MDPI - Publisher of Open Access Journals

11 pages, 278 KB

Open AccessArticle

Perturbed Dirichlet Series and the Difference Operator

by Nianlian Wang, Jay Mehta and Shigeru Kanemitsu

Axioms 2026, 15(4), 277; https://doi.org/10.3390/axioms15040277 - 10 Apr 2026

The most well-known perturbed Dirichlet series is the Hurwitz zeta-function. Its analytic continuation via the binomial expansion has been studied extensively, beginning with Wilton’s work. In this paper, we shall provide, above all things, two striking instances of the binomial expansion. One is [...] Read more.

The most well-known perturbed Dirichlet series is the Hurwitz zeta-function. Its analytic continuation via the binomial expansion has been studied extensively, beginning with Wilton’s work. In this paper, we shall provide, above all things, two striking instances of the binomial expansion. One is elucidation of Mikolás an integral formula for the Hurwitz zeta-function valid in the critical strip to the effect that it is a manifestation of the picking-up principle of the values at the poles of the gamma function of the binomial expansion. The other is a new proof of Hasse’s formula by the binomial expansion. Also, we show the effectiveness of the difference operator in dealing with a series of the form

\sum_{n = 0}^{\infty} {(n + a_{1})}^{- s_{1}} {(n + a_{2})}^{- s_{2}} {(n + a_{3})}^{- s_{3}} \dots, Re s_{j} > 2, j = 1, 2, \dots

where

0 < a_{j} \leq 1

or

a_{j} \in H

(in the upper half-plane). Furthermore, elucidation of the above results is made in the light of the Hardy–Hecke transform. Full article

(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)

19 pages, 327 KB

Open AccessFeature PaperArticle

Analytic Continuation of the Hurwitz Transform

by Namhoon Kim

Mathematics 2026, 14(2), 271; https://doi.org/10.3390/math14020271 - 10 Jan 2026

Viewed by 422

Abstract

An integral over the unit interval of the product of a given function and the Hurwitz zeta function is known as the Hurwitz transform of the function. We give sufficient conditions on the function for the Hurwitz transform to continue meromorphically to a [...] Read more.

An integral over the unit interval of the product of a given function and the Hurwitz zeta function is known as the Hurwitz transform of the function. We give sufficient conditions on the function for the Hurwitz transform to continue meromorphically to a larger region and to the complex plane. Integral representations for the meromorphic extension of the Hurwitz transform are given, and some new integral identities involving the Hurwitz zeta function are derived. Full article

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

22 pages, 363 KB

Open AccessFeature PaperArticle

Joint Discrete Approximation by Shifts of Hurwitz Zeta-Function: The Case of Short Intervals

by Antanas Laurinčikas and Darius Šiaučiūnas

Mathematics 2025, 13(22), 3654; https://doi.org/10.3390/math13223654 - 14 Nov 2025

Viewed by 653

Abstract

Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip [...] Read more.

Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip

D = {s = σ + i t : 1 / 2 < σ < 1}

. However, such an approximation causes efficiency problems, and applying short intervals is one of the measures to make that approximation more effective. In this paper, we consider the simultaneous approximation of a tuple of analytic functions in the strip

D

by discrete shifts

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

with positive

h_{1}, \dots, h_{r}

of Hurwitz zeta-functions in the interval

[N, N + M]

with

M = {max}_{1 ⩽ j ⩽ r} (h_{j}^{- 1}, {(N h_{j})}^{23 / 70})

. Two cases are considered:

1^{°}

the set

{(h_{j} log (m + α_{j}), m \in N_{0}, j = 1, \dots, r), 2 π}

is linearly independent over

Q

; and

2^{°}

a general case, where

α_{j}

and

h_{j}

are arbitrary. In case

1^{°}

, we obtain that the set of approximating shifts has a positive lower density (and density) for every tuple of analytic functions. In case

2^{°}

, the set of approximated functions forms a certain closed set. For the proof, an approach based on new limit theorems on weakly convergent probability measures in the space of analytic functions in short intervals is applied. The power

η = 23 / 70

comes from a new mean square estimate for the Hurwitz zeta-function. Full article

14 pages, 305 KB

Open AccessArticle

Some Properties of Meromorphic Functions Defined by the Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Nicoleta Breaz and Abeer M. Albalahi

Mathematics 2025, 13(21), 3430; https://doi.org/10.3390/math13213430 - 28 Oct 2025

Cited by 1 | Viewed by 591

Abstract

The findings of this study are connected with geometric function theory and were acquired using subordination-based techniques in conjunction with the Hurwitz–Lerch Zeta function. We used the Hurwitz–Lerch Zeta function to investigate certain properties of multivalent meromorphic functions. The primary objective of this [...] Read more.

The findings of this study are connected with geometric function theory and were acquired using subordination-based techniques in conjunction with the Hurwitz–Lerch Zeta function. We used the Hurwitz–Lerch Zeta function to investigate certain properties of multivalent meromorphic functions. The primary objective of this study is to provide an investigation on the argument properties of multivalent meromorphic functions in a punctured open unit disc and to obtain some results for its subclass. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

22 pages, 360 KB

Open AccessArticle

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

by Antanas Laurinčikas and Darius Šiaučiūnas

Symmetry 2025, 17(10), 1662; https://doi.org/10.3390/sym17101662 - 5 Oct 2025

Viewed by 725

Abstract

In this paper, we prove the theorems on the simultaneous approximation of a pair of analytic functions by discrete shifts

(ζ (s + i k h_{1}), ζ (s + i k h_{2}, α))

[...] Read more.

