Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# 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

Version 1 : Received: 15 March 2017 / Approved: 16 March 2017 / Online: 16 March 2017 (16:59:29 CET)
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 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). 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

Euler’s product formula over the primes and Euler’s 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 ηl(s), in which every lth 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 ηl (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

Riemann Hypothesis; Dirichlet eta function; Zeta function; prime numbers; number theory; critical-line; critical-strip; symmetry breaking

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