Inspired by Euler's solution to the Basel problem, consider the function $$\begin{align} f(x) &= \left(1-\frac{x^2}{2^2}\right)\left(1-\frac{x^2}{3^2}\right)\left(1-\frac{x^2}{5^2}\right)\cdots \\ &= \prod_{\text{$p$ prime}}\left(1-\frac{x^2}{p^2}\right). \end{align}$$ This is a function whose zeros are precisely the prime numbers and their negations. Its oscillations grow quite rapidly:
A semi-log plot of $|f(x)|$ shows that the growth is superlinear but apparently subexponential.
Question: What is the asymptotic rate of growth of the oscillations?
(N.B. To create the plots I approximated the function using the first $1000$ primes, i.e. up to $p=7919$.)