Abstract
In the paper, we prove a joint universality theorem for the Riemann zeta-function and a collection of Lerch zeta-functions with parameters algebraically independent over the field of rational numbers.
Highlights
Let λ ∈ R and α, 0 < α 1, be fixed parameters
It is well known that the Lerch zeta-function L(λ, α, s) with transcendental parameter α is universal
In [9], a joint universality theorem for the Riemann zeta-function ζ(s) and periodic
Summary
Let λ ∈ R and α, 0 < α 1, be fixed parameters. The Lerch zeta-function L(λ, α, s), s = σ + it, is defined, for σ > 1, by. In [9], a joint universality theorem for the Riemann zeta-function ζ(s) and periodic. We call the approximation property of the functions (1) in Theorem 3 a mixed joint universality because the function ζ(s) and the functions ζ(s, αj; Ajl) are of different types: the function ζ(s) has Euler product, while the functions ζ(s, αj; Ajl) with transcendental αj do not have Euler product over primes This is reflected in the approximated functions: the function f (s) must be non-vanishing on K, while the functions fjl are arbitrary continuous functions on Kjl. The first mixed joint universality theorem has been obtained by Mishou [10] for the Riemann zeta-function and Hurwitz zeta-function ζ(s, α) with transcendental parameter α. The first mixed joint universality theorem has been obtained by Mishou [10] for the Riemann zeta-function and Hurwitz zeta-function ζ(s, α) with transcendental parameter α This result in [11] has been generalized for a periodic zeta-function and a periodic Hurwitz zeta-function. The proof of Theorem 4 is based on a joint limit theorem on weakly convergent probability measures in the space of analytic functions
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