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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. Odlyzkoconjectured this to be true. This is far from being proved, even assuming theRiemann 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 thePrime Number Theorem is sufficient to prove that the mean value is $0$, but therate 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 alsoput forward the interesting question: will the uniform distribution of thenormalized zeros be compatible with the GUE hypothesis?  Let $\rho=\frac12+i\alpha$ run through the complex zeros of zeta. We do notassume the RH so that $\alpha$ may be complex. For $0<\kappa<\frac65$ we provethat \[\lim_{T\to\infty}\frac{1}{N(T)}\sum_{0<\Re\alpha\leT}e^{2i\kappa\vartheta(\alpha)}=0\] where $\vartheta(t)$ is the phase of$\zeta(\frac12+it)=e^{-i\vartheta(t)}Z(t)$.

On the distribution (mod 1) of the normalized zeros of the Riemann Zeta-function