For any complex-valued function \(f:{\mathbb{R}}\to {\mathbb{C}}\) and any rectangle \(R\subset {\mathbb{C}}\), let \(P_T\) denote the proportion function: \[ P_T(f(t)\in R) = \frac{1}{T}\operatorname{meas}\{t\in [T,2T]\mid f(t)\in R\}. \] Let \(X=\{X(p)\mid p \text{ prime}\}\) denote a sequence of independent random variables, uniformly distributed on the complex unit circle, parameterized by the set of prime numbers. Denote the asociated random Euler product by \[ \zeta(s,X)=\prod_p (1-X(p)p^{-s})^{-1},\quad s=\sigma+it,\ \sigma>1/2. \] It’s know that this product converges almost surely. Let \(P(\log \zeta(s,X)\in R)\) denote the probability that the random variable \(\log \zeta(s,X)\) belongs to \(R\). Define the descrepancy to be \[ D_\sigma(T)=\sup_R |P_T(\log \zeta(\sigma+it)\in R)-P(\log \zeta(s,X)\in R)|, \] where the sup is taken over all rectangles whose sides are parallel to the coordinate axes.
The main results of the article under review are as follows.

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For \(1/2<\sigma <1\), we have \(D_\sigma(T)<<(\log T)^{-\sigma},\) where the implied constant depends on \(\sigma\). A similar estimate holds for \(\sigma=1\).

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For \(1/2<\sigma <1\) and for each sufficiently small \(\epsilon>0\), we have \(D_\sigma(T)=\Omega (T^{1-2\sigma-\epsilon}),\) where the implied constant depends on \(\sigma,\epsilon\).

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For \(a\in {\mathbb{C}}^\times\) and \(1/2<\sigma_1<\sigma_2<1\), let \(N_a(\sigma_1,\sigma_2,T)=\#\{(\sigma,t)\mid \sigma_1<\sigma<\sigma_2, T<t<2T, \zeta(\sigma+it)=a\}\) denote the number of \(a\)-points in the stated rectangle. There is a constant \(c(a,\sigma_1,\sigma_2) >0\) such that \[ N_a(\sigma_1,\sigma_2,T)=c(a,\sigma_1,\sigma_2)T+ O(T\frac{\log\log T}{(\log T)^{\sigma_1/2}}). \]

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For \(1/2<\sigma <1\) there is a constant \(b(\sigma)>0\), such that for all \(3<\tau<b(\sigma)(\log T)^{1-\sigma}(\log \log T)^{1-\frac{1}{\sigma}}\), \[ P_T(Re\{\log \zeta(\sigma+it)\}>\tau) =P(Re\{\log \zeta(s,X)\}>\tau)+\text{ error}, \] where the lower order error term is explicitly estimated. A similar estimate is obtained by replacing the real part \(Re\) by the imaginary part \(Im\).

More carefully stated versions of these results are given in the paper under review.