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15 pages, 298 KiB

Open AccessFeature PaperArticle

The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

by Antanas Laurinčikas

Axioms 2025, 14(6), 472; https://doi.org/10.3390/axioms14060472 - 17 Jun 2025

Viewed by 298

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where [...] Read more.

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where

H \in [T^{27 / 82}, T^{1 / 2}]

. The cases of all parameters,

λ, α \in (0, 1]

, are considered. If the set

{\log (m + α) : m \in N_{0}}

is linearly independent over

Q

, then every analytic function in the strip

{s = σ + i t \in C : σ \in (1 / 2, 1)}

is approximated by the above shifts. Full article

12 pages, 291 KiB

Open AccessArticle

A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions

by Antanas Laurinčikas and Renata Macaitienė

Symmetry 2025, 17(6), 814; https://doi.org/10.3390/sym17060814 - 23 May 2025

Viewed by 240

Abstract

For

j = 1, \dots, r

, let

Q_{j}

be a positive definite

n_{j} \times n_{j}

matrix, and

ζ (s_{j}; Q_{j})

denote the corresponding Epstein zeta-function. In this paper, assuming that [...] Read more.

For

j = 1, \dots, r

, let

Q_{j}

be a positive definite

n_{j} \times n_{j}

matrix, and

ζ (s_{j}; Q_{j})

denote the corresponding Epstein zeta-function. In this paper, assuming that

n_{j} ⩾ 4

is even and

{\underset{̲}{x}}^{T} Q_{j} \underset{̲}{x} \in Z

,

\underset{̲}{x} \in Z^{r} ∖ {\underset{̲}{0}}

, a joint limit theorem of Bohr–Jessen type for the functions

ζ (s_{1}; Q_{1}), \dots, ζ (s_{r}; Q_{r})

, by using generalizing shifts

ζ (σ_{1} + i φ_{1} (t); Q_{1}), \dots, ζ (σ_{r} + i φ_{r} (t); Q_{r})

, is proved. Here, the functions

φ_{1} (t), \dots, φ_{r} (t)

are increasing to

+ \infty

, with monotonic derivatives

φ_{j}^{'} (t)

satisfying the asymptotic growth conditions:

φ_{j} (t) ≪ t φ_{j}^{'} (t)

, and

φ_{j}^{'} (t) = o (φ_{j + 1}^{'} (t))

as

t \to \infty

. An explicit form of the limit measure is given. This theorem extends and generalizes the previous result on the joint value-distribution of Epstein zeta-functions. Full article

13 pages, 269 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On Equivalents of the Riemann Hypothesis Connected to the Approximation Properties of the Zeta Function

by Antanas Laurinčikas

Axioms 2025, 14(3), 169; https://doi.org/10.3390/axioms14030169 - 26 Feb 2025

Viewed by 799

Abstract

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

,

m \in N

) lie on the critical line

σ = 1 / 2

. [...] Read more.

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

,

m \in N

) lie on the critical line

σ = 1 / 2

. In this paper, combining the universality property of

ζ (s)

with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts

ζ (s + i t_{τ})

approximating the function

ζ (s)

. Here,

t_{τ}

denotes the Gram function, which is a continuous extension of the Gram points. Full article

17 pages, 312 KiB

Open AccessFeature PaperArticle

On Discrete Shifts of Some Beurling Zeta Functions

by Antanas Laurinčikas and Darius Šiaučiūnas

Mathematics 2025, 13(1), 48; https://doi.org/10.3390/math13010048 - 26 Dec 2024

Viewed by 967

Abstract

We consider the Beurling zeta function

ζ_{P} (s)

,

s = σ + i t

, of the system of generalized prime numbers

P

with generalized integers m satisfying the condition [...] Read more.

We consider the Beurling zeta function

ζ_{P} (s)

,

s = σ + i t

, of the system of generalized prime numbers

P

with generalized integers m satisfying the condition

\sum_{m ⩽ x} 1 = a x + O (x^{δ})

,

a > 0

,

0 ⩽ δ < 1

, and suppose that

ζ_{P} (s)

has a bounded mean square for

σ > σ_{P}

with some

σ_{P} < 1

. Then, we prove that, for every

h > 0

, there exists a closed non-empty set of analytic functions that are approximated by discrete shifts

ζ_{P} (s + i l h)

. This set shifts has a positive density. For the proof, a weak convergence of probability measures in the space of analytic functions is applied. Full article

16 pages, 312 KiB

Open AccessArticle

Joint Approximation by the Riemann and Hurwitz Zeta-Functions in Short Intervals

by Antanas Laurinčikas

Symmetry 2024, 16(12), 1707; https://doi.org/10.3390/sym16121707 - 23 Dec 2024

Cited by 1 | Viewed by 786

Abstract

In this study, the approximation of a pair of analytic functions defined on the strip

{s = σ + i t \in C : 1 / 2 < σ < 1}

by shifts [...] Read more.

