Joint discrete universality for $L$-functions from the Selberg class and periodic Hurwitz zeta-functions

A. Balčiūnas ; R. Macaitienė ; D. Šiaučiūnas

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 46-65.

Résumé

The Selberg class $\mathcal{S}$ contains Dirichlet series $$ \mathcal{L}(s)= \sum_{m=1}^\infty \frac{a(m)}{m^s}, \quad s=\sigma+it, $$ such that, for every $\varepsilon>0$, $a(m)\ll_\varepsilon m^\varepsilon$; there exists an integer $k\geqslant 0$ such that $(s-1)^k \mathcal{L}(s)$ is an entire function of finite order; the functions $\mathcal{L}$ satisfy a functional equation connecting $s$ with $1-s$, and have a product representation over prime numbers. Steuding introduced a subclass $\widetilde{\mathcal{S}}$ of $\mathcal{S}$ with additional condition $$ \lim_{x\to\infty} \left(\sum_{p\leqslant x} 1\right)^{-1} \sum_{p\leqslant x}|a(p)|^2=\kappa>0, $$ where $p$ runs prime numbers. Let $\alpha$, $0\alpha\leqslant 1$, be a fixed parameter, and $\mathfrak{a}=\{a_m: m\in \mathbb{N}_0\}$ be a periodic sequence of complex numbers. The second object of the paper is the periodic Hurwitz zeta-function $\zeta(s,\alpha;\mathfrak{a})$ which is defined, for $\sigma>1$, by the Dirichlet series $$ \zeta(s,\alpha; \mathfrak{a})=\sum_{m=0}^\infty \frac{a_m}{(m+\alpha)^s}, $$ and is meromorphically continued to the whole complex plane. The paper is devoted to the discrete universality of the collection $$ \left(\mathcal{L}(\widetilde{s}), \zeta(s,\alpha_1; \mathfrak{a}_{11}), \dots,\zeta(s,\alpha_1; \mathfrak{a}_{1l_1}), \dots, \zeta(s,\alpha_r; \mathfrak{a}_{r1}), \dots, \zeta(s,\alpha_r; \mathfrak{a}_{rl_r})\right), $$ where $\mathcal{L}(\widetilde{s})\in \widetilde{S}$, and $\zeta(s,\alpha_j; \mathfrak{a}_{jl_j})$ are periodic Hurwitz zeta-functions, i. e., to the simultaneous approximation of a collection $$ \left(f(\widetilde{s}), f_{11}(s),\dots, f_{1l_1}(s), \dots, f_{r1}(s), \dots, f_{rl_r}(s)\right) $$ of analytic functions from a wide class by a collection of shifts \begin{align*} \big(\mathcal{L}(\widetilde{s}+ikh), \zeta(s+ikh_1,\alpha_1; \mathfrak{a}_{11}), \dots,\zeta(s+ikh_1,\alpha_1; \mathfrak{a}_{1l_1}), \dots, \\ \zeta(s+ikh_r,\alpha_r; \mathfrak{a}_{r1}), \dots, \zeta(s+ikh_r,\alpha_r; \mathfrak{a}_{rl_r})\big), \end{align*} where $h, h_1, \dots, h_r$ are positive numbers, is considered. For this, the linear independence over the field of rational numbers for the set $$ \left\{\left(h\log p: p\in \mathbb{P}\right), \left( h_j\log(m+\alpha_j): m\in \mathbb{N}_0,\, j=1,\dots, r\right), 2\pi\right\}, $$ where $\mathbb{P}$ denotes the set of all prime numbers, is applied.

Détail

Keywords: Dirichlet series, Hurwitz zeta-function, periodic Hurwitz zeta-function, Selberg class, universality, weak convergence.

