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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]

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]

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]

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]

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]

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]

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]

Let $h(n)$ denote the largest product of distinct primes whose sum does not exceed $n$. The main result of this paper is that the property for all $n\geq 1$ , we have $\log h(n) (where $\text{li}^{-1}$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

Let f be a full-level cusp form for GLm(ℤ) with Fourier coefficients Af(cm-2,..., c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|⩽H | ΣX

Let Q1,... , Qt ∈ℝ[x] be polynomials with no constant term for which each linear combination m1Q1(x) + · · ·+mtQt (x) with m1,... , mt ∈ℤand not all 0 always has an irrational coefficient. Let I1,... , It be sets included in the interval [0, 1), each being a union of finitely many subintervals of [0, 1). Furthermore, let T be the set of positive integers n for which {Q1(n)} ∈ I1,... , {Qt(n)} ∈ It simultaneously, where {y} stands for the fractional part of y. Let t1, t2,... be a uniformly summable sequence of complex numbers and set T(x) = ∑ n ≤ xtn and T x T = ∑ n ≤ x , n ∈ T t n . We prove that, as x→∞, T x / x ∼ T x T / λ I 1 ⋯ λ I t x , where λ(I) is the Lebesgue measure of a set I. [ABSTRACT FROM AUTHOR]

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

A simple, but very useful concept in number theory is that of an arithmetic function. On the other hand, hyperstructure theory, first introduced by Marty, is a generalization of group theory and it is connected to different fields of mathematics. In our paper, we present a link between the theories of algebraic structures, hyperstructures and arithmetic functions. More precisely, we present hyperstructures of arithmetic functions and determine their fundamental groups and fundamental rings. Moreover, we study their complete parts and strongly regular relations. [ABSTRACT FROM AUTHOR]

Let be a Dirichlet character (mod ) with conductor . In a quite recent paper Zhao and Cao deduced the identity , which reduces to Menon's identity if is the principal character (mod ). We generalize the above identity by considering even functions (mod ), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums. [ABSTRACT FROM AUTHOR]

Let n> 1 be an intrger, κ(e)(n) denote the maximal e-squarefree e-divisor of n. In this paper, we shall establish a short interval result for the function κ(e)(n). [ABSTRACT FROM AUTHOR]

In this paper we consider the mean value of hybrid arithmetic function of the form λ6f (n)σb(n)Øc(n), where λf (n) is the n-th Fourier coefficients of the holomorphic cusp form f, σ(n) is the sum-of-divisors function, Ø(n) is the Euler's totient function. In detail, we prove that for any e > 0, ∑ n⩽x λ6f (n)σb(n)Øc(n) = x(b+c+1)P4(log x) + O(xb+c+631/652+ε); where b; c ∊ ℝ. [ABSTRACT FROM AUTHOR]

We obtain the asymptotic expansion of the sequence with general term AnGn , where An and Gn are the arithmetic and geometric means of the numbers d(1); d(2), ..., d(n), with d(n) denoting the number of positive divisors of n. Also, we obtain some explicit bounds concerning Gn and AnGn. [ABSTRACT FROM AUTHOR]

Let be a positive integer, and define for , where denotes the number of distinct prime factors of , and represents the total number of prime factors of (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that which is similar to the known identity equivalent to the Prime Number Theorem. For , we prove that We also raise a hypothesis on the parities of which implies the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

