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

TL;DR

This paper uncovers a symmetry in the partial sums of the Riemann zeta function, providing new insights into its properties, functional equation, and zeros, with implications for understanding the distribution of primes.

Contribution

It introduces a novel symmetry in the partial sums of $n^{-s}$, offering an alternative perspective on key results related to the Riemann zeta function and its zeros.

Findings

Symmetry relates sums over conjugate regions in the complex plane.

Distances from key points are equal only at $\sigma=1/2$, explaining the critical line.

Algorithms demonstrate the symmetry clarifies the zeros' occurrence.

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.