of their negatives.
- $f(x)$ is continuous.
- $-\infty < f(x) < \infty \quad {\small \text{This stems from the previous two properties.}}$
Now, what I have so far is the following:
$$ \hat{f}(\xi) = \int_{-\infty}^{\infty} e^{-2\pi i x\xi}\left(\prod_{p\text{ is prime}} \left(1 - \frac{x^2}{p^2}\right)\right)\ dx $$
I can't manage to get any further.
Is there a way to get a closed form for the Fourier transform of this function?
Additional information:
$f(x)$ is known to have at least one non-zero point.
$$f(1) = \zeta(2)^{-1} = \frac{6}{\pi^2} \approx 0.6079$$