Mathematics

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12 pages, 257 KiB

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 (registering DOI) - 29 Apr 2025

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)

14 pages, 266 KiB

Open AccessFeature PaperArticle

Mellin Transform of Weierstrass Zeta Function and Integral Representations of Some Lambert Series

by Namhoon Kim

Mathematics 2025, 13(4), 582; https://doi.org/10.3390/math13040582 - 10 Feb 2025

Cited by 1 | Viewed by 373

Abstract

We consider a series which combines two Dirichlet series constructed from the coefficients of a Laurent series and derive a general integral representation of the series as a Mellin transform. As an application, we obtain a family of Mellin integral identities involving the [...] Read more.

We consider a series which combines two Dirichlet series constructed from the coefficients of a Laurent series and derive a general integral representation of the series as a Mellin transform. As an application, we obtain a family of Mellin integral identities involving the Weierstrass elliptic functions and some Lambert series. These identities are used to derive some of the properties of the Lambert series. Full article

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

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 2 | Viewed by 953

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)

14 pages, 325 KiB

Open AccessArticle

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

by Darius Šiaučiūnas and Monika Tekorė

Mathematics 2023, 11(22), 4615; https://doi.org/10.3390/math11224615 - 10 Nov 2023

Cited by 1 | Viewed by 1071

Abstract

Let

a = {a_{m} : m \in N}

be a periodic multiplicative sequence of complex numbers and

L (s; a)

,

s = σ + i t

a Dirichlet series with coefficients

a_{m}

. In the [...] Read more.

Let

a = {a_{m} : m \in N}

be a periodic multiplicative sequence of complex numbers and

L (s; a)

,

s = σ + i t

a Dirichlet series with coefficients

a_{m}

. In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip

1 / 2 < σ < 1

via discrete shifts

L (s + i h t_{k}; a)

,

h > 0

,

k \in N

, where

{t_{k} : k \in N}

is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied. Full article

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

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 1721

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)

15 pages, 303 KiB

Open AccessFeature PaperArticle

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

by Virginija Garbaliauskienė, Antanas Laurinčikas and Darius Šiaučiūnas

Mathematics 2023, 11(10), 2315; https://doi.org/10.3390/math11102315 - 16 May 2023

Cited by 1 | Viewed by 1206

Abstract

In the paper, it is obtained that there are infinite discrete shifts

Ξ (s + i k h)

,

h > 0

,

k \in N_{0}

of the Mellin transform

Ξ (s)

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

In the paper, it is obtained that there are infinite discrete shifts

Ξ (s + i k h)

,

h > 0

,

k \in N_{0}

of the Mellin transform

Ξ (s)

of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a 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 Analytic Methods in Number Theory and Allied Fields)

10 pages, 295 KiB

Open AccessFeature PaperArticle

On the Mishou Theorem for Zeta-Functions with Periodic Coefficients

by Aidas Balčiūnas, Mindaugas Jasas, Renata Macaitienė and Darius Šiaučiūnas

Mathematics 2023, 11(9), 2042; https://doi.org/10.3390/math11092042 - 25 Apr 2023

Cited by 2 | Viewed by 1295

Abstract

Let

a = {a_{m}}

and

b = {b_{m}}

be two periodic sequences of complex numbers, and, additionally,

a

is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts [...] Read more.

Let

a = {a_{m}}

and

b = {b_{m}}

be two periodic sequences of complex numbers, and, additionally,

a

is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts

(ζ_{n_{T}} (s + i τ; a), ζ_{n_{T}} (s + i τ, α; b))

of absolutely convergent Dirichlet series

ζ_{n_{T}} (s; a)

and

ζ_{n_{T}} (s, α; b)

involving the sequences

a

and

b

is considered. Here,

n_{T} \to \infty

and

n_{T} ≪ T^{2}

as

T \to \infty

. The coefficients of these series tend to

a_{m}

and

b_{m}

, respectively. It is proved that the set of the above shifts in the interval

[0, T]

has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions. Full article

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

9 pages, 264 KiB

Open AccessArticle

Hurwitz Zeta Function Is Prime

by Marius Dundulis, Ramūnas Garunkštis, Erikas Karikovas and Raivydas Šimėnas

Mathematics 2023, 11(5), 1150; https://doi.org/10.3390/math11051150 - 25 Feb 2023

Viewed by 1478

Abstract

We proved that the Hurwitz zeta function is prime. In addition, we derived the Nevanlinna characteristic for this function. Full article

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

24 pages, 420 KiB

Open AccessArticle

Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms

by Nianliang Wang, Takako Kuzumaki and Shigeru Kanemitsu

Mathematics 2023, 11(4), 917; https://doi.org/10.3390/math11040917 - 11 Feb 2023

Viewed by 1564

Abstract

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma [...] Read more.

