scholarly journals On the coprimality of some arithmetic functions

2010 ◽  
Vol 87 (101) ◽  
pp. 121-128
Author(s):  
Koninck De ◽  
Imre Kátai
Keyword(s):  
Positive Integer ◽  
Asymptotic Estimate ◽  
Counting Function ◽  
Arithmetic Functions ◽  
Euler Function ◽  
Number Of Divisors

Let ? stand for the Euler function. Given a positive integer n, let ?(n) stand for the sum of the positive divisors of n and let ?(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd (?(n), ?(n)) = gcd(?(n), ?(n)) = 1}. Moreover, setting l(n) : = gcd(?(n), ?(n+1)), we provide an asymptotic estimate for the size of #{n ? x: l(n) = 1}.

Asymptotic formulas for certain arithmetic functions

2008 ◽  
Vol 58 (3) ◽  
Author(s):  
M. Garaev ◽  
M. Kühleitner ◽  
F. Luca ◽  
W. Nowak
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Arithmetic Functions ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
Second Moments

AbstractThis is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

scholarly journals A generalization of the practical numbers

2018 ◽  
Vol 14 (05) ◽  
pp. 1487-1503
Author(s):  
Nicholas Schwab ◽  
Lola Thompson
Keyword(s):  
Positive Integer ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Asymptotic Density ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
First Time

A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].

scholarly journals The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

2021 ◽  
pp. 15-22
Author(s):  
Xu Yifan ◽  
Shen Zhongyan
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Euler Function ◽  
Prime Factors ◽  
Integer Solutions

By using the properties of Euler function, an upper bound of solutions of Euler function equation  is given, where  is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation  are given, where is the number of distinct prime factors of n.

scholarly journals Partitions into distinct large parts

1994 ◽  
Vol 57 (3) ◽  
pp. 386-416 ◽  
Author(s):  
Gregory A. Freiman ◽  
Jane Pitman
Keyword(s):  
Positive Integer ◽  
Asymptotic Estimate

AbstractAn asymptotic estimate is obtained for the number of partitions of the positive integer n into distinct parts, each of which is at least m. The estimate holds uniformly with respect to positive m such that m = o(n(log n)−9), as n → ∞.

scholarly journals Partitions into large unequal parts from a general sequence

2006 ◽  
Vol 80 (1) ◽  
pp. 13-44
Author(s):  
Kevin John Fergusson
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Asymptotic Estimate ◽  
General Sequence

AbstractAn asymptotic estimate is obtained for the number of partitions of the positive integer n into unequal parts coming from a sequence u, with each part greater than m, under suitable conditions on the sequence u. The estimate holds uniformly with respect to integers m such that 0 ≤ m ≤ n1−δ, as n → ∞, where δ is a given real number, such that 0 < δ < 1.

Corrigendum: `On the average number of divisors of the Euler function'

2016 ◽  
Vol 89 (1-2) ◽  
pp. 257-260
Author(s):  
FLORIAN LUCA ◽  
CARL POMERANCE
Keyword(s):  
Euler Function ◽  
Number Of Divisors

scholarly journals On the Nature of γ-th Arithmetic Zeta Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 790 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Positive Integer ◽  
Complex Number ◽  
Symmetric Functions ◽  
Zeta Functions ◽  
Arithmetic Functions ◽  
Ancient Greek ◽  
Exceptional Set ◽  
Elementary Symmetric Functions ◽  
Main Components ◽  
Schanuel's Conjecture

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of e α , for algebraic values of α ) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as e γ log   n . This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called γ-th arithmetic zeta function ζ γ ( n ) : = n γ ( = e γ log   n ), for a positive integer n and a complex number γ . Moreover, we raise a conjecture about the exceptional set of ζ γ , in the case in which γ is transcendental, and we connect it to the famous Schanuel’s conjecture.

scholarly journals A note on the value distribution of $\phi f^2 f^{(k)}-1$

2021 ◽  
Vol 55 (1) ◽  
pp. 64-75
Author(s):  
P. Sahoo ◽  
G. Biswas
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Characteristic Function ◽  
Counting Function ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Small Function ◽  
Nevanlinna Characteristic ◽  
Transcendental Meromorphic Function

In this paper, we study the value distribution of the differential polynomial $\varphi f^2f^{(k)}-1$, where $f(z)$ is a transcendental meromorphic function, $\varphi (z)\;(\not\equiv 0)$ is a small function of $f(z)$ and $k\;(\geq 2)$ is a positive integer. We obtain an inequality concerning the Nevanlinna Characteristic function $T(r,f)$ estimated by reduced counting function only. Our result extends the result due to J.F. Xu and H.X. Yi [J. Math. Inequal., 10 (2016), 971-976].

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