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Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.

We obtain global explicit numerical bounds, with best possible constants, for the differences 1 n ∑ k ⩽ n ω (k) - log log n and 1 n ∑ k ⩽ n Ω (k) - log log n , where ω (k) and Ω (k) refer to the number of distinct prime divisors, and the total number of prime divisors of k, respectively. [ABSTRACT FROM AUTHOR]

Copyright of Journal Of Sichuan University (Natural Sciences Division) / Sichuan Daxue Xuebao-Ziran Kexueban is the property of Editorial Department of Journal of Sichuan University Natural Science Edition and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let 𝑐, 𝛼, 𝛽 ∈ R be such that 1 1 is irrational and with bounded partial quotients, 𝛽 ∈ [0, 𝛼). In this paper, we study asymptotic behaviour of the summations of the form Σ_(n≤N) (f([n^c]))/([n^c]) and Σ_(n≤N) (f([𝛼n+𝛽]))/([𝛼n+𝛽]) , where f is the Euler totient function 𝜑, Dedekind function Ψ, sum-of-divisors function 𝜎, or the alternating sum-of-divisors function 𝜎_(𝑎𝑙𝑡).

The author's last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For positive integers 𝑘, 𝑚, and 𝑛, let 𝑆𝑘 𝑚(𝑛) be the sum of all elements in the finite set {𝑥 𝑘 : 1 ≤ 𝑥 ≤ 𝑛⁄𝑚 , (𝑥, 𝑛) = 1}. The formula for 𝑆𝑘 𝑚(𝑛) is established and simpler formulae for 𝑆𝑘 𝑚(𝑛) under some conditions on 𝑚 and 𝑛 are verified. The explicit formulae for 𝑆1 2 𝑎 (𝑛) and 𝑆2 2 𝑎 (𝑛), where 2 𝑎 |𝑛 and 𝑎 ≥ 1, are also provided.

Number theory is an area of mathematics not unknown to students majoring in mathematics teaching. As early as in junior high, they may encounter basic number theory concepts such as prime numbers, multiples, divisors, or even the fundamental theorem of arithmetic—the proof of which, however, will only be included in the university curriculum of mathematics teacher students. When teaching number theory, in addition to inspiring and motivating students, it is important to have them ask the right questions. Proofs are important, but so is the proper formulation of mathematical conjectures, not to mention intuitions regarding mathematical properties and relationships. These processes are facilitated by the software used, developed to visualize functions related to number theory. Our study aims to assess the relevance of visualization in the teaching and understanding of certain arithmetic functions among mathematics teacher students. Furthermore, we aim to investigate the practical usability of our visualization software, relying on student feedback. According to our results, students found the software useful, as it helped them to discover previously unknown relationships in the examined arithmetic functions. We may conclude our software fills a niche, with student feedback indicating its usefulness. [ABSTRACT FROM AUTHOR]

The Sandor-Smarandache function, denoted by SS(n), is a newly-introduced Smarandache-type arithmetic function. This paper focuses on the functions SS(30p), SS(60p), SS(210p), SS(420p) and SS(840p), where p (≥2) is a prime. At the end of the paper, four tables, giving the values of SS(30p), SS(60p), SS(210p) and SS(420p) for the first 200 primes, calculated on a computer, are given. [ABSTRACT FROM AUTHOR]

We prove mean-value results for Schemmel's function (generalalizing Euler's totient function). This funtion was originally defined to count the number of set of m consecutive integers each being ≤ n and coprime to n. We generalize this to all integers m and consider its various mean-values. [ABSTRACT FROM AUTHOR]

Any positive integer n ≥ 2 can be written uniquely as n = m r where m is square-full, r is square-free and gcd (m , r) = 1. We denote the square-full part and the square-free part of n by pow (n) and sq (n) , respectively. In this paper, we use an elementary method to study the distribution of square-free and square-full part of an integer. Moreover, we also study the distribution of primitive roots modulo a prime p that are square-full number. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to define a new additive function related to generalized divisors of an integer and to study some of its properties. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Sandor-Smarandache function, denoted by SS(n), is a recently posed Smarandache-type arithmetic function. This paper concentrates on the function SS(210m), where m (≥ 1) is an integer. At the end of the paper, a table giving values of SS(210m) for m = 1(1)100, calculated on a computer, is appended. [ABSTRACT FROM AUTHOR]

