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. Applying the Piatetski-Shapiro 11/12 Theorem we show that, for 0<κ<6/5, the mean value 1N∑n≤Nexp⁡(2πiκxn) tends to zero. In the case κ=1 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. Let ρ=12+iα run through the complex zeros of zeta. We do not assume the RH so that α may be complex. For 0<κ<65 we prove thatlimT→∞⁡1N(T)∑0<Reα≤Te2iκϑ(α)=0 where ϑ(t) is the phase of ζ(12+it)=e−iϑ(t)Z(t).