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The exact behaviour of the Reimann Zeta conjugate pair ratio function

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posted on 2016-07-22, 15:01 authored by John MartinJohn Martin
Graphical examination of the Riemann Zeta conjugate pair ratio function ζ(s)/ζ(1−s) reveals a simple AM-FM lineshape for the real and imaginary components with (i) the FM terms present in the exact terms of the Riemann Siegel θ ( t ) function and Γ(0 . 5 + it ) and (ii) only the AM term is dependent on the distance of the real value of s from 0.5. The envelope function abs (ζ(s)/ζ(1−s) ) is smooth and increases (decreases) with increasing distance from the critical line for Re ( s ) < 0 . 5 ( Re ( s ) > 0 . 5) indicating no zeroes away from the critical line in agreement with the Riemann Hypothesis.