For a fixed nonnegative integer u and positive integer n , we investigate the symmetric function ∑ d | n (c d (n d)) u p d n d , where p n denotes the n th power sum symmetric function, and c d (r) is a Ramanujan sum , equal to the sum of the r th powers of all the primitive d th roots of unity. We establish the Schur positivity of these functions for u = 0 and u = 1 , showing that, in each case, the associated representation of the symmetric group S n decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case u = 2. [ABSTRACT FROM AUTHOR]

Number theory is an area of mathematics not unknown to students majoring in mathematics teaching. As early as in junior high, they may encounter basic number theory concepts such as prime numbers, multiples, divisors, or even the fundamental theorem of arithmetic—the proof of which, however, will only be included in the university curriculum of mathematics teacher students. When teaching number theory, in addition to inspiring and motivating students, it is important to have them ask the right questions. Proofs are important, but so is the proper formulation of mathematical conjectures, not to mention intuitions regarding mathematical properties and relationships. These processes are facilitated by the software used, developed to visualize functions related to number theory. Our study aims to assess the relevance of visualization in the teaching and understanding of certain arithmetic functions among mathematics teacher students. Furthermore, we aim to investigate the practical usability of our visualization software, relying on student feedback. According to our results, students found the software useful, as it helped them to discover previously unknown relationships in the examined arithmetic functions. We may conclude our software fills a niche, with student feedback indicating its usefulness.

An innovative number theoretic function (arithmetic function) is introduced having analogous and additional new properties to the most commonly occurring arithmetic functions in the analytic number theory. This new function involves the general exponential term of the form V (n) = aΩ(n) where Ω(n) is the total number of prime factors of n counted with multiplicity, and a is any integer. Many existing arithmetic functions become special cases of V (n) as it is exciting to note that on the set of square-free positive integers and for a = -1, V (n) = μ (n) where μ (n) is the Möbius μ-function. Further, for a = -1, and for all n, V (n) = ζ (n), thus V (n) reduces to the traditional Liouville ζ-function. In this respect, function V (n) may be thought of as an extension of the Liouville and Möbius functions. Some new results are also formulated and studied using this general exponential arithmetic functionV (n). [ABSTRACT FROM AUTHOR]

Let h be an arithmetical function. We define arithmetic function graph Gh(G) of a given finite group G with respect to h, as a graph with vertex set V (Gh(G)) = G and any two distinct vertices a and b are adjacent in Gh(G) if and only if h(|a||b|) = h(|a|)h(|b|). It is observed that the order prime graph of a finite group is nothing but the arithmetic function graph with respect to the Euler's ɸ-function. In this paper, we investigate some results regrading arithmetic function graphs of finite groups. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new Menon-type identity:... where λj(b) := exp(2πiwjb/n) is an additive character of Z/nZ for 1 ⩽ j ⩽ r, (c1, . . .,cs) ∈ Zs is a fixed vector such that gcd(c1, . . .,cs,n) = 1, ϕs(n) = nsΠp|n(1- p-s) is the Jordan's totient function and σr(n) = Σd|n dr is the rth divisor function. This extends Rao's identity ([9]) and Sury's identity ([11]) to additive characters. Following the method of T'oth [14], we also generalize the above identity to arbitrary arithmetic function. [ABSTRACT FROM AUTHOR]

