Spirals of Riemann’s Zeta-Function — Curvature, Denseness and Universality

This paper deals with applications of Voronin's universality theorem for the Riemann zeta-function ζ. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values ζ(σ+it)for real t where σ∈(1/2,1)is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from ζ(σ+it)when σ>1/2and we show that there is a connection with the zeros of ζ'(σ+it). Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.