Zero density theorems for short intervals

Abstract

We provide explicit bounds on the number of zeros for $\zeta$ in short intervals near the edge of the critical strip. Ultimately we show that even in short intervals a positive proportion of zeros will not be near the edge of the critical strip. For example with $h>100$ and $t>10^{100}$ the proportion of zeros of $\zeta$ with imaginary part in $[t-h,t+h]$ which have real part in $[\frac{1}{16},\frac{15}{16}]$ is at least $13\%$ and a positive proportion is already obtained at $t>10^{43}$.