Effective uniform approximation by the Riemann zeta-function. (English) Zbl 1223.11101

The authors first discuss the celebrated universality theorem of S. M. Voronin [Izv. Akad. Nauk SSSR Ser. Mat. 39, 475–486 (1975; Zbl 0315.10037)], that essentially any non-vanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the Riemann zeta-function \(\zeta(s)\). The authors prove an effective uniform approximation of Voronin’s theorem. Let \[ {\mathcal K} = \Bigl\{\,s\in{\mathbb C}\;:\; |s-s_0| \leq r\,\Bigr\}, \] where \(r>0\) and \(s_0 = \sigma_0+it_0,\,1/2 <\sigma_0<1.\) Suppose that \(g : {\mathcal K}\,\to\,\mathbb C\) is continuous, \(g(s_0)\neq0\) and \(g(s)\) is analytic for \(|s-s_0| \leq r\). Then, for any \(\varepsilon \in (0,|g(s_0)|)\), there exist real numbers \(\tau \in [T,\,2T]\) and \(\delta(\varepsilon,g,\tau)>0\) such that \[ \max_{|s-s_0|\leq\delta r}|\zeta(s+i\tau) - g(s)| < \varepsilon. \] The quantities \(T\) and \(\delta\) are explicitly defined in the text. It is also shown that, under certain conditions, one may assume even that \(g(s_0) = 0\). The authors remark that the above assertion holds also for the derivatives \(\zeta^{(j)}(s)\) in place of \(\zeta(s)\). They also discuss the case of effective approximations by linear combinations of \(\zeta^{(j)}(s)\) for \(j = 0,\dots, J\). The key ingredient in the proofs is an effective multidimensional \(\Omega\)-result of S. M. Voronin [Izv. Akad. Nauk SSSR Ser. Mat. 52, No. 2, 424–436 (1988; Zbl 0651.10026)].


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
30E10 Approximation in the complex plane
Full Text: DOI Euclid


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