Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$

Question

Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.

What will be the significance of proving that there is an $H$ such that for $T\geq H$, $$N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$$ and $$N(T+1)-N(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$$ Are the above results (especially the first one) known? Please explain the importance of the above results.

edit So if the above two results are proved we can conclude that $$\liminf_{T\to \infty}\frac{N_0(T+1)-N_0(T)}{N(T+1)-N(T)}=1$$ Thank you.

Answer 1

The first two displays (together) are in between the Lindelöf hypothesis and the Riemann hypothesis. That is, they imply the Lindelöf hypothesis, while they follow from the Riemann hypothesis. They are not known unconditionally. See Sections 13.5-13.6 in Titchmarsh: The theory of the Riemann zeta-function.

It is straightforward that the first two displays imply the third display in the stronger form $$\lim_{T\to \infty}\frac{N_0(T+1)-N_0(T)}{N(T+1)-N(T)}=1.$$ To see this, divide both the numerator and the denominator by $\frac{1}{2\pi}\log \frac{T}{2\pi}$, and observe that the new numerator and denominator tend to $1$ by assumption.