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

摘要

We consider the problem whether the ordinates of the nontrivial zeros of zeta(s) are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros (x(n)) are uniformly distributed modulo 1. Applying the Piatetski-Shapiro 11/12 Theorem we show that, for 0 < kappa < 6/5, the mean value 1/N Sigma(n <= N) exp(2 pi i kappa x(n)) tends to zero. In the case kappa = 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 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. Let rho = 1/2 + i alpha run through the complex zeros of zeta. We do not assume the RH so that alpha may be complex. For 0 < kappa < 6/5 we prove that