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. Full article

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

13 pages, 317 KiB

Open AccessFeature PaperArticle

A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function

by Antanas Laurinčikas and Renata Macaitienė

Mathematics 2023, 11(4), 799; https://doi.org/10.3390/math11040799 - 4 Feb 2023

Cited by 1 | Viewed by 1324

Abstract

Suppose that Q is a positive defined

n \times n

matrix, and

Q [\underset{̲}{x}] = {\underset{̲}{x}}^{T} Q \underset{̲}{x}

with

\underset{̲}{x} \in Z^{n}

. The Epstein zeta-function

ζ (s; Q)

, [...] Read more.

Suppose that Q is a positive defined

n \times n

matrix, and

Q [\underset{̲}{x}] = {\underset{̲}{x}}^{T} Q \underset{̲}{x}

with

\underset{̲}{x} \in Z^{n}

. The Epstein zeta-function

ζ (s; Q)

,

s = σ + i t

, is defined, for

σ > \frac{n}{2}

, by the series

ζ (s; Q) = \sum_{\underset{̲}{x} \in Z^{n} ∖ {\underset{̲}{0}}} {(Q [\underset{̲}{x}])}^{- s},

and it has a meromorphic continuation to the whole complex plane. Let

n ⩾ 4

be even, while

φ (t)

is an increasing differentiable function with a continuous monotonic bounded derivative

φ^{'} (t)

such that

φ (2 t) {(φ^{'} (t))}^{- 1} ≪ t

, and the sequence

{a φ (k)}

is uniformly distributed modulo 1. In the paper, it is obtained that

\frac{1}{N} # \{N ⩽ k ⩽ 2 N : ζ (σ + i φ (k); Q) \in A\}

,

A \in B (C)

, for

σ > \frac{n - 1}{2}

, converges weakly to an explicitly given probability measure on

(C, B (C))

as

N \to \infty

. Full article

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

12 pages, 312 KiB

Open AccessArticle

Joint Approximation of Analytic Functions by Shifts of Lerch Zeta-Functions

by Antanas Laurinčikas, Toma Mikalauskaitė and Darius Šiaučiūnas

Mathematics 2023, 11(3), 752; https://doi.org/10.3390/math11030752 - 2 Feb 2023

Cited by 2 | Viewed by 1083

Abstract

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical [...] Read more.

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical strip, which is approximated by the above tuples of shifts. Further, a generalization for some compositions of tuples of Lerch zeta-functions is given. Full article

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

15 pages, 317 KiB

Open AccessArticle

On Joint Universality in the Selberg–Steuding Class

by Roma Kačinskaitė, Antanas Laurinčikas and Brigita Žemaitienė

Mathematics 2023, 11(3), 737; https://doi.org/10.3390/math11030737 - 1 Feb 2023

Cited by 1 | Viewed by 1301

Abstract

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class

S

is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In [...] Read more.

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class

S

is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class

S

on the approximation of a collection of analytic functions by shifts

(L (s + i a_{1} τ), \dots, L (s + i a_{r} τ))

, where

a_{1}, \dots, a_{r}

are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials. Full article

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

7 pages, 251 KiB

Open AccessArticle

On the Order of Growth of Lerch Zeta Functions

by Jörn Steuding and Janyarak Tongsomporn

Mathematics 2023, 11(3), 723; https://doi.org/10.3390/math11030723 - 1 Feb 2023

Viewed by 1469

Abstract

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → [...] Read more.

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t^ϵ (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros. Full article

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

21 pages, 377 KiB

Open AccessArticle

Unification of Chowla’s Problem and Maillet–Demyanenko Determinants

by Nianliang Wang, Kalyan Chakraborty and Shigeru Kanemitsu

Mathematics 2023, 11(3), 655; https://doi.org/10.3390/math11030655 - 28 Jan 2023

Cited by 1 | Viewed by 1109

Abstract

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of

L (1, χ)

= \sum_{n = 1}^{\infty} \frac{χ (n)}{n}

. On the other hand, we refer to [...] Read more.

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of

L (1, χ)

= \sum_{n = 1}^{\infty} \frac{χ (n)}{n}

. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula. Full article

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

32 pages, 495 KiB

Open AccessArticle

A Unifying Principle in the Theory of Modular Relations

by Guodong Liu, Kalyan Chakraborty and Shigeru Kanemitsu

Mathematics 2023, 11(3), 535; https://doi.org/10.3390/math11030535 - 19 Jan 2023

Viewed by 1824

Abstract

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which [...] Read more.

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function

Z (s)

satisfying a suitable functional equation, one can generalize it to

Z_{f} (s)

in the form of an integral involving the Mellin transform

F (s)

of a certain suitable function

f (x)

and process it further as

{\tilde{Z}}_{f} (s)

. Under the condition that

F (s)

is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for

{\tilde{Z}}_{f} (s)

. Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion). Full article

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

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