I have been playing with the following function: $$ f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k} $$ It is hard to get correct numerical values. I'll start with the basic. Can you confirm that (1) we have absolute convergence, (2) the limit exists if $x$ is an integer, (3) if $x$ is a composite number then $f(x)=0$, and if $x$ is prime, $f(x)\neq 0$.

For simplicity, let's consider the absolute value of $f(x)$. Now the interesting part. If $x$ is not too close to a composite number, it sounds like $$ |f(x)|\sim \exp(-\lambda x) $$ for some $\lambda >0$, possibly $\lambda\approx 4.5$. Is there an asymptotic formula that can be easily derived? Since $f(x)=0$ if $x$ is composite, that formula would be valid only for some values of $x$, for instance if the fractional part is within some range, or if $x$ is prime, which seems to be where the formula works best.

If this was true, you could approximately compute the number of primes $<n$ as $$\pi(n)\approx \sum_{k=2}^n e^{\lambda k} |f(k)|. $$ The formula is useless for computational purposes, but I am wondering if it might have some theoretical interest. Anyway, my question is this: can you get some asymptotic formula as $x\rightarrow\infty$ depending on the fractional part of $x$ or if $x$ is prime? Mine might not be correct. Even better, what is the value of $\lambda$ assuming it ever exists?