Here is a prime counting function constructed by interfering trigonometric functions.
\[
\Pi_\varepsilon^\varrho(x) = \sum_{s=0}^{\varrho} \left(1 + e^{2\varepsilon\left(s - x\right)}\right)^{-1}
\max \left\{\sigma(\varepsilon,s), 0 \right\}
\\ \sigma(\varepsilon,s)=
\frac{\cos\left(\pi s\right)^{2\varepsilon}}{1 + e^{\varepsilon\left(6 - 4s\right)}} - \sum_{q=2}^{\varrho}
\frac{\cos\left(\frac{\pi s}{q}\right)^{2\varepsilon}}{1 + e^{\varepsilon\left(6q - 4s\right)}}
\]
In the positive infinite limit this parametrised approximation tends to the true PCF function.
\[
\Pi(x)=\lim_{\varrho,\varepsilon\rightarrow\infty}\Pi_\varepsilon^\varrho(x) \ \text{where} \ \varrho,\varepsilon \in \mathbb{N}
\]