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 ζ(s) are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros (xn) 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 < κ < 6/5, the mean value 1 N P n≤N exp(2πiκxn) tends to zero. The case κ = 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 κ. Also the case κ = 1 seems to contradict the behavior of the first two million zeros of ζ(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 ρ = 1 2 + iα run through the complex zeros of zeta. We do not assume the RH so that α may be complex. For 0 < κ < 6 5 we prove that lim T→∞ 1 N(T) X 0<Re α≤T e 2iκϑ(α) = 0 where ϑ(t) is the phase of ζ( 1 2 + it) = e −iϑ(t)Z(t).