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.

\[ \Pi(x)=\lim_{\varrho,\varepsilon\rightarrow\infty}\Pi_\varepsilon^\varrho(x) \ \text{where} \ \varrho,\varepsilon \in \mathbb{N} \]

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