Geometry of the Riemann Zeta Function

TL;DR

This paper explores the geometric structure of the Riemann zeta function in the complex plane, revealing symmetries, a new interpretation of analytic continuation, and conditions for zeros, with implications for number theory.

Contribution

It introduces a geometric perspective on the zeta function, including a limacon-based construction and analysis of prime step effects, advancing understanding of zeros and analytic continuation.

Findings

Identifies symmetry in the zeta function's geometric representation

Proposes a limacon-based model for analytic continuation

Highlights the role of prime steps in zero distribution

Abstract

The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional results of analytic number theory. A new interpretation of analytic continuation allows a construction, based on the limacon, which shows the conditions under which zeros can occur. The special importance of prime steps in the decrease of the real part from 1 to 0 when zeta equals zero is shown. Discussion of bounds and extension to generalized zeta functions are included.