In this paper, we prove the theorems on the simultaneous approximation of a pair of analytic functions by discrete shifts

(ζ (s + i k h_{1}), ζ (s + i k h_{2}, α))

,

h_{1} > 0

,

h_{2} > 0

of the Riemann zeta function

ζ (s)

and Hurwitz zeta function

ζ (s, α)

. The lower density and density of the above approximating shifts are considered in short intervals

[N, N + M]

as

N \to \infty

with

M = o (N)

. If the set

{(h_{1} log p : p \in P), (h_{2} log (m + α) : m \in N_{0}), 2 π}

is linearly independent over

Q

, the class of approximated pairs is explicitly given. If

α

and

h_{1}

,

h_{2}

are arbitrary, then it is known that the set of approximated pairs is a certain non-empty closed subset of

H^{2} (Δ)

, where

H (Δ)

is the space of analytic functions on the strip

Δ = {s \in C : 1 / 2 < Re s < 1}

. For the proof, limit theorems on weakly convergent probability measures in the space

H^{2} (Δ)

are applied. Full article

(This article belongs to the Section Mathematics)

15 pages, 307 KB

Open AccessArticle

Fuzzy Treatment for Meromorphic Classes of Admissible Functions Connected to Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Rabab Sidaoui

Axioms 2025, 14(7), 523; https://doi.org/10.3390/axioms14070523 - 8 Jul 2025

Cited by 3 | Viewed by 602

Abstract

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are [...] Read more.

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are related to a class of complex parameter operators. Complex analysis ideas from geometric function theory are used to derive fuzzy differential subordination conclusions. Due to the compositional structure of the operator, some pertinent classes of admissible functions are studied through the application of fuzzy differential subordination. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

15 pages, 298 KB

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 728

Abstract

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

23 pages, 340 KB

Open AccessArticle

Third-Order Fuzzy Subordination and Superordination on Analytic Functions on Punctured Unit Disk

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah and Abeer M. Albalahi

Axioms 2025, 14(5), 378; https://doi.org/10.3390/axioms14050378 - 17 May 2025

Cited by 1 | Viewed by 605

Abstract

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best [...] Read more.

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best subordinant for the third-order fuzzy differential subordinations and superordinations, respectively. The investigation concludes with the assertion of sandwich-type theorems connecting the conclusions of the studies conducted using the particular methods of the theories of the third-order fuzzy differential subordination and superordination, respectively. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

12 pages, 257 KB

Open AccessArticle

Partial Sums of the Hurwitz and Allied Functions and Their Special Values

by Nianliang Wang, Ruiyang Li and Takako Kuzumaki

Mathematics 2025, 13(9), 1469; https://doi.org/10.3390/math13091469 - 29 Apr 2025

Cited by 1 | Viewed by 795

Abstract

We supplement the formulas for partial sums of the Hurwitz zeta-function and its derivatives, producing more integral representations and generic definitions of important constants. Then, these are used, coupled with the functional equation for the completed zeta-function to clarify the results of Choudhury, [...] Read more.

We supplement the formulas for partial sums of the Hurwitz zeta-function and its derivatives, producing more integral representations and generic definitions of important constants. Then, these are used, coupled with the functional equation for the completed zeta-function to clarify the results of Choudhury, giving rise to closed expressions for the Riemann zeta-function and its derivatives. Full article

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

16 pages, 312 KB

Open AccessFeature PaperArticle

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 2 | Viewed by 1121

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)

14 pages, 293 KB

Open AccessArticle

Fuzzy Subordination Results for Meromorphic Functions Associated with Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi and Marwa Ennaceur

Mathematics 2024, 12(23), 3721; https://doi.org/10.3390/math12233721 - 27 Nov 2024

Cited by 3 | Viewed by 1229

Abstract

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using [...] Read more.

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function for obtaining new fuzzy differential subordination results. Furthermore, the best fuzzy dominants are provided for each of the fuzzy differential subordinations investigated. The results presented enhance the approach to fuzzy differential subordination theory by giving new results involving operators in the study, for which starlikeness and convexity properties are revealed using the fuzzy differential subordination theory. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

12 pages, 274 KB

Open AccessFeature PaperArticle

Series over Bessel Functions as Series in Terms of Riemann’s Zeta Function

by Slobodan B. Tričković and Miomir S. Stanković

Mathematics 2024, 12(19), 3000; https://doi.org/10.3390/math12193000 - 26 Sep 2024

Viewed by 940

Abstract

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann [...] Read more.

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions. Full article

15 pages, 295 KB

Open AccessFeature PaperArticle

On Closed Forms of Some Trigonometric Series

by Slobodan B. Tričković and Miomir S. Stanković

Axioms 2024, 13(9), 631; https://doi.org/10.3390/axioms13090631 - 14 Sep 2024

Cited by 1 | Viewed by 1075

Abstract

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these [...] Read more.

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative. Full article

(This article belongs to the Special Issue Special Functions and Related Topics)

23 pages, 539 KB

Open AccessArticle

On Convoluted Forms of Multivariate Legendre-Hermite Polynomials with Algebraic Matrix Based Approach

by Mumtaz Riyasat, Amal S. Alali, Shahid Ahmad Wani and Subuhi Khan

Mathematics 2024, 12(17), 2662; https://doi.org/10.3390/math12172662 - 27 Aug 2024

Viewed by 1424

Abstract

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations [...] Read more.

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and

λ

-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

13 pages, 283 KB

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 9 | Viewed by 1424

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)

Search Results (63)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (63)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI