Abstract
We prove that, for arbitrary Dirichlet L-functions L(s;chi _1),ldots ,L(s;chi _n) (including the case when chi _j is equivalent to chi _k for jne k), suitable shifts of type L(s+ialpha _jt^{a_j}log ^{b_j}t;chi _j) can simultaneously approximate any given set of analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs (a_j,b_j) are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where t runs over the set of positive integers.
Highlights
In 1977, Voronin [20] proved the so-called joint universality which, roughly speaking, states that any collection of Dirichlet L-functions associated with non-equivalent characters can simultaneously and uniformly approximate non-vanishing analytic functions in the above sense
In order to approximate a collection of non-vanishing continuous functions on some compact subset of {s ∈ C: 1/2 < Re(s) < 1} with connected complement, which are analytic in the interior, it is sufficient to take twists of the Riemann zeta function with non-equivalent Dirichlet characters
This idea was extended by Šleževiciene [17] to certain L-functions associated with multiplicative functions, by Laurincikas and Matsumoto [9] to L-functions associated with newforms twisted by non-equivalent characters, and by Steuding in [18, Sect. 12.3] to a wide class of L-functions with Euler product, which can be compared to the well-known Selberg class
Summary
In 1975, Voronin [19] discovered a universality property for the Riemann zeta function ζ (s), namely he proved that for every compact set K ⊂ {s ∈ C: 1/2 < Re(s) < 1} with connected complement, any non-vanishing continuous function f (s) on K , analytic in the interior of K , and every ε > 0, we have. In 1977, Voronin [20] proved the so-called joint universality which, roughly speaking, states that any collection of Dirichlet L-functions associated with non-equivalent characters can simultaneously and uniformly approximate non-vanishing analytic functions in the above sense. In order to approximate a collection of non-vanishing continuous functions on some compact subset of {s ∈ C: 1/2 < Re(s) < 1} with connected complement, which are analytic in the interior, it is sufficient to take twists of the Riemann zeta function with non-equivalent Dirichlet characters. One possible way to approximate a collection of analytic functions by a given L-function is to consider its twists with sufficiently many non-equivalent characters Another possibility to obtain a joint universality theorem by considering only one L-function was observed by Kaczorowski et al [5]. We might consider an L-function, a compact set K ⊂ {s ∈ C: 1/2 < Re(s) < 1} with connected complement, and nonvanishing continuous functions f1, . . . , fn on K , analytic in the interior of K , and ask for functions γ1, . . . , γn : R → R satisfying
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