Abstract
The zeta-function ζ(s, F), s = σ + it of a cusp form F of weight κ in the half-plane σ > (κ + 1)/2 is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form F. The compositions V(ζ(s,F)) with an operator V on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators V is proved.
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References
O. Hölder, “Über die Eigenschaft der Gammafunktion keiner algebraischen Differentialgleichung zu genugen,” Math. Ann. 28, 1–13 (1887).
D. Hilbert, “Paper delivered at the Mathematical Congress of 1900,” in Hilbert’s Problems (Nauka, Moscow, 1969) [in Russian].
D. D. Mordukhai-Boltovskii, “On the Hilbert problem,” Izv. Varshavsk. Polytekhn. Inst. 2, 1–16 (1914).
A. Ostrowski, “U¨ber 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 the Hilbert problems,” Dokl. Akad. Nauk SSSR 107 (4), 512–515 (1956).
S. M. Voronin, “The differential independence of ζ-functions,” Dokl. Akad. Nauk SSSR 206 (6), 1264–1266 (1973).
S. M. Voronin, Selected Works. Mathematics (Izd. MGTU, Moscow, 2006) [in Russian].
S. M. Voronin and A. A. Karatsuba, The Riemann Zeta-Function (Fizmatlit, Moscow, 1994) [in Russian].
S. M. Voronin, “A theorem on the “universality” of the Riemann zeta-function,” Dokl. Akad. Nauk SSSR. 39 (3), 475–486 (1975).
S. M. Voronin, “On the functional independence of Dirichlet L-functions,” Acta Arith. 27, 493–503 (1975).
A. Laurinčikas, K. Matsumoto, and J. Steuding, “The universality of L-functions associated with new forms,” Izv. RAN. Ser. Mat. 67 (1), 83–98 (2003) [Izv. Math. 67 (1), 77–90 (2003)].
A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function (Kluwer Acad. Publ., Dordrecht, 1996).
A. Laurinčikas and K. Matsumoto, “The universality of zeta-functions attached to certain cusp forms,” Acta Arith. 98 (4), 345–359 (2001).
A. Laurinčikas, K. Matsumoto, and J. Steuding, “Universality of some functions related to zeta-functions of certain cusp forms,” Osaka J. Math. 50 (4), 1021–1037 (2013).
Acknowledgments
The author wishes to express gratitude to the referee for the painstaking verification of the paper and some useful remarks.
Funding
The research is funded by the European Social Fund according to the activity “Improvement of Researchers’ qualification by implementing world-class R&D projects” of Measure No. 09.3.3-LMT-K-712-01-0037.
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Laurinčikas, A. On the Functional Independence of Zeta-Functions of Certain Cusp Forms. Math Notes 107, 609–617 (2020). https://doi.org/10.1134/S0001434620030281
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DOI: https://doi.org/10.1134/S0001434620030281