Shifting the ordinates of zeros of the Riemann zeta function

TL;DR

Under the Riemann Hypothesis, the paper proves that for large T, there exists a zero of the zeta function whose ordinate can be shifted by a small amount y within a specific interval, demonstrating a form of zero distribution flexibility.

Contribution

The paper establishes a new result on the distribution of zeros of the Riemann zeta function, showing the existence of zeros with shifted ordinates within certain intervals under the Riemann Hypothesis.

Findings

Existence of zeros with shifted ordinates in specified intervals

Zero distribution properties depend on parameters y and C

Results hold assuming the Riemann Hypothesis

Abstract

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