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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.

Part I of this article includes: Introduction; A Bridge over Turbulent Waters; Chasing the Critical Points and their Parameter Planes; and A: The Additive World - 1: Far East - the Asymptotically-Critical Plateau; 2: Real Critical Points, from Miniscule to Vast; and (3) Shang-ri-La – The Unreal Criticals.

Part II of this article includes: B: The Multiplicative Universe; and Appendix: Fractal Geography of Eta, Xi and Dirichlet L-functions; and Introduction for Non-mathematicians.