Universality of an absolutely convergent Dirichlet series with modified shifts

Authors: ANTANAS LAURINCIKAS, RENATA MACAITIENE, DARIUS SIAUCIUNAS

Abstract: In the paper, a theorem on approximation of a wide class of analytic functions by generalized shifts $\zeta_{u_T}(s+i\varphi(\tau))$ of an absolutely convergent Dirichlet series $\zeta_{u_T}(s)$ which in the mean is close to the Riemann zeta-function is obtained. Here $\varphi(\tau)$ is a monotonically increasing differentiable function having a monotonic continuous derivative such that $\varphi(2\tau)\max\limits_{\tau\leqslant t\leqslant 2\tau} \frac{1}{\varphi'(t)} \ll \tau$ as $\tau\to\infty$, and $u_T\to\infty$ and $u_T\ll T^2$ as $T\to\infty$.

Keywords: Haar measure, Mergelyan theorem, Riemann zeta-function, universality, weak convergence

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