Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415386 | Journal of Number Theory | 2015 | 17 Pages |
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 1Nânâ¤Nexpâ¡(2Ïiκxn) 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 Ï=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 thatlimTâââ¡1N(T)â0