Department of Physics and Computer Science
The prime-number counting function π(n), which is significant in the prime number theorem, is derived by analyzing the region of convergence of the real-part of the Riemann- Zeta function using the unilateral z-transform. In order to satisfy the stability criteria of the z-transform, it is found that the real part of the Riemann-Zeta function must converge to the prime-counting function.
Harney, M., Haranas, I. A Derivation of π(n) Based on a Stability Analysis of the Riemann-Zeta Function. Progress in Physics 2010(2), 55.