Let n be a positive integer and let S = { x 1 ,... , x n } be a set of n distinct positive integers. For x ∈ S , one defines G S (x) = { d ∈ S : d < x , d | x and (d | y | x , y ∈ S) ⇒ y ∈ { d , x } }. For any arithmetic function f and any positive integer a , we define power arithmetic function f a by f a (x) = (f (x)) a for any positive integer x. We denote by (f a (S)) and (f a [ S ]) the n × n power matrices having f a evaluated at the greatest common divisor and the least common multiple of x i and x j as its (i , j) -entry, respectively. By | T | we denote the number of elements of any finite set T. In this paper, we show that if S is gcd closed (i.e. gcd ⁡ (x i , x j) ∈ S for all integers i and j with 1 ≤ i , j ≤ n) such that max x ∈ S ⁡ { | G S (x) | } = 1 , then for arbitrary positive integers a and b with a | b , the power matrix (f b (S)) is divisible by the power matrix (f a (S)) if f ∈ C S = { f : (f ⁎ μ) (d) ∈ Z whenever d | lcm (S) } with lcm (S) being the least common multiple of all the elements of S , and the power matrix (f b [ S ]) is divisible by the power matrix (f a (S)) if f ∈ D S = { f ∈ C S : f (x) | f (y) whenever x | y and x , y ∈ S } is multiplicative. Our results extend the theorem of Hong gotten in 2008, that of Li and Tan obtained in 2011 and Zhu's theorem attained recently. [ABSTRACT FROM AUTHOR]

Let f:ℕ→ℕ0 be a multiplicative arithmetic function such that for all primes p and positive integers α, f(pα)< pα and f(p)|f(pα). Suppose also that any prime that divides f(pα) also divides pf(p). Define f (0)= 0, and let H(n)= limm→∞ fm(n), where fm denotes the mth iterate of f . We prove that the function H is completely multiplicative. [ABSTRACT FROM AUTHOR]

For each of the functions f ∈ {φ, σ, ω, τ} and every natural number K, we show that there are infinitely many solutions to the inequalities f(pn - 1) < f(pn+1 - 1) < ⋯ < f(pn+K - 1), and similarly for f(pn -1) > f(pn+1 - 1) > ⋯ > f(pn+K -1). We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length. [ABSTRACT FROM AUTHOR]

Let S = { x 1 , … , x n } be a set of n distinct positive integers and f be an arithmetic function. We use f ( S ) = f ( x i , x j ) (resp. f [ S ] = f [ x i , x j ] ) to denote the n × n matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of x i and x j as its i , j -entry. The set S is called a divisor chain if there is a permutation σ of { 1 , … , n } such that x σ ( 1 ) | … | x σ ( n ) . If S can be partitioned as S = ⋃ i = 1 k S i with all S i ( 1 ⩽ i ⩽ k ) being divisor chains and (max ( S i ) , max ( S j ) ) = gcd( S ) for 1 ⩽ i ≠ j ⩽ k , then we say that S consists of finitely many quasi-coprime divisor chains. In this paper, we introduce a new method to give the formulas for the determinants of the matrices ( f ( S ) ) and ( f [ S ] ) on finitely many quasi-coprime divisor chains S . We show also that det ( f ( S ) ) | det ( f [ S ] ) holds under some natural conditions. These extend the results obtained by Tan and Lin (2010) and Tan et al. (2013), respectively. [ABSTRACT FROM AUTHOR]

The aim of this survey is to give a compact view of the evolution of Probabilistic Number Theory, starting from the very beginnings and following to the last decades. Far from pretending to be complete, we concentrate mostly on the ideas and insights J. Kubilius contributed to or influenced further development. [ABSTRACT FROM AUTHOR]

Let n = p p ⋯ p, where p (1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ω k( n) = P( p+ p) P( p+ p) ⋯ P( p k−1+ p) P( p + p), where P( n) is the largest prime factor of n. Define w( n) = n and w( n) = w( w( n)) for all integers i ≥ 1. The smallest integer s for which there exists a positive integer t such that Ω( n) = Ω( n) is called the index of periodicity of n. The authors investigate the index of periodicity of n. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas for the mean values of a number of arithmetic functions f = f( n) on a 'short' interval of the form x < n ≤ x + h, where the length h differs from x by a power-law decrease: h= x, 0 < δ < 1. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of representations of arithmetic functions, and corresponding operator-theoretic and free probabilistic properties. We associate given arithmetic functions f to certain matrices αn(f). [ABSTRACT FROM AUTHOR]

This note deals with the exponent function associated to a finite group. Some classes of finite groups determined by basic properties of this function are investigated. [ABSTRACT FROM AUTHOR]

