JOINT DISCTRETE UNIVERSALITY OF DIRICHLET L-FUNCTIONS. II

Abstract

In 1975, S. M. Voronin obtained the universality of Dirichlet L-functions L(s, χ), s = σ +it. This means that, for every compact K of the strip {s ∈ C : 1 2 < σ < 1}, every continuous non-vanishing function on K which is analytic in the interior of K can be approximated uniformly on K by shifts L(s+iτ, χ), τ ∈ R. Also, S. M. Voronin investigating the functional independence of Dirichlet L-functions obtained the joint universality. In this case, a collection of analytic functions is approximated simultaneously by shifts L(s + iτ, χ1), . . . , L(s + iτ, χr), where χ1, . . . , χr are pairwise non-equivalent Dirichlet characters. The above universality is of continuous type. Also, a joint discrete universality for Dirichlet L-functions is known. In this case, a collection of analytic functions is approximated by discrete shifts L(s + ikh, χ1), . . . , L(s + ikh, χr), where h > 0 is a fixed number and k ∈ N0 = N ∪ {0}, and was proposed by B. Bagchi in 1981. For joint discrete universality of Dirichlet L-functions, a more general setting is possible. In [3], the approximation by shifts L(s + ikh1, χ1), . . . , L(s+ikhr, χr) with different h1 > 0, . . . , hr > 0 was considered. This paper is devoted to approximation by shifts L(s + ikh1, χ1), . . . , L(s + ikhr1 , χr1 ), L(s + ikh, χr1+1), . . . , L(s + ikh, χr), with different h1, . . . , hr1 , h. For this, the linear independence over Q of the set L(h1, . . . , hr1 , h; π) = { (h1 log p : p ∈ P), . . . ,(hr1 log p : p ∈ P), (h log p : p ∈ P); π } , where P denotes the set of all prime numbers, is applied.