On the vertical distribution of values of L-functions in the Selberg class

We prove explicit formulae for [Formula: see text]-points of [Formula: see text]-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function [Formula: see text] to the case of [Formula: see text]-points of the aforementioned [Formula: see text]-functions. This result implies the uniform distribution of subsequences of [Formula: see text]-points and from this a discrete universality theorem in the spirit of Voronin is derived.

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

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.

ON MIXED JOINT DISCRETE UNIVERSALITY FOR A CLASS OF ZETA-FUNCTIONS: A FURTHER GENERALIZATION

We present the most general at this moment results on the discrete mixed joint value-distribution (Theorems 5 and 6) and the universality property (Theorems 3 and 4) for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.

Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function

Let 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.

WEIGHTED DISCRETE UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION

It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iτ), τ ϵ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if τ take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when τ takes values from the arithmetic progression {kh : k ϵN} with arbitrary fixed h > 0. For this, two types of h are considered.

On joint approximation of analytic functions by nonlinear shifts of zeta-functions of certain cusp forms

In the paper, joint discrete universality theorems on the simultaneous approximation of a collection of analytic functions by a collection of discrete shifts of zeta-functions attached to normalized Hecke-eigen cusp forms are obtained. These shifts are defined by means of nonlinear differentiable functions that satisfy certain growth conditions, and their combination on positive integers is uniformly distributed modulo 1.

Discrete universality of the Riemann zeta-function in short intervals

We consider the approximation of analytic functions by shifts of the Riemann zeta-function ?(s+ikh) with fixed h > 0 when positive integers k run over the interval [N,N+M], where N1/3(logN)26=15 ? M ? N, and prove that those k have a positive lower density as N ? ?. The same is true for some compositions. Two types of h > 0 are discussed separately.