Modification of the Mishou theorem

The Mishou theorem asserts that a pair of analytic functions from a wide class can be approximated by shifts of the Riemann zeta and Hurwitz zeta-functions $(\zeta(s+i\tau), \zeta(s+i\tau, \alpha))$ with transcendental $\alpha$, $\tau\in\mathbb{R}$, and that the set of such $\tau$ has a positive lower density. In the paper, we prove that the above set has a positive density for all but at most countably many $\varepsilon>0$, where $\varepsilon$ is the accuracy of approximation. We also obtain similar results for composite functions $F(\zeta(s),\zeta(s,\alpha))$ for some classes of operator $F$.
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