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

Michael Harney, Ioannis Haranas
2010 unpublished
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.