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A Derivation of π(n) Based on a Stability Analysis of the Riemann-Zeta Function
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by
Michael Harney,
Ioannis Haranas
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2010
Abstract
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.
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