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The values of the Riemann zeta-function on discrete sets. (English) Zbl 1456.11156

Mishou, Hidehiko (ed.) et al., Various aspects of multiple zeta functions – in honor of Professor Kohji Matsumoto’s 60th birthday. Proceedings of the international conference, Nagoya University, Nagoya, Japan August 21–25, 2020. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 84, 315-334 (2020).
For positive real numbers \(\delta_1, \delta_2\), define \[ \mathcal{A}_i = \left\{\delta_i\frac{\log q}{2 \pi} : q \in \mathbb{Q}_+\right\}, \quad i=1, 2, \quad \mathcal{A} = \mathcal{A}_1 \times \mathcal{A}_2 \setminus \{(0, 0)\},\] \[\mathcal{L}(\delta_1, \delta_2) := \left\{\alpha \in \mathbb{R}: \text{ for all } (\theta_1, \theta_2) \in \mathcal{A} : 1, \alpha, \alpha', \alpha\theta_1 + \alpha'\theta_2 \text{ are linear independent over }\mathbb{Q} \right\}\] and let \(\mathcal{D}:=\{s \in \mathbb{C} : \operatorname{Re} (s) \in (1/2, 1)\}\).
Theorem 1. Let \(t_1, t_2\) be arbitrary real numbers and \(\delta_1, \delta_2\) be positive real numbers. Assume \[\zeta(s + i(t_1 + \delta_1n)) = \zeta(s + i(t_2+\delta_2\sigma(n)))\quad for \ n = 1, 2, \dots ,\] where \(\sigma : \mathbb{N} \to \mathbb{N}\) is bijective. Then we have \(t_1 = t_2\), \(\delta_1 = \delta_2\), \(\sigma = id\) or \(\operatorname{Re}(s) \leq b\), where \(b\) is a constant depending only on \(t_i, \delta_i\) (for \(i = 1, 2\)) and \(\sigma\).
Theorem 2. Let \(t_1, t_2\) be arbitrary real numbers and \(\delta_1, \delta_2\) be positive numbers such that \(\delta_1, \delta_2 \not = 2\pi r / \log q\) for any \(r, q \in \mathbb Q_+\) with \(q\not = 1\). Let also \(\alpha \in \mathcal{L}(\delta_1, \delta_2) \cap (1, \infty)\), \(K\) be a compact set of \(\mathcal{D}\) with connected complement and \(f, g\) continuous non-vanishing functions on \(K\), which are analytic in the interior of \(K\). Then, for every \(\varepsilon > 0\), \[ \liminf\limits_{N\to\infty}\dfrac{1}{N}\#\left\{1\leq n\leq N:\begin{array}{ll} \max\limits_{s\in K}|\zeta(s+i(t_1+\delta_1\lfloor n\alpha\rfloor))-f(s)|<\varepsilon\\ \max\limits_{s\in K}|\zeta(s+i(t_2+\delta_2\lfloor n\alpha'\rfloor))-g(s)|<\varepsilon \end{array}\right\}>0.\hspace*{1cm} \]
Now, define \[\mathbb{M}:=\{\mathsf{M} \subset \mathbb{C}\ \vert \ \mathsf{M}^\circ \not = \emptyset\},\] where \(A^\circ\) denotes the interior of a set \(A\) in \(\hat{\mathbb{C}}: = \mathbb{C} \cup \{\infty\}\) and, for \(i=1, 2\) \[L(s, f_i):=\sum_{m \geq 1}f_i(m)m^{-s}\] Theorem 3. Let \(h\) be a positive real number, \(l \in \mathbb{N}\) and \(\mathsf{M} \in \mathbb{M}\). If \(\operatorname{Re} (s) \in (1/2, 1)\), then there exists an infinite subset \(\mathsf{N} \subset \mathbb{N}\) such that \[\zeta(s + ih(n+k-1)) \in \mathsf{M}\] for any \(n \in \mathsf{N}\) and \(k\in\{1, 2, \dots, l\}\).
Theorem 4. Let \(f\) be a bounded arithmetical function, supported on a set of positive integers, having at least two distinct prime divisors and let \(t_1, t_2\) be arbitrary real numbers and \(\delta_1, \delta_2\) be arbitrary positive real numbers. Assume \[L(s + i(t_1 + \delta_1n), f) = L(s + i(t_2+\delta_2\sigma(n)), f)\quad for \ n = 1, 2, \dots ,\] where \(\sigma : \mathbb{N} \to \mathbb{N}\) is bijective. Then we have \(t_1 = t_2\), \(\delta_1 = \delta_2\), \(\sigma = id\) or \(\operatorname{Re} (s) \leq b\), where \(b\) is a constant depending only on \(t_i, \delta_i\) (for \(i = 1, 2\)) and \(\sigma\).
For the entire collection see [Zbl 1446.11004].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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