We characterize all functions f: ℕ → ℂ such that f( m + n) = f( m) + f( n) for all m, n ∈ ℕ. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f( n) = ± n (subject to some restrictions on when the choice of the sign is possible) and f( n) = ± $\tfrac{1} {2} $ (again subject to some restrictions on when the choice of the sign is possible). [ABSTRACT FROM AUTHOR]

The distribution of values of Pillai's function over the ring of Gauss integers $ \mathbb{Z}\left[ i \right] $ is studied. The asymptotic formulas for the summators $ {\varSigma_{{N\left( \alpha \right)\leq x}}}\frac{{g\left( \alpha \right)}}{{{N^{\alpha }}\left( \alpha \right)}} $ , where $ a\in \mathbb{R} $ , are found. [ABSTRACT FROM AUTHOR]

Let k be any given positive integer. We define the arithmetic function gk for any positive integer n by \[ g_{k}(n):=\frac {\prod _{i=0}^k ((n+i)^2+1)}{{\rm lcm}_{0\le i\le k}\{(n+i)^2+1\}}. \] We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}. [ABSTRACT FROM PUBLISHER]

In this note, we generalize a one variable inequality of Klamkin to the case of two variables. [ABSTRACT FROM AUTHOR]

Let P( n) stand for the largest prime factor of n ≥ 2 and set P(1) = 1. For each integer n ≥ 2, let δδ( n) be the distance to the nearest P( n)-smooth number, that is, to the nearest integer whose largest prime factor is no larger than that of n. We provide a heuristic argument showing that ΣΣ n≤≤ x 1/ δδ( n) = (4 log 2 - 2 + o(1)) x as x →→ ∞. Moreover, given an arbitrary real-valued arithmetic function ƒ, we study the behavior of the more general function δδƒ( n) defined by δδƒ( n) = min1≤≤ m≠≠ n, ƒ( m)≤≤ƒ( n) | n - m| for n ≥ 2, and δδƒ (1) = 1. In particular, given any positive integers a < b, we show that ΣΣ a≤≤ n< b 1/ δδƒ( n) ≥ 2( b - a)/3 and that if ƒ( n) ≥ ƒ( a) for all n ∈ [ a, b[, then one has ΣΣ a< n< b δδƒ( n) ≤≤ ( b - a) log( b - a)/(2 log 2). [ABSTRACT FROM AUTHOR]

Let t(e)(n) denote the number of e-squarefree e-divisor of n. The aim of this paper is to establish a short interval result for the function t(e)(n). This enriches the properties of the e-squarefree e-divisor function. [ABSTRACT FROM AUTHOR]

For f one of the classical arithmetic functions d, ϕ and σ, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity. [ABSTRACT FROM AUTHOR]

An integer d = Πs i=1 pbi i is called the exponential divisor of n = Πs i=1 pai i > 1 if bijai for every i ϵ {1, 2, …, s}. Let τ (e)(n) denote the number of exponential divisors of n, where τ(e)(1) = 1 by convention. The aim of this paper is to establish a short interval result for for —r-th power of the function τ(e) for any fixed integer r≥1. [ABSTRACT FROM AUTHOR]

Let m0, m1, m2, ... be positive integers with m i ≥ 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as where 0 ≤ a i ≤ m i − 1 for all i and a t ≠ 0. Let each f i be a function defined on “0, 1, 2, ..., m i −1” with f i(0) = 0. Write . In this paper, we give the asymptotic formula for 1/ x Σ n≤ x S( n) k, where k is a positive integer. [ABSTRACT FROM AUTHOR]

An asymptotic formula is obtained for the number of representations of an element of a finite field as a weighted sum of two prescribed powers of primitive elements. This generalises previous work on sums of primitive elements, including that relating to some conjectures of Golomb. [Copyright &y& Elsevier]

Denote the nth prime number by pn. In this note we show a limit theorem involving the product of powers of prime numbers: If a sequence an has the limit a, under some conditions, we show that pnan / n√p1a1···pnan has the limit ea. [ABSTRACT FROM AUTHOR]