  We consider the problem whether the ordinates of the non-trivial zeros of
$\zeta(s)$ are uniformly distributed modulo the Gram points, or equivalently,
if the normalized zeros $(x_n)$ are uniformly distributed modulo 1. Odlyzko
conjectured this to be true. This is far from being proved, even assuming the
Riemann hypothesis (RH, for short).
  Applying the Piatetski-Shapiro $11/12$ Theorem we are able to show that, for
$0<\kappa<6/5$, the mean value $\frac1N\sum_{n\le N}\exp(2\pi i \kappa x_n)$
tends to zero. The case $\kappa=1$ is especially interesting. In this case the
Prime Number Theorem is sufficient to prove that the mean value is $0$, but the
rate of convergence is slower than for other values of $\kappa$. Also the case
$\kappa=1$ seems to contradict the behavior of the first two million zeros of
$\zeta(s)$.
  We make an effort not to use the RH. So our Theorems are absolute. We also
put forward the interesting question: will the uniform distribution of the
normalized zeros be compatible with the GUE hypothesis?
  Let $\rho=\frac12+i\alpha$ run through the complex zeros of zeta. We do not
assume the RH so that $\alpha$ may be complex. For $0<\kappa<\frac65$ we prove
that \[\lim_{T\to\infty}\frac{1}{N(T)}\sum_{0<\Re\alpha\le
T}e^{2i\kappa\vartheta(\alpha)}=0\] where $\vartheta(t)$ is the phase of
$\zeta(\frac12+it)=e^{-i\vartheta(t)}Z(t)$.
