Periodic Hurwitz Zeta-function Research Articles

A DISCRETE VERSION OF THE MISHOU THEOREM RELATED TO PERIODIC ZETA-FUNCTIONS

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts $\zeta_{u_N}(s+ikh_1; \ga)$ and $\zeta_{u_N}(s+ikh_2, \alpha; \gb)$ of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence $\ga$, and the periodic Hurwitz zeta-function, respectively. We suppose that $u_N\to\infty$ and $u_N\ll N^2$ as $N\to\infty$, and the set $\{(h_1\log p:\! p\in\! \PP), (h_2\log(m+\alpha): m\in \NN_0), 2\pi\}$ is linearly independent over $\QQ$.

On Mixed Joint Discrete Universality for a Class of Zeta-functions: One More Case

We prove a new case of mixed discrete joint universality theorem on approximation of certain target couple of analytic functions by the shifts of a pair consisting of the function $\varphi(s)$ belonging to wide class of Matsumoto zeta-functions and the periodic Hurwitz zeta-function $\zeta(s,\alpha;\mathfrak{B})$. We work under the condition that the common difference of arithmetical progression $h > 0$ is such that $\exp \{2\pi/h\}$ is a rational number and the parameter $\alpha$ is a transcendental number. The essential difference from the result in our previous article [10] is that here we do not study the class of partial zeta-functions $\varphi_{h}(s)$, but work with the class of the original functions $\varphi(s)$.

ON A DIRICHLET SERIES CONNECTED TO A PERIODIC HURWITZ ZETA-FUNCTION WITH TRANSCENDENTAL AND RATIONAL PARAMETER

In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.

A Modification of the Mixed Joint Universality Theorem for a Class of Zeta Functions

The property of zeta functions on mixed joint universality in the Voronin’s sense states that any two holomorphic functions can be approximated simultaneously with an accuracy of ε>0 by suitable vertical shifts of the pair consisting the Riemann and Hurwitz zeta functions. A rather general result can be obtained for the classes of zeta functions, particularly when an approximating pair is composed of the Matsumoto zeta functions’ class and the periodic Hurwitz zeta function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably ε>0. Moreover, we provide concluding remarks on certain more general mixed tuples of zeta functions.

JOINT DISCRETE APPROXIMATION OF A PAIR OF ANALYTIC FUNCTIONS BY PERIODIC ZETA-FUNCTIONS

In the paper, the problem of simultaneous approximation of a pair of analytic functions by a pair of discrete shifts of the periodic and periodic Hurwitz zeta-function is considered. The above shifts are defined by using the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function. For the proof of approximation theorems, a weak form of the Montgomery pair correlation conjecture is applied.

On mixed joint discrete universality for a class of zeta-functions. II

We study analytic properties of the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with a transcendental parameter. We first survey briefly previous results, and then investigate the mixed joint discrete value distribution and prove the mixed joint discrete universality theorem for these functions, in the case when common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are necessary.

A generalized limit theorem for the periodic Hurwitz zeta-function

A generalized limit theorem for the periodic Hurwitz zeta-function

ON APPROXIMATION OF ANALYTIC FUNCTIONS BY PERIODIC HURWITZ ZETA-FUNCTIONS

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental orrational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).

Universality of the Periodic Hurwitz Zeta-Function with Rational Parameter

The periodic Hurwitz zeta-function, a generalization of the classical Hurwitz zeta-function, is defined by a Dirichlet series with periodic coefficients and depends on a fixed parameter. We show that a wide class of analytic functions is approximated by shifts of a periodic zeta-function with rational parameter.

Periodinės Hurvico dzeta funkcijos reikšmių pasiskirstymas su svoriu

Straipsnyje yra įrodyta diskrečioji ribinė teorema su svoriu periodinei Hurvico dzeta funkcijai ζ(s, α; a) su transcendenčiuoju parametru α, 0 < α ≤ 1, kompleksinėje plokštumoje, kai papildomai yra naudojama svorio funkcija w(l).

A discrete limit theorem for the periodic Hurwitz zeta-function. II

In the paper, we prove a limit theorem of discrete type on the weak convergence of probability measures on the complex plane for the periodic Hurwitz zeta-function.

REMARKS ON THE MIXED JOINT UNIVERSALITY FOR A CLASS OF ZETA FUNCTIONS

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when $\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

Self-approximation of periodic Hurwitz zeta-functions

Let (s;!;A) be the periodic Hurwitz zeta-function. We look for real numbers and for which there exist many real numbers such that the shifts (s + i;! ;A) and (s + i;! ;A) are near each other.

A discrete limit theorem for the periodic Hurwitz zeta-function

In the paper, we prove a limit theorem of discrete type on the weak convergence of probability measures on the complex plane for the periodic Hurwitz zeta-function.

A weighted limit theorem for periodic Hurwitz zeta-function

In the paper, a weighted limit theorem for weakly convergent probability measures on the complex plane for the periodic Hurwitz zeta function is obtained.

Joint discrete universality of periodic zeta-functions

In this paper, we obtain the joint universality in the Voronin sense for the collection of Dirichlet L-function and periodic Hurwitz zeta-function.

The discrete universality of the periodic Hurwitz zeta function

The periodic Hurwitz zeta function , s=σ+it, 0<α≤1, is defined, for σ>1, by and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In this paper, a discrete universality theorem for the function with a transcendental parameter α is proved. Roughly speaking, this means that every analytic function can be approximated uniformly on compact sets by shifts , where m is a non-negative integer and h is a fixed positive number such that is rational.

Universality of the periodic Hurwitz zeta-function

In this article, the universality in the Voronin sense for the Hurwitz zeta-function with periodic coefficients is proved.

On the periodic Hurwitz zeta-function.

In this paper, an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved.