A generalization of the infinitary divisibility relation: Algebraic and analytic properties

2019 ◽  
Vol 15 (09) ◽  
pp. 1771-1792
Author(s):  
Joseph Vade Burnett ◽  
Otto Vaughn Osterman
Keyword(s):  
Unique Factorization ◽  
Summatory Function ◽  
Algebraic Properties ◽  
Positive Integers ◽  
Generalized Type

We consider a generalized type of unique factorization of the positive integers with restrictions on the exponents and view them as a family of arithmetic convolutions and divisibility relations, similar to the convolutions defined by Narkewicz [On a class of arithmetical convolutions, Colloq. Math. 10 (1963) 81–94]. We introduce special types of multiplicativity corresponding to these convolutions, and discuss algebraic properties of the associated arithmetic convolutions and analogs of the Möbius functions. We also prove asymptotics for analogs of the totient function, totient summatory function, and divisor summatory function.

scholarly journals LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

2019 ◽  
Vol 24 (3) ◽  
pp. 404-421
Author(s):  
Lahoucine Elaissaoui ◽  
Zine El-Abidine Guennoun
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integrable Function ◽  
Harmonic Series ◽  
Algebraic Properties ◽  
Tangent Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Square Integrable Function ◽  
Positive Integers

We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.

scholarly journals Exponentials and Logarithms Properties in an Extended Complex Number Field

2021 ◽  
Author(s):  
Daniel Tischhauser
Keyword(s):  
Complex Number ◽  
Number System ◽  
Complex Numbers ◽  
Binary Operations ◽  
Algebraic Properties ◽  
Extended Complex ◽  
Multivalued Functions ◽  
Positive Integers ◽  
Complex Exponential ◽  
Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we prove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

scholarly journals Topological monoids of monotone injective partial selfmaps of ℕ with cofinite domain and image

2011 ◽  
Vol 48 (3) ◽  
pp. 342-353 ◽  
Author(s):  
Oleg Gutik ◽  
Dušan Repovš
Keyword(s):  
Inverse Semigroup ◽  
Topological Semigroup ◽  
Locally Compact ◽  
Bicyclic Semigroup ◽  
Trivial Group ◽  
Topological Inverse Semigroup ◽  
Compact Topology ◽  
Algebraic Properties ◽  
Positive Integers

In this paper we study the semigroup ℐ ∞↗ (ℕ) of partial cofinal monotone bijective transformations of the set of positive integers ℕ. We show that the semigroup ℐ ∞↗ (ℕ) has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology τ on ℐ ∞↗ (ℕ) such that (ℐ ∞↗ (ℕ); τ) is a topological inverse semigroup, is discrete. Finally, we describe the closure of (ℐ ∞↗ (ℕ); τ) in a topological semigroup.

scholarly journals On the average order of the gcd-sum function over arbitrary sets of integers

2021 ◽  
Vol 27 (3) ◽  
pp. 16-28
Author(s):  
V. Siva Rama Prasad ◽  
◽  
P. Anantha Reddy ◽  
Keyword(s):  
Asymptotic Formula ◽  
Greatest Common Divisor ◽  
Average Order ◽  
Summatory Function ◽  
Asymptotic Formulae ◽  
Error Terms ◽  
Positive Integers

Let \mathbb{N} denote the set of all positive integers and for j,n \in \mathbb{N}, let (j,n) denote their greatest common divisor. For any S\subseteq \mathbb{N}, we define P_{S}(n) to be the sum of those (j,n) \in S, where j \in \{1,2,3, \ldots, n\}. An asymptotic formula for the summatory function of P_{S}(n) is obtained in this paper which is applicable to a variety of sets S. Also the formula given by Bordellès for the summatory function of P_{\mathbb{N}}(n) can be derived from our result. Further, depending on the structure of S, the asymptotic formulae obtained from our theorem give better error terms than those deducible from a theorem of Bordellès (see Remark 4.4).

7. Conjectures and theorems

2020 ◽  
pp. 112-129
Author(s):  
Robin Wilson
Keyword(s):  
Prime Numbers ◽  
Unique Factorization ◽  
The 1950S ◽  
Positive Integers ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

‘Conjectures and theorems’ investigates a number of topics, such as the distribution of prime numbers, and two unsolved problems, Goldbach’s conjecture and the twin prime conjecture. The factorization of positive integers into primes is unique, but this does not hold for certain other systems of numbers. A more in-depth look at unique factorization gives deeper results, including a proposed result of Gauss. Mathematicians in the 1950s and 1960s confirmed that he was correct, as shown in the so-called ‘Baker-Heegner-Stark theorem’.

On a Class of Archimedean Integral Domains

1976 ◽  
Vol 28 (2) ◽  
pp. 365-375 ◽  
Author(s):  
Raymond A. Beauregard ◽  
David E. Dobbs
Keyword(s):  
Number Theory ◽  
Integral Domain ◽  
Unique Factorization ◽  
Elementary Number Theory ◽  
Integral Domains ◽  
Elementary Number ◽  
Unique Factorization Domain ◽  
Starting Point ◽  
Positive Integers

Our starting point is an observation in elementary number theory [10, Exercise 26, p. 17]: if a and b are positive integers such that each number in the sequence a, b2, a3, b4, … divides the next, then a = b. Its proof depends only on Z being a unique factorization domain (UFD) whose units are 1, —1. Accordingly, we abstract and say that a (commutative integral) domain R satisfies (*) in case, whenever nonzero elements a and b in R are such that each element in the sequence a, b2, a3, b4, … divides the next, then a and b are associates in R (that is, a = bu for some unit u of R). The main objective of this paper is the study of the class of domains satisfying (*).

scholarly journals On class (n, mBQ) Operators

2021 ◽  
Vol 11 (2) ◽  
pp. 053-057
Author(s):  
Wanjala Victor ◽  
Beatrice Adhiambo Obiero
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Algebraic Properties ◽  
Positive Integers ◽  
Power Class

In this paper, we introduce the class of (n, mBQ) operators acting on a complex Hilbert space H. An operator if T ∈ B (H) is said to belong to class (n, mBQ) if T ∗2mT 2n commutes with (T ∗mTn ) 2 equivalently [T ∗2mT 2n, (T ∗mTn)2] = 0, for a positive integers n and m. We investigate algebraic properties that this class enjoys. Have. We analyze the relation of this class to (n,m)-power class (Q) operators.

A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS

2008 ◽  
Vol 04 (04) ◽  
pp. 549-561 ◽  
Author(s):  
EMIL DANIEL SCHWAB ◽  
PENTTI HAUKKANEN
Keyword(s):  
Fundamental Theorem ◽  
Factorization Theorem ◽  
Unique Factorization ◽  
Multiplicative Semigroup ◽  
Arithmetical Functions ◽  
Bicyclic Semigroup ◽  
Fundamental Theorem Of Arithmetic ◽  
Positive Integers ◽  
Special Case

We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

scholarly journals Endomorphism rings of p-local finite spectra are semi-perfect

2009 ◽  
Vol 139 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Petar Pavešić
Keyword(s):  
Endomorphism Ring ◽  
Jacobson Radical ◽  
Strong Form ◽  
Main Step ◽  
Unique Factorization ◽  
Endomorphism Rings ◽  
Finite Spectrum ◽  
Geometric Properties ◽  
Perfect Ring ◽  
Algebraic Properties

Let X be a finite spectrum. We prove that R(X(p)), the endomorphism ring of the p-localization of X, is a semi-perfect ring. This implies, among other things, a strong form of unique factorization for finite p-local spectra. The main step in the proof is that the Jacobson radical of R(X(p)) is idempotent-lifting, which is proved by a combination of geometric properties of finite spectra and algebraic properties of the p-localization.

scholarly journals Exponentials and Logarithms Properties in an Extended Complex Number Field

2021 ◽  
Author(s):  
Daniel Tischhauser
Keyword(s):  
Complex Number ◽  
Number System ◽  
Complex Numbers ◽  
Binary Operations ◽  
Algebraic Properties ◽  
Extended Complex ◽  
Multivalued Functions ◽  
Positive Integers ◽  
Complex Exponential ◽  
Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we proove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

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