## 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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### References:

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