On the Mishou Theorem for Zeta-Functions with Periodic Coefficients

Aidas Balčiūnas, Mindaugas Jasas, Renata Macaitienė and Darius Šiaučiūnas ()
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Aidas Balčiūnas: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
Mindaugas Jasas: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
Renata Macaitienė: Institute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania
Darius Šiaučiūnas: Institute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania

Mathematics, 2023, vol. 11, issue 9, 1-10

Abstract: Let a = { a m } and b = { b m } be two periodic sequences of complex numbers, and, additionally, a is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts ( ζ n T ( s + i τ ; a ) , ζ n T ( s + i τ , α ; b ) ) of absolutely convergent Dirichlet series ζ n T ( s ; a ) and ζ n T ( s , α ; b ) involving the sequences a and b is considered. Here, n T → ∞ and n T ≪ T 2 as T → ∞ . The coefficients of these series tend to a m and b m , respectively. It is proved that the set of the above shifts in the interval [ 0 , T ] has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions.

Keywords: Hurwitz zeta-function; joint universality; periodic Hurwitz zeta-function; periodic zeta-function; universality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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