A Derivation of $pi(n)$ Based on a Stability Analysis of the Riemann-Zeta Function

The prime-number counting function $pi(n)$, 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.