|DATE||October 14 (Thu), 2021|
|TITLE||[GS_M_NT] 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 $Re(s) =1/2$. We find an asymptotic formula for the number of a-values in the rectangle $ 1/2 + h_1 / (logT)^\theta \leq Re(s) \leq 1/2 + h_2 / (logT)^\theta$, $T \leq Im(s) \leq 2T $ for fixed $h_1 , h_2>0$ and $0 < \theta <1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwiłł’s recent results on the discrepancy between the distribution of the Riemann zeta function 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 $ Re(s) = 1/2 +1/(logT)^\theta$.