Fractal Geography of Riemann Zeta and Related Functions

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. This paper completes a discovery process I began in 2009, using computational applications I had developed, looking at the ‘dark hearts ’ 1- the Mandelbrot parameter planes- of a wide variety of complex functions, including the zeta function, to explore the world of complex functions as widely as possible and elucidate universal properties. This year, as I began to re-explore th