In this study, the approximation of a pair of analytic functions defined on the strip

{s = σ + i t \in C : 1 / 2 < σ < 1}

by shifts

(ζ (s + i τ), ζ (s + i τ, α))

,

τ \in R

, of the Riemann and Hurwitz zeta-functions with transcendental

α

in the interval

[T, T + H]

with

T^{27 / 82} ⩽ H ⩽ T^{1 / 2}

was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval

[0, T]

, and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied. Full article

(This article belongs to the Section Mathematics)

7 pages, 245 KiB

Open AccessArticle

Remarks on the Connection of the Riemann Hypothesis to Self-Approximation

by Antanas Laurinčikas

Computation 2024, 12(8), 164; https://doi.org/10.3390/computation12080164 - 14 Aug 2024

Cited by 1 | Viewed by 1300

Abstract

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function

ζ (s)

by shifts

ζ (s + i τ)

. In this paper, it is determined [...] Read more.

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function

ζ (s)

by shifts

ζ (s + i τ)

. In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating

ζ (s)

with all but at most countably many accuracies

ε > 0

. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed. Full article

13 pages, 283 KiB

Open AccessArticle

The Mean Square of the Hurwitz Zeta-Function in Short Intervals

by Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2024, 13(8), 510; https://doi.org/10.3390/axioms13080510 - 28 Jul 2024

Cited by 4 | Viewed by 904

Abstract

The Hurwitz zeta-function

ζ (s, α)

,

s = σ + i t

, with parameter

0 < α ⩽ 1

is a generalization of the Riemann zeta-function

ζ (s)

( [...] Read more.

The Hurwitz zeta-function

ζ (s, α)

,

s = σ + i t

, with parameter

0 < α ⩽ 1

is a generalization of the Riemann zeta-function

ζ (s)

(

ζ (s, 1) = ζ (s)

) and was introduced at the end of the 19th century. The function

ζ (s, α)

plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function

ζ (s, α)

is the main example of zeta-functions without Euler’s product (except for the cases

α = 1

,

α = 1 / 2

), and its value distribution is governed by arithmetical properties of

α

. For the majority of zeta-functions,

ζ (s, α)

for some

α

is universal, i.e., its shifts

ζ (s + i τ, α)

,

τ \in R

, approximate every analytic function defined in the strip

{s : 1 / 2 < σ < 1}

. For needs of effectivization of the universality property for

ζ (s, α)

, the interval for

τ

must be as short as possible, and this can be achieved by using the mean square estimate for

ζ (σ + i t, α)

in short intervals. In this paper, we obtain the bound

O (H)

for that mean square over the interval

[T - H, T + H]

, with

T^{27 / 82} ⩽ H ⩽ T^{σ}

and

1 / 2 < σ ⩽ 7 / 12

. This is the first result on the mean square for

ζ (s, α)

in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for

ζ (s, α)

and other zeta-functions in short intervals. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

15 pages, 293 KiB

Open AccessArticle

A Joint Limit Theorem for Epstein and Hurwitz Zeta-Functions

by Hany Gerges, Antanas Laurinčikas and Renata Macaitienė

Mathematics 2024, 12(13), 1922; https://doi.org/10.3390/math12131922 - 21 Jun 2024

Cited by 2 | Viewed by 906

Abstract

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on

C^{2}

defined by means of the Epstein

ζ (s; Q)

and Hurwitz

ζ (s, α)

zeta-functions. The [...] Read more.

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on

C^{2}

defined by means of the Epstein

ζ (s; Q)

and Hurwitz

ζ (s, α)

zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter

α

are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. Full article

17 pages, 303 KiB

Open AccessArticle

Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function

by Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2024, 13(4), 251; https://doi.org/10.3390/axioms13040251 - 10 Apr 2024

Cited by 3 | Viewed by 1436

Abstract

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform

Z (s)

,

s = σ + i t

, with fixed [...] Read more.

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform

Z (s)

,

s = σ + i t

, with fixed

1 / 2 < σ < 1

, of the square

{| ζ (1 / 2 + i t) |}^{2}

of the Riemann zeta-function. We consider probability measures defined by means of

Z (σ + i φ (t))

, where

φ (t)

,

t ⩾ t_{0} > 0

, is an increasing to

+ \infty

differentiable function with monotonically decreasing derivative

φ^{'} (t)

satisfying a certain normalizing estimate related to the mean square of the function

Z (σ + i φ (t))

. This allows us to extend the distribution laws for

Z (s)

. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

23 pages, 343 KiB

Open AccessArticle

On Universality of Some Beurling Zeta-Functions

by Andrius Geštautas and Antanas Laurinčikas

Axioms 2024, 13(3), 145; https://doi.org/10.3390/axioms13030145 - 23 Feb 2024

Viewed by 1396

Abstract

Let

P

be the set of generalized prime numbers, and

ζ_{P} (s)

,

s = σ + i t

, denote the Beurling zeta-function associated with

P

. In the paper, we consider the approximation of analytic functions by using [...] Read more.

