Shifting the ordinates of zeros of the Riemann zeta function

Abstract

Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_{\ast }>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_{\ast }$, both \[ ζ(\tfrac12+iγ)=0 \quad\text{and}\quadζ(\tfrac12+i(γ+y))\ne 0 \] hold for at least one $\gamma$ in the interval $[T,T(1+\epsilon )]$, where $\epsilon :=T^{-C/\log \log T}$.

15 pages; new introduction; some flaws have been fixed