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On the value distribution of shifts of universal Dirichlet series. (English) Zbl 1116.11069

The strongest version of an universality theorem by S. M. Voronin [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 475–486 (1975; Zbl 0315.10037)] is
Theorem 1. Suppose that \(\mathcal K\) is a compact subset of \(\{s \in {\mathbb C} : {1\over 2}<\operatorname{Re}\,s<1\}\) with connected complement, and let \(g(s)\) be a non-vanishing continuous function on \(\mathcal K\) which is analytic in the interior of \(\mathcal K\). Then, for any \(\epsilon >0\), \[ \liminf_{T \to \infty}{1\over T} \text{meas} \{\tau \in [0,T] : \max_{s\in \mathcal K}| \zeta (s+i\tau) - g(s)| <\epsilon \} >0. \] A proof has been given by B. Bagchi [Math. Z. 181, 319–334 (1982; Zbl 0479.10028)].
The authors are concerned with a multi-dimensional analogue of theorem 1, the joint universality for shifts of universal Dirichlet series, and consequences on the value-distribution.They assume in the theorems of this paper that \({\mathcal F}(s)\) is analytic and universal in \({\mathcal S}:={\mathcal S}(\sigma_1,\sigma_2):=\{s \in {\mathbb C} : \sigma_1 < \operatorname{Re}\, s <\sigma_2\}\), \[ \begin{aligned} \lambda_j \in {\mathbb C} \;\text{for}\; 1\leq j \leq n ,\quad \lambda_j \neq \lambda_k \,\text{for}\, 1 \leq j<k \leq n, \;\Lambda :=\\ \max_{1\leq j \leq n}\operatorname{Re}\,\lambda_j, \lambda := \min_{1\leq j \leq n}\operatorname{Re}\,\lambda_j,\,\Lambda - \lambda < \sigma_2 - \sigma_1\end{aligned} \] and prove the following important consequences of universality.
Theorem 2. For every \(l \in {\mathbb N},\,s_0 \in {\mathcal S}(\sigma_1-\lambda,\sigma_2-\Lambda)\), the image of the curve \[ \bigl({\mathcal F}(\lambda_1+s_0 +it),\dots,{\mathcal F}(\lambda_n+s_0 +it),\dots,{\mathcal F}^{(l)}(\lambda_1+s_0 +it),\dots,{\mathcal F}^{(l)}(\lambda_n+s_0 +it)\bigr)_{t \in{\mathbb R}} \] is dense in \({\mathbb C}^{n(l+1)}\).
Theorem 2 implies the general concept of functional independence which is an extension of a result of A. Ostrowski [Math. Z. 8, 241–298 (1920; JFM 47.0292.01)]:
Theorem 3. Let \(f_0,\dots,f_m\) be continuous functions on \({\mathbb C}^{n(l+1)}\).If \[ \sum_{k=0}^ms^kf_k\bigl({\mathcal F}(\lambda_1+s),\dots,{\mathcal F}(\lambda_n+s),\dots,{\mathcal F}^{(l)}(\lambda_1+s),\dots,{\mathcal F}^{(l)}(\lambda_n+s)\bigr)=0 \] for all \(s \in {\mathcal S}(\sigma_1-\lambda,\sigma_2-\Lambda)\), then \(f_k \equiv 0 \;\text{for}\,k=0,\dots,m.\)
A direct generalization of Theorem 1 is
Theorem 4. Let \(n\geq 2\) , \(a_1,\dots,a_n \in {\mathbb C},\neq 0\), and \({\mathcal Z}(s):=\sum _{j=1}^n a_j {\mathcal F}(\lambda_j +s)\). Let \({\mathcal K} \neq \emptyset,\,\subset {\mathcal S}\),compact with connected complement such that the sets \(\{\lambda_j +s:s\in {\mathcal K} \}\) are disjoint subsets of \({\mathcal S}\). Let \(g(s)\) be a continuous function on \({\mathcal K}\) which is analytic in the interior of \({\mathcal K}\). Then, for any \(\varepsilon >0\), \[ \liminf_{T \to \infty}{1\over T} \text{meas} \{\tau \in [0,T] : \max_{s\in \mathcal K}| {\mathcal Z}(s+i\tau)-g(s)| <\varepsilon\} >0. \] Using Rouché’s theorem and Littlewood’s lemma from the theory of functions, the authors obtain from theorem 4 estimations for the the number \(N(\sigma,T)\) of zeros of \({\mathcal Z}(s)\) in \(\{s \in {\mathbb C} : \operatorname{Re}\, s>\sigma,0<\operatorname{Im}\, s <T\}\) under the further assumptions that \({\mathcal F}(s)=\sum_{n=1}^\infty {f(n)\over{n^s}} \,\text{for}\, \operatorname{Re}\, s>\sigma_2,\) \({\mathcal F}(s)\not\equiv 0,\) and is regular and of finite order in \({\mathcal S}(\sigma_1,\sigma_2)\) with \[ \limsup_{T \to \infty}{1\over T}\int_0^T | {\mathcal F}(\sigma+it| ^2\,dt<\infty. \]
Theorem 5. For fixed \(\sigma \in (\sigma_1-\lambda,\sigma_2-\Lambda)\), there exist positive constants \(c_1, c_2\), depending only on \(\sigma\) and \({\mathcal Z}(s)\), such that \[ c_1 \leq \liminf_{T \to \infty}{1\over T}N(\sigma,T)\leq\limsup_{T \to \infty}{1\over T}N(\sigma,T)\leq c_2. \]

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions
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References:

[1] Bagchi B (1981) The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series. PhD Thesis, Indian Statistical Institute, Calcutta
[2] Karatsuba AA, Voronin SM (1992) The Riemann Zeta-Function. Berlin: de Gruyter
[6] Reich A (1977) Universelle Wertverteilung von Eulerprodukten. Nachr Akad Wiss Göttingen, Math-Phys Kl 1–17 · Zbl 0379.10025
[7] Titchmarsh EC (1939) The Theory of Functions, 2nd ed. Oxford: Univ Press
[8] Voronin SM (1975) Theorem on the ’Universality’ of the Riemann Zeta-Function. Izv Akad Nauk SSSR, Ser Matem 39: 475–486 (in Russian); Engl Transl Math USSR Izv 9: 443–445 (1975) · Zbl 0315.10037
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