On the zeros of the function $\zeta(s)$ in the neighborhood of the critical line

The following theorem is proved. If $H\geqslant T^a$, where $T>T_0>0$ and $a>27/82$, then for $1/2<\sigma\leqslant1$ the estimate
$$ N(\sigma,T+H)-N(\sigma,T)=O\biggl(\frac{H}{\sigma-0.5}\biggr) $$
holds uniformly in $\sigma$, where $N(\sigma_1,t)$ denotes the number of zeros $s=\sigma+it$, with $\sigma>\sigma_1$ and $0<t<T$, of the Riemann zeta-function $\zeta(s)$.
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