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 Res=1/2. We find an asymptotic formula for the number of a-values in the rectangle 1/2+h1/(log⁡T)θ≤Res≤1/2+h2/(log⁡T)θ, T≤Ims≤2T for fixed h1,h2>0 and 0<θ<1/13. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwiłł's recent results on the discrepancy between the distribution of ζ(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 Res=1/2+1/(log⁡T)θ.