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We apply an effective multidimensional Ω-result of Voronin in order to obtain effective universality-type theorems for the Riemann zeta-function. We further use this approach to study approximation properties of linear combinations of derivatives of the zeta-function.

Given any complex number a, we prove that there are infinitely many simple roots of the equation ζ(s) = a with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve t → ζ( 1 2 + it). Moreover, we show that the curve R t → (ζ( 1 2 + it), ζ ( 1 2 + it)) is not dense in C 2 .

Wenzhi Luo studied the distribution of nontrivial zeros of the derivatives of Selberg zeta-functions on cocompact hyperbolic surfaces, and obtained an asymptotic formula for zero density with bounded height. Then he related the distribution of zeros to the multiplicities of Laplacian eigenvalues. Using better bounds for the growth of the Selberg zeta-functions we improve some of the above results. Mathematics Subject Classification (2000). Primary 11M36; Secondary 30F35.

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