The $a$-values of the Riemann zeta function near the critical line

We study the value distribution of the Riemann zeta function near the line $\mathrm{\Re }s=1\mathrm{/}2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $,$T \leq \Im s \leq 2T$forfixed$h_1, h_2>0$and$ 0 < \theta <1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwi\l\l’s recent results on the discrepancy between the distribution of $\zeta (s)$ and its random model. We also propose the secondary main term for the Selberg’s central limit theorem by providing sharper estimates on the line $\mathrm{\Re }s=1\mathrm{/}2+1\mathrm{/}(\log T{)}^{\theta }$.