One hundred years uniform distribution modulo one and recent applications to Riemann’s Zeta-function

Abstract

We start with a brief account of the theory of uniform distribution modulo one founded by Weyl and others around 100 years ago (which is neither supposed to be complete nor historically depleting the topic). We present a few classical implications to diophantine approximation. However, our main focus is on applications to the Riemann zeta-function. Following Rademacher and Hlawka, we show that the ordinates of the nontrivial zeros of the zeta-function ζ(s) are uniformly distributed modulo one. We conclude with recent investigations concerning the distribution of the roots of the equation ζ(s) = a, where a is any complex number, and further questions about such uniformly distributed sequences.