On the Hurwitz Zeta-Function with Algebraic Irrational Parameter. II

A. Laurinčikas

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 134-144.

Résumé

It is known that the Hurwitz zeta-function $\zeta (s,\alpha )$ with transcendental or rational parameter $\alpha $ has a discrete universality property; i.e., the shifts $\zeta (s+ikh,\alpha )$, $k\in \mathbb N_0$, $h> 0$, approximate a wide class of analytic functions. The case of algebraic irrational $\alpha $ is a complicated open problem. In the paper, some progress in this problem is achieved. It is proved that there exists a nonempty closed set $F_{\alpha ,h}$ of analytic functions such that the functions in $F_{\alpha ,h}$ are approximated by the above shifts. Also, the case of certain compositions $\Phi (\zeta (s,\alpha ))$ is discussed.

Bibliographie

[1] Bagchi B., The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, PhD Thesis, Indian Stat. Inst., Calcutta, 1981

[2] A. Balčiūnas, A. Dubickas, and A. Laurinčikas, “On the Hurwitz zeta functions with algebraic irrational parameter”, Math. Notes, 105:1–2 (2019), 173–179 | MR | Zbl

[3] Billingsley P., Convergence of probability measures, J. Willey Sons, New York, 1968 | MR | Zbl

[4] Cassels J.W.S., “Footnote to a note of Davenport and Heilbronn”, J. London Math. Soc., 36 (1961), 177–184 | DOI | MR | Zbl

[5] Dubickas A., Laurinčikas A., “Distribution modulo 1 and the discrete universality of the Riemann zeta-function”, Abh. Math. Semin. Univ. Hamburg, 86:1 (2016), 79–87 | DOI | MR | Zbl

[6] Gonek S.M., Analytic properties of zeta and $L$-functions, PhD Thesis, Univ. Michigan, Ann Arbor, 1979 | MR

[7] Hurwitz A., “Einige Eigenschaften der Dirichlet'schen Funktionen $F(s)=\sum \bigl (\frac {D}{n}\bigr )\cdot \frac {1}{n^s}$, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten”, Z. Math. Phys., 27 (1882), 86–101

[8] Laurinčikas A., “A discrete universality theorem for the Hurwitz zeta-function”, J. Number Theory, 143 (2014), 232–247 | DOI | MR | Zbl

[9] Laurinčikas A., “On discrete universality of the Hurwitz zeta-function”, Result. Math., 72:1–2 (2017), 907–917 | MR | Zbl

[10] A. Laurinčikas, “Discrete universality of the Riemann zeta-function and uniform distribution modulo 1”, St. Petersburg Math. J., 30:1 (2019), 103–110 | MR | Zbl

[11] Laurinčikas A., ““Almost” universality of the Lerch zeta-function”, Math. Commun., 24:1 (2019), 107–118 | MR | Zbl

[12] Laurinčikas A., Garunkštis R., The Lerch zeta-function, Kluwer, Dordrecht, 2002 | MR | Zbl

[13] Laurinčikas A., Macaitienė R., “The discrete universality of the periodic Hurwitz zeta function”, Integral Transforms Spec. Funct., 20 (2009), 673–686 | DOI | MR | Zbl

[14] Laurinčikas A., Macaitienė R., “Universality results on Hurwitz zeta-functions”, Algebra, logic and number theory (Proc. Conf., Bȩdlewo, 2014), Banach Cent. Publ., 108, Inst. Math. Pol. Acad. Sci., Warsaw, 2016, 161–172 | DOI | MR | Zbl

[15] Laurinčikas A., Macaitienė R., Šiaučiūnas D., “Uniform distribution modulo 1 and the joint universality of Dirichlet $L$-functions”, Lith. Math. J., 56:4 (2016), 529–539 | DOI | MR | Zbl

[16] Macaitienė R., “On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts”, Arch. Math., 108:3 (2017), 271–281 | DOI | MR | Zbl

[17] Montgomery H.L., Topics in multiplicative number theory, Lect. Notes Math., 227, Springer, Berlin, 1971 | DOI | MR | Zbl

[18] Pańkowski Ł., “Joint universality for dependent $L$-functions”, Ramanujan J., 45:1 (2018), 181–195 | DOI | MR | Zbl

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

[20] S. M. Voronin, “Theorem on the ‘universality’ of the Riemann zeta-function”, Math. USSR, Izv., 9:3 (1975), 443–453 | DOI | MR

[21] S. M. Voronin, Analytic properties of generating Dirichlet functions of arithmetical objects, Doctoral (Phys.–Math.) Dissertation, Steklov Math. Inst., Moscow, 1977 | MR