Let

P

be the set of generalized prime numbers, and

ζ_{P} (s)

,

s = σ + i t

, denote the Beurling zeta-function associated with

P

. In the paper, we consider the approximation of analytic functions by using shifts

ζ_{P} (s + i τ)

,

τ \in R

. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set

{log p : p \in P}

, and the existence of a bounded mean square for

ζ_{P} (s)

. Under the above hypotheses, we obtain the universality of the function

ζ_{P} (s)

. This means that the set of shifts

ζ_{P} (s + i τ)

approximating a given analytic function defined on a certain strip

\hat{σ} < σ < 1

has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

15 pages, 298 KiB

Open AccessFeature PaperArticle

On Value Distribution of Certain Beurling Zeta-Functions

by Antanas Laurinčikas

Mathematics 2024, 12(3), 459; https://doi.org/10.3390/math12030459 - 31 Jan 2024

Cited by 3 | Viewed by 1045

Abstract

In this paper, the approximation of analytic functions by shifts

ζ_{P} (s + i τ)

of Beurling zeta-functions

ζ_{P} (s)

of certain systems

P

of generalized prime numbers is discussed. It is required that the system of [...] Read more.

In this paper, the approximation of analytic functions by shifts

ζ_{P} (s + i τ)

of Beurling zeta-functions

ζ_{P} (s)

of certain systems

P

of generalized prime numbers is discussed. It is required that the system of generalized integers

N_{P}

generated by

P

satisfies

\sum_{m ⩽ x, m \in N} 1 = a x + O (x^{δ})

,

a > 0

,

0 ⩽ δ < 1

, and the function

ζ_{P} (s)

in some strip lying in

\hat{σ} < σ < 1

,

\hat{σ} > δ

, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

12 pages, 297 KiB

Open AccessArticle

On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform

by Antanas Laurinčikas

Axioms 2023, 12(8), 789; https://doi.org/10.3390/axioms12080789 - 14 Aug 2023

Cited by 2 | Viewed by 1295

Abstract

In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical [...] Read more.

In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip

{s \in C : 1 / 2 < σ < 1}

. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

14 pages, 309 KiB

Open AccessArticle

Joint Discrete Universality in the Selberg–Steuding Class

by Roma Kačinskaitė, Antanas Laurinčikas and Brigita Žemaitienė

Axioms 2023, 12(7), 674; https://doi.org/10.3390/axioms12070674 - 8 Jul 2023

Viewed by 1022

Abstract

In the paper, we consider the approximation of analytic functions by shifts from the wide class

\tilde{S}

of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality [...] Read more.

In the paper, we consider the approximation of analytic functions by shifts from the wide class

\tilde{S}

of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality theorem for the function

L (s) \in \tilde{S}

. Using the linear independence over

Q

of the multiset

\{(h_{j} log p : p \in P), j = 1, \dots, r; 2 π\}

for positive

h_{j}

, we obtain that there are many infinite shifts

(L (s + i k h_{1}), \dots, L (s + i k h_{r}))

,

k = 0, 1, \dots

, approximating every collection

(f_{1} (s), \dots, f_{r} (s))

of analytic non-vanishing functions defined in the strip

{s \in C : σ_{L} < σ < 1}

, where

σ_{L}

is a degree of the function

L (s)

. For the proof, the probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

19 pages, 351 KiB

Open AccessArticle

On the Approximation by Mellin Transform of the Riemann Zeta-Function

by Maxim Korolev and Antanas Laurinčikas

Axioms 2023, 12(6), 520; https://doi.org/10.3390/axioms12060520 - 25 May 2023

Cited by 4 | Viewed by 1533

Abstract

This paper is devoted to the approximation of a certain class of analytic functions by shifts

Z (s + i τ)

,

τ \in R

, of the modified Mellin transform

Z (s)

of the square of the Riemann [...] Read more.

This paper is devoted to the approximation of a certain class of analytic functions by shifts

Z (s + i τ)

,

τ \in R

, of the modified Mellin transform

Z (s)

of the square of the Riemann zeta-function

ζ (1 / 2 + i t)

. More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts

Z (s + i τ)

, which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure. Full article

12 pages, 322 KiB

Open AccessFeature PaperArticle

Generalized Universality for Compositions of the Riemann Zeta-Function in Short Intervals

by Antanas Laurinčikas and Renata Macaitienė

Mathematics 2023, 11(11), 2436; https://doi.org/10.3390/math11112436 - 24 May 2023

Cited by 1 | Viewed by 1808

Abstract

In the paper, the approximation of analytic functions on compact sets of the strip

{s = σ + i t \in C ∣ 1 / 2 < σ < 1}

by shifts [...] Read more.

In the paper, the approximation of analytic functions on compact sets of the strip

{s = σ + i t \in C ∣ 1 / 2 < σ < 1}

by shifts

F (ζ (s + i u_{1} (τ)), \dots, ζ (s + i u_{r} (τ)))

, where

ζ (s)

is the Riemann zeta-function,

u_{1}, \dots, u_{r}

are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval

[T, T + H]

with

H = o (T)

,

T \to \infty

, has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

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