# Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Z*eta* Function to Re(*s*) = 1/2 and the Zeros of its Derivative to Re(*s*) > 1/2

Version 2 : Received: 22 March 2017 / Approved: 22 March 2017 / Online: 22 March 2017 (04:48:57 CET)

How to cite:
Lander, A. Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Z*eta* Function to Re(*s*) = 1/2 and the Zeros of its Derivative to Re(*s*) > 1/2. *Preprints* **2017**, 2017030121 (doi: 10.20944/preprints201703.0121.v2).
Lander, A. Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Zeta Function to Re(s) = 1/2 and the Zeros of its Derivative to Re(s) > 1/2. Preprints 2017, 2017030121 (doi: 10.20944/preprints201703.0121.v2).

## Abstract

*zeta*function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤1 Euler’s

*zeta*diverges. Dirichlet’s

*eta*function

*η*(

*s*), negates alternate terms of

*zeta*, permitting convergence when s∈C and Re(

*s*) < 1, and its non-trivial zeros {

*ρ*}, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re(

*ρ*) = 1/2. This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re(

*s*) ≠ 1/2. Analytical continuation generates a modified eta series

*η*(

_{l}*s*), in which every

*l*

^{th}term is multiplied by (1-

*l*). If the integer

*l*is appropriately determined by the Im(

*s*), similar paired finite vector series have a difference that closely follows

*η*(

_{l}*s*) and their terminal vectors intersect in a unique way permitting zeros only when Re(

*s*) = 1/2. Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re(

*s*) > 1/2.

## Subject Areas

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