Wastewater is a complex mixture of organic and inorganic impurities, suspended solids, nutrients, saprotrophic and disease-causing bacteria and other microbes. When untreated wastewater is discharged into the rivers, seas, or oceans, it causes harm to human and aquatic life. Therefore, it is necessary to monitor the wastewater quality before diverting it into receiving water. The main aim of this study was the selection of an appropriate aggregation function to propose the wastewater quality index (WWQI) which is suitable for monitoring the spatial and seasonal variations of wastewater quality in the stormwater flow channel (SWF) located in Kolkata, India. Two different aggregation functions, i.e., weighted geometric mean function and weighted arithmetic mean function, were tested and analysed with primary data. Twelve wastewater quality parameters were selected. And, sampling was done seasonally for a 1-year period from December 2018 to December 2019 at eight selected locations along the SWF channel. The relative weights of each parameter were calculated based on their relative significance. Sub-index rating curves were generated to each parameter for WWQI calculations. The results revealed that weighted arithmetic mean function produced higher values of WWQI than geometric mean function. Furthermore, sensitivity analysis was carried out to evaluate the aggregation function that best suits the data of the SWF channel. The sensitivity analysis also revealed that weighted arithmetic mean function, being linear and free from ambiguity and less eclipsing, was more appropriate for measuring the WWQI for the SWF channel. Therefore, the study concluded that weighted arithmetic mean function was the most suitable aggregation function to calculate the WWQI for the SWF channel. [ABSTRACT FROM AUTHOR]

Recently the author and Derbal introduced and studied some elementary properties of arithmetic functions related the greatest common divisor. New properties of them are given in this paper. [ABSTRACT FROM AUTHOR]

We examine an arithmetical function defined by recursion relations on the sequence {f(pk)}k∈ℕ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic p ≥ 0 , and let K [ T ] + be the set of monic polynomials in the polynomial ring K[T]. We define the Dirichlet product of F-valued arithmetic functions on K [ T ] + and consider the ring of F-valued arithmetic functions. The purpose of this paper is to prove an analogue of the ABC conjecture in this ring. [ABSTRACT FROM AUTHOR]

In this article, we present a detailed study of the logarithmic, iterated logarithmic and derived logarithmic densities of a bounded and positive arithmetic function and obtain as particular cases results on subset E of ℕ*, and give some applications to classical number theory. Some new existence criteria are established. This mathematical concept gives a prolongation of the results got in ([1], [3], [4], [5], [6], [7], [8]) [ABSTRACT FROM AUTHOR]

Let n > 1 be an integer. Φ(e)(n) is a multiplicative function. The integer d = Пsi=1 pbii is called an exponential divisor of n = Пsi=1 paii, if bi|ai for every i ∊ {1; 2; ...; s}. Let Φ(e)(n) denote the number of divisors d of n such that d and n have no common exponential divisors. The aim of the paper is to establish a short interval for r-th power of the function Φ(e)(n) for any fixed integer r ≥ 1. [ABSTRACT FROM AUTHOR]

Sandor introduced a new Smarandache-type function, denoted by SS(n), and is called the Sandor-Smarandache function. When n is an odd (positive) integer, then SS(n) has a very simple form, as has been derived by Sandor himself. However, when n is even, then the form of SS(n) is not simple, and remains an open problem. This paper finds SS(n) for some special cases of n. Particular attention is given to values of the general forms SS(2mp), SS(6mp), SS(60mp) and SS(420mp), where m is any (positive) integer and p is an odd prime. Some particular cases have been treated in detail. In Section 4, some remarks are observed. [ABSTRACT FROM AUTHOR]

In this paper, we study some important means of Jordan's totient function, especially, we obtain asymptotic formula for geometric mean and harmonic mean. We also study alternating sums of Jordan's totient function and Carleman's inequality for this function. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas with remainder terms for the hyperbolic summations ∑ m n ≤ x f ((m , n)) and ∑ m n ≤ x f ([ m , n ]) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions f (n) = τ (n) , log n , ω (n) and Ω (n) . We also define a common generalization of the latter three functions, and prove a corresponding result. [ABSTRACT FROM AUTHOR]

A homothetic arithmetic function of ratio $K$ is a function $f \mathbb{N}\rightarrow R$ such that $f(Kn)=f(n)$ for every $n\in\mathbb{N}$. Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f(\mathbb{N})$ in terms of the period and the ratio of $f$.