Abstract
We prove functional independence and joint functional independence for a set of Hurwitz zeta functions with periodic coefficients and parameters algebraically independent over the field of rational numbers.
Similar content being viewed by others
References
Hilbert’s Problems: Collection (Nauka, Moscow, 1969) [in Russian].
A. Ostrowski, “Über Dirichletsche Reihen und algebraische Differentialgleichungen,” Math. Z. 8(3–4), 241–298 (1920).
A. G. Postnikov, “On the differential independence of Dirichlet series,” Dokl. Akad. Nauk SSSR 66(4), 561–564 (1949).
A. G. Postnikov, “On the generalization of one of Hilbert’s problems,” Dokl. Akad. Nauk SSSR 107(4), 512–515 (1956).
S.M. Voronin, “On differential independence of ζ-functions,” Dokl. Akad. Nauk SSSR 209(6), 1264–1266 (1973) [Soviet Math. Dokl. 14, 607–609 (1973)].
S.M. Voronin and A. A. Karatsuba, The Riemann Zeta Function (Fizmatlit, Moscow, 1994) [in Russian].
S.M. Voronin, Selected Works: Mathematics, Ed. by A. A. Karatsuba (Moscow, 2006) [in Russian].
S. M. Voronin, “On the functional independence of the Dirichlet L-function,” Acta Arith. 27, 493–503 (1975).
A. Laurinčikas, “Distribution des valeurs de certaines séries de Dirichlet,” C. R. Acad. Sci. Paris. Sér. A 289, 43–45 (1979).
A. Laurinčikas, “Sur les séries de Dirichlet et les polynômes trigonométriques,” in Séminaire de Théorie des Nombres, 1978–1979 (CNRS, Talence), Exp. no. 24.
A. P. Laurinchikas, “Distribution of values of generating Dirichlet series of multiplicative functions,” Litovsk. Mat. Sb. 22(1), 101–111 (1982) [Lithuanian Math. J. 22 (1), 56–63 (1982)].
A. Laurinčikas, Limit Theorems for the Riemann Zeta Function, in Mathematics and its Applications (Kluwer Academic Publishers Group, Dordrecht, 1996), Vol. 352.
A. Reich, “Zetafunktionen und Differenzen-Differentialgleichungen,” Arch. Math. (Basel) 38(1), 226–235 (1982).
R. Garunkštis and A. Laurinčikas, “On one Hilbert’s problem for the Lerch zeta function,” Publ. Inst.Math. (Beograd) (N. S.) 65(79), 63–68 (1999).
R. Garunkštis and A. Laurinčikas, “The Lerch zeta function,” Integral Transform. Spec. Funct. 10(3–4), 211–226 (2000); in Analytical Methods of Analysis and Differential Equations (Minsk, 1999) [in Russian].
A. Laurinčikas and K. Matsumoto, “The joint universality and the functional independence for Lerch zeta functions,” Nagoya Math. J. 157, 211–227 (2000).
A. Laurinčikas and R. Garunkštis, The Lerch Zeta Function (Kluwer Academic Publishers, Dordrecht, 2002).
A. Laurinčikas and K. Matsumoto, “Joint value-distribution theorems on Lerch zeta functions. II,” Liet. Mat. Rink. 46(3), 332–350 (2006) [Lithuanian Math. J. 46 (3), 271–286 (2006)].
A. Laurinčikas and K. Matsumoto, “Joint value-distribution theorems on Lerch zeta functions. III,” in Proc. 4th Palanga Conf. (in press).
V. Garbaliauskienė and A. Laurinčikas, “Some analytic properties for L-functions of elliptic curves,” in Proc. Institute of Mathematics (National Academy of Sciences of Belarus, 2005), Vol. 13, No. 1, pp. 1–8 [in Russian].
J. Steuding, Value-Distribution of L-Functions and Allied Zeta Functions — with an Emphasis on Aspects of Universality, Habilitationschrift (J. W. Goethe-Universität, Frankfurt, 2003).
A. Laurinchikas, Joint Universality of Periodic Hurwitz Zeta Functions, Preprint (Vilnius University, Vilnius, 2006) [in Russian].
S. M. Voronin, “A theorem on the distribution of values of the Riemann zeta function,” Dokl. Akad. Nauk SSSR 221(4), 771 (1975) [Soviet Math. Dokl. 16, 410 (1975)].
S. M. Voronin, “Theorem on the ‘universality’ of the Riemann zeta function,” Izv. Akad. Nauk SSSR Ser. Mat. 39(3), 475–486 (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. Laurinchikas, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 1, pp. 69–78.
Rights and permissions
About this article
Cite this article
Laurinchikas, A. Functional independence of periodic Hurwitz zeta functions. Math Notes 83, 65–71 (2008). https://doi.org/10.1134/S0001434608010082
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434608010082