The mixed joint functional independence of the Riemann zeta- and periodic Hurwitz zeta-functions

R. Kačinskaitė, S. Rapimbergaitė

Šiauliai University

Abstract: The functional independence of zeta-functions is an interesting nowadays problem. This problem comes back to D. Hilbert. In 1900, at the International Congress of Mathematicians in Paris, he conjectured that the Riemman zeta-function does not satisfy any algebraic-differential equation. This conjecture was solved by A. Ostrowski. In 1975, S.M. Voronin proved the functional independence of the Riemann zeta-function. After that many mathematicians obtained the functional independence of certain zeta- and $L$-functions.
In the present paper, the joint functional independence of a collection consisting of the Riemann zeta-function and several periodic Hurwitz zeta-functions with parameters algebraically independent over the field of rational numbers is obtained. Such type of functional independence is called as “mixed functional independence” since the Riemann zeta-function has Euler product expansion over primes while the periodic Hurwitz zeta-functions do not have Euler product.
Bibliography: 17 titles.

Keywords: functional independence, Hurwitz zeta-function, periodic coefficients, Riemann zeta-function, universality.

UDC: 519.14

Received: 10.06.2016
Accepted: 12.12.2016

DOI: 10.22405/2226-8383-2016-17-4-57-64

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