Fractal Geography of the Riemann Zeta Function

TL;DR

This paper explores the complex and fractal nature of the Riemann Zeta function by analyzing its Mandelbrot and Julia sets, revealing its chaotic and intricate geometric structures.

Contribution

It introduces a novel analysis of the Zeta function's Mandelbrot and Julia sets, uncovering its fractal and chaotic geography in complex function space.

Findings

Revealed the fractal structures of the Zeta function's Julia sets.

Mapped the chaotic regions of the Zeta function's spectrum.

Unified the understanding of Mandelbrot and Julia sets for Zeta.

Abstract

The quadratic Mandelbrot set has been referred to as the most complex and beautiful object in mathematics and the Riemann Zeta function takes the prize for the most complicated and enigmatic function. Here we elucidate the spectrum of Mandelbrot and Julia sets of Zeta, to unearth the geography of its chaotic and fractal diversities, combining these two extremes into one intrepid journey into the deepest abyss of complex function space.