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

Abstract

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 $\zeta(\frac12+it)=e^{-i\vartheta(t)}Z(t)$.