On an analytical formula suitable for counting the number of primes contained in a given interval. (Q1524790)

Sei: \[ \begin{aligned} S(x) &= \sum_{m=1}^\infty \frac{x^m}{1-x^m} - \frac{2x^2}{1-x} - x,\\ P(z) &= \frac1{2\pi i} \int_0^{2\pi} \fracwithdelims()12^z e^{i\theta z}S(\frac12e^{i\theta})d\theta;\end{aligned} \] seien \(\alpha\), \(\beta\) zwei nicht ganze Zahlen, \(\beta>\alpha>12\); sei \(\beta'=2^{-E(\beta)-2}\), so ist die Anzahl der Primzahlen im Intervall \((\alpha\dots\beta)\): \[ N_{\alpha\beta} = \frac{\beta'}{2\pi} \int_0^{2\pi} e^{i\theta} \sum_{h=E(\alpha)+1}^{E(\beta)} \frac{P'(\beta'e^{i\theta}-h)}{P(\beta'e^{i\theta}-h)} d\theta. \]