Bibliographie

[1] Bagchi B., The statistical behavior and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Indian Statistical Institute, Calcutta, 1981

[2] Billingsley P., Convergence of Probability Measures, Wiley, New York, 1968 | MR | Zbl

[3] Buivydas E., Laurinčikas A., “A discrete version of the Mishou theorem”, Ramanujan J., 38:2 (2015), 331–347 | MR | Zbl

[4] Buivydas E., Laurinčikas A., “A generalized joint discrete universality theorem for the Riemann and Hurwitz zeta-function”, Lith. Math. J., 55:2 (2015), 193–206 | MR | Zbl

[5] J. B. Conway, Functions of one complex variable, Springer, Berlin–Heidelberg–New York, 1978 | MR

[6] Janulis K., Mixed joint universality for Dirichlet $L$-functions and Hurwitz type zeta-functions, Doctoral dissertation, Vilnius University, Vilnius, 2015

[7] Kačinskaitė R., Laurinčikas A., “The joint distribution of periodic zeta-functions”, Stud. Sci. Math. Hung., 48 (2011), 257–279 | MR

[8] Kačinskaitė R., Matsumoto K., “The mixed joint universality for a class of zeta-functions”, Math. Nachr., 288:16 (2015), 1900–1909 | MR

[9] Kačinskaitė R., Matsumoto K., “Remarks on the mixed joint universality for a class of zeta-functions”, Bull. Austral. Math. Soc., 95:2 (2017), 187–198 | MR

[10] Kačinskaitė R., Matsumoto K., “On mixed joint discrete universality for a class of zeta-functions”, Anal. and Probab. Methods in Number Theory, eds. A. Dubickas et al., Vilnius University, Vilnius, 2017, 51–66 | MR

[11] Karatsuba A. A., Voronin S. M., The Riemann zeta-function, Walter de Gruyter, Berlin, 1992 | MR | MR

[12] Laurinčikas A., “The joint universality for periodic Hurwitz zeta-functions”, Analysis (Munich), 26:3 (2006), 419–428 | MR | Zbl

[13] Laurinčikas A., “Joint universality of zeta-functions with periodic coefficients”, Izv. Math., 74:3 (2010), 515–539 | MR | Zbl

[14] Laurinčikas A., “The joint discrete universality of periodic zeta-functions”, From Arithmetic to Zeta-Functions, Springer, 2016, 231–246 | Zbl

[15] Laurinčikas A., “Joint discrete universality for periodic zeta-functions”, Quaest. Math. | DOI

[16] Laurinčikas A., Macaitienė R., Proc. Steklov Inst. Math., 299:1 (2017), Discrete universality in the Selberg class

[17] Macaitienė R., “Joint universality for $L$-functions from the Selberg class and periodic Hurwitz zeta-functions”, Ukrain. Math. J., 70:5 (2018), 655–671 | MR | Zbl

[18] Mergelyan S. N., “Uniform approximations to functions of a complex variable”, Usp. Matem. Nauk, 7:2 (1952,), 31–122 (in Russian) | MR | Zbl

[19] Mishou H., “The joint value-distribution of the Riemann zeta-function and Hurwitz zeta-function”, Lith. Math. J., 47 (2007), 32–47 | MR | Zbl

[20] H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math., 227, Springer-Verlag, Berlin, 1971 | MR | Zbl

[21] H. Nagoshi, J. Steuding, “Universality for $L$-functions in the Selberg class”, Lith. Math. J., 50 (2010), 293–311 | MR | Zbl

[22] Račkauskienė S., Joint universality of zeta-functions with periodic coefficients, Doctoral dissertation, Vilnius University, Vilnius, 2012

[23] Reich A., “Werteverteilung von Zetafunktionen”, Arch. Math., 34 (1980), 440–451 | MR | Zbl

[24] Selberg A., “Old and new conjectures and results about a class of Dirichlet series”, Proc. of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), eds. E. Bombieri, Univ. Salerno, Salerno, 1992, 367–395 | MR

[25] J. Steuding, “On the universality for functions in the Selberg class”, Proc. of the Session in Analytic Number Theory and Diophantine Equations (Bonn, 2002), Bonner Math. Sciften., 360, eds. D. R. Heath-Brown, B. Z. Moroz, Bonn, 2003, 22 pp. | MR | Zbl

[26] Steuding J., Value-Distribution of $L$-Functions, Lecture Notes Math., 1877, Springer-Verlag, Berlin–Heidelberg, 2007 | MR | Zbl

[27] Voronin S. M., “Theorem on the “universality” of the Riemann zeta-function”, Math. USSR Izv., 9 (1975), 443–453 | MR | Zbl

[28] Voronin S. M., “On the functional independence of Dirichlet $L$-functions”, Acta Arith., 27 (1975), 493–503 | Zbl

[29] Voronin S. M., Analytic properties of generating functions of arithmetical objects, Diss. Doctor Phys. matem. nauk, Matem. Institute V. A. Steklov, M., 1977