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:

 [1] S. Feng, Zeros of the Riemann zeta function on the critical line,J. Number Theory,132(2012), 511-542. · Zbl 1333.11086 [2] G. H. Hardy, Sur les z´eros de la fonctionζ(s) de Riemann,Comptes Rendus Acad. Sci. Paris,158(1914), 1012-1014. · JFM 45.0716.04 [3] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Dover Publications, 2006. · Zbl 0281.10001 [4] N. M. Korobov, Estimates of trigonometric sums and their applications, Uspehi Mat. Nauk,13(1958), 185-192 (Russian). [5] M. L. Lapidus and M. van Frankenhuijsen, Fractal geometry, complex dimensions, and zeta functions (second edition), Springer, New YorkBerlin, 2013. · Zbl 1261.28011 [6] N. Levinson, Almost all roots ofζ(s) =aare arbitrarily close toσ= 1/2, Proc. Nat. Acad. Sci. U.S.A.,72(1975), 1322-1324. · Zbl 0301.10033 [7] K. Matsumoto, An Introduction to the Value-Distribution Theory of ZetaFunctions,Siauliai Math. Semin.ˇ,1(2006), 61-83. · Zbl 1126.11042 [8] K. Matsumoto, A survey on the theory of universality for zeta and Lfunctions, in: Number theory. Plowing and starring through high wave forms,Proceedings of the7th China-Japan Seminar, Fukuoka, Japan, October28- November1, 2013, M. Kaneko (ed.) et al., World Scientiﬁc 2015, Series on Number Theory and Its Applications 11, 95-144. [9] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge University Press, 2006. · Zbl 1142.11001 [10] R. Nevanlinna, Eindeutige analytische Funktionen, Springer 1933. · Zbl 0050.30302 [11] L. Pa´nkowski, Joint universality for dependentL-functions,The Ramanujan Journal, not assigned in an issue, 2017. [12] C. R. Putnam, On the Non-Periodicity of the Zeros of the Riemann ZetaFunction,American Journal of Mathematics,76, (1954), 97-99. · Zbl 0055.06904 [13] C. R. Putnam, Remarks on Periodic Sequences and the Riemann ZetaFunction,American Journal of Mathematics,76, (1954), 828-830. · Zbl 0056.29503 [14] A. Reich, Wertverteilung von Zetafunktionen,Arch. Math.,34(1980), 440- 451. · Zbl 0431.10025 [15] A. Reich, Dirichletsche Reihen auf arithmetischen Progressionen,Monatsh. Math.,93(1982), 33-37. · Zbl 0464.10031 [16] B. Riemann, ¨Uber die Anzahl der Primzahlen unterhalb einer gegebenen Gr¨osse,Monatsber. Preuss. Akad. Wiss. Berlin(1859), 671-680. [17] J. Steuding, Value-distribution ofL-functions, Springer, Berlin, 2007. · Zbl 1130.11044 [18] I. M. Vinogradov, A new estimate for the functionζ(1 +it),Izv. Akad. Nauk SSSR, Ser. Mat.,22(1958), 161-164 (Russian). · Zbl 0097.26302 [19] S. M. Voronin, Theorem on the ‘universality’ of the Riemann zeta-function, Izv. Akad. Nauk SSSR, Ser. Matem.,39(1975), 475-486 (Russian);Math. USSR Izv.,9(1975), 443-445. · Zbl 0333.30023 [20] H.
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