Joint universality of periodic zeta-functions with multiplicative coefficients. II

Abstract

In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence f kg of imaginary parts of nontrivial zeros of the Riemann zeta-function is obtained. For its proof, a weak form of the Montgomery pair correlation conjecture is used. The paper is a continuation of [A. Laurinčikas, M. Tekorė, Joint universality of periodic zeta-functions with multiplicative coefficients, Nonlinear Anal. Model. Control, 25(5):860–883, 2020] using nonlinear shifts for approximation of analytic functions.

Gram Points in the Universality of the Dirichlet Series with Periodic Coefficients

“…This estimate is the weak form of the Montgomery pair correlation conjecture [20]. Then, in [21], it was obtained that, under (4), for a fixed number h > 0, multiplicative sequence a,…”

Approximation of Analytic Functions by Shifts of Certain Compositions

“…Later, universality for compositions of other zeta-functions was obtained; for example, the paper [25] is devoted to compositions of zeta-functions of normalized Hecke cusp forms. Now we recall the main result of [22]. For j = 1, .…”

“…Joint universality theorems involving the function ζ(s, α) were studied in [15][16][17][18][19][20]. The papers [21,22] are devoted to joint approximation of analytic functions by generalized non-linear shifts of periodic zetafunctions. The aim of this paper is universality theorems for compositions of collections of periodic zeta-functions studied in [22].…”

“…, r, let a j = {a jm : m ∈ N} be a periodic sequences of complex numbers, and let ζ(s; a j ) be the corresponding periodic zeta-function. In [22], for shifts of ζ(s; a j ), the sequence {γ k : k ∈ N, γ k > 0} of imaginary parts of non-trivial zeros of the Riemann zeta-function is used. Moreover, it was required that the estimate…”

“…(Note that (1) follows from the Montgomery pair correlation hypothesis [26]). Then the main result of [22] is the following statement. Let #A denote the cardinality of the set A, and N runs over the set of natural numbers N.…”