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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:

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A semi-log plot of $|f(x)|$ shows that the growth is superlinear but apparently subexponential.

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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 $p=7919$.)

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    Kind of reminds of the prime counting function, but not quite. :D – 2017-01-10
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    Come to think of it, $\exp(x/\log(x+1))$ does match the growth rather well: https://i.stack.imgur.com/hDvJz.png – 2017-01-11
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    Not about asymptotes, but related: https://en.wikipedia.org/wiki/Matsumoto_zeta_function – 2017-01-11
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    Hm, what do you mean by "What is the asymptotic rate of growth of the oscillations?"? Are you asking for the asymptotic behavior of the top of the humps? – 2017-01-11
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    Yes, exactly that. – 2017-01-11

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