%0 Journal Article
%T A Derivation of $pi(n)$ Based on a Stability Analysis of the Riemann-Zeta Function
%A Harney M.
%A Haranas I. I.
%J Progress in Physics
%D 2010
%I HEXIS (Arizona)
%X 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.
%K Riemann-zeta function
%K prime numbers
%K stability analysis
%U http://ptep-online.com/index_files/2010/PP-21-12.PDF