Glasnik Matematicki, Vol. 55, No. 1 (2020), 55-65.

EXTENSION OF THE FUNCTIONAL INDEPENDENCE OF THE RIEMANN ZETA-FUNCTION

Antanas Laurinčikas

Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania
e-mail: antanas.laurincikas@mif.vu.lt

Abstract.   In 1972, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i. e., if the functions φ_j are continuous in ℂ^N and φ₀(ζ(s), …, ζ^(N-1)(s))+ ∙∙∙ + sⁿ φ_n(ζ(s), …, ζ^(N-1)(s)) ≡ 0, then φ_j≡ 0 for j=0,…, n. The problem goes back to Hilbert who obtained the algebraic-differential independence of ζ(s). In the paper, the functional independence of compositions F(ζ(s)) for some classes of operators F in the space of analytic functions is proved. For example, as a particular case, the functional independence of the function cosζ(s) follows.

2010 Mathematics Subject Classification.   11M06

Key words and phrases.   Algebraic-differential independence, functional independence, Riemann zeta-function, space of analytic functions, universality

https://doi.org/10.3336/gm.55.1.05

References:

1. D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437-479.
   MathSciNet     CrossRef

2. O. Hölder, Über die Eigenschaft der Gammafunktion keiner algebraischen Differentialgleichung zu genügen, Math. Ann. 28 (1887), 1-13.

3. A. A. Karatsuba and S. M. Voronin, Walter de Gruyter, Berlin, 1992.
   MathSciNet     CrossRef

4. A. Laurinčikas, Limit theorems for the Riemann Zeta-function, Kluwer, Dordrecht, Boston, London, 1996.
   MathSciNet     CrossRef

5. A. Laurinčikas, Universality of the Riemann zeta-function, J. Number Theory 130 (2010), 2323-2331.
   MathSciNet     CrossRef

6. A. Laurinčikas, Universality of composite functions, in: Functions in number theory and their probabilistic aspects, RIMS Kôkyûroku Bessatsu B34 Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. 191-204.
   MathSciNet    

7. A. Laurinčikas, On zeros of some analytic functions related to the Riemann zeta-function, Glasn. Mat. Ser III. 48 (2013), 59-65.
   MathSciNet     CrossRef

8. A. Ostrowski, Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), 241-298.
   MathSciNet     CrossRef

9. A. G. Postnikov, On the differential independence of Dirichlet series, Doklady Akad. Nauk SSSR (N.S.) 66 (1949) 561-564 (Russian).
   MathSciNet    

10. A. G. Postnikov, A generalization of one of the Hilbert problems, Doklady Akad. Nauk SSSR (N.S.) 107 (1956), 512-515 (Russian).
    MathSciNet    

11. S. M. Voronin, The distribution of the nonzero values of the Riemann ζ-function, Trudy Math. Inst. Steklov 128 (1972), 131-150 (Russian).
    MathSciNet    

12. S. M. Voronin, The differential independence of ζ-functions, Doklady Akad. Nauk SSSR (N.S.) 209 (1973), 1264-1266 (Russian).
    MathSciNet    

13. S. M. Voronin, A theorem on the ``universality'' of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 475-486 (Russian).
    MathSciNet    

14. S. M. Voronin, The functional independence of Dirichlet L-functions, Acta Arith. 27 (1975), 493-503 (Russian).
    MathSciNet    

15. S. M. Voronin, Selected works: Mathematics, (ed. A. A. Karatsuba), Publishing House MGTU Im. N. E. Baumana, Moscow, 2006 (Russian).