  We use subsequence and moving average ergodic theorems applied to Boole's
transformation and its variants and their invariant measures on the real line
to give new characterisations of the Lindelh{\"o}f Hypothesis and the Riemann
hypothesis. These ideas are then used to study the value distribution of
Dirichlet L series, and the zeta functions of Dedekind, Hurwitz and Riemann and
their derivatives. This builds on earlier work of R. L. using Birkhoff's
ergodic theorem and probability theory.
