Symmetry in Partial Sums of $n^{-s}$

Abstract

A detailed, internal symmetry exists between individual terms $n^{-s}$, where $n \in P$ is less than a particular value $n_p$, and sums over conjugate regions consisting of adjoining steps $n$ greater than $n_p$. The boundaries of the conjugate regions are where first angle differences $\delta \theta_n = -tlog((n+1)/n)$ equal odd multiples of $\pi$. Two significant points in the complex plane are defined by this symmetry: O'(s), conjugate to the origin O, and which equals $\zeta(s)$ for $\sigma \in (0,1)$; and $P(s)$, conjugate to itself, which gives Riemann's correction to the discrete sum in the Riemann-Siegel equation. The distances from P to O and P to O' are equal only for $\sigma = 1/2$, where superposition of O and O' results under the single-parameter condition that $\angle OP$ and $\angle PO'$ are opposed. Analysis of this symmetry allows an alternate understanding of many of the results of number theory relating to $\zeta (s)$, including its functional equation, analytic continuation, the Riemann-Siegel equation, and its zeros. Discussion of three explicit computational algorithms illustrates that the apparent peculiarity of the occurrence of zeros when $\sigma = 1/2$ is removed by direct recognition of the symmetry.