DOCSLIB.ORG

On a Certain Kind of Generalized Number-Theoretical Möbius Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On a Certain Kind of Generalized Number-Theoretical Möbius Function On a Certain Kind of Generalized Number-Theoretical Möbius Function Tom C. Brown,£ Leetsch C. Hsu,† Jun Wang† and Peter Jau-Shyong Shiue‡ Citation data: T.C. Brown, C. Hsu Leetsch, Jun Wang, and Peter J.-S. Shiue, On a certain kind of generalized number-theoretical Moebius function, Math. Scientist 25 (2000), 72–77. Abstract The classical Möbius function appears in many places in number theory and in combinatorial the- ory. Several different generalizations of this function have been studied. We wish to bring to the attention of a wider audience a particular generalization which has some attractive applications. We give some new examples and applications, and mention some known results. Keywords: Generalized Möbius functions; arithmetical functions; Möbius-type functions; multiplica- tive functions AMS 2000 Subject Classification: Primary 11A25; 05A10 Secondary 11B75; 20K99 This paper is dedicated to the memory of Professor Gian-Carlo Rota 1 Introduction We define a generalized Möbius function ma for each complex number a. (When a = 1, m1 is the classical Möbius function.) We show that the set of functions ma forms an Abelian group with respect to the Dirichlet product, and then give a number of examples and applications, including a generalized Möbius inversion formula and a generalized Euler function. Special cases of the generalized Möbius functions studied here have been used in [6–8]. For other generalizations see [1,5]. For interesting survey articles, see [2,3, 11]. £Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC Canada V5A 1S6. [email protected]. †Department of Mathematics, Dalian University of Technology, Dalian 116024, China. ‡Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154-4020, USA. 1 Let us recall that the classical Möbius function m(n) is defined for positive integers n in the following way: m(1) = 1. If n is not square free then m(n) = 0. If n is square free and r is the number of distinct primes dividing n, then m(n) = ( 1)r [9]. For any integer r, a Möbius function of order r may be defined by using binomial coefficients, namely for each positive integer n, r ( ) = ( )¶p(n) mr n ∏ ( ) 1 pjn ¶p n where p runs through all the prime divisors of n, and ¶p(n) = ordpn denotes the highest power k of p k such that p divides n. Obviously m1(n) = m(n). For more details, see [7]. We now define a generalized Möbius function ma for each complex number a, by setting a ( ) = ( )¶p(n) ma n ∏ ( ) 1 pjn ¶p n At the end of the paper, we mention a particularly interesting application of the case where a is real. 2 Group-theoretic properties Recall that the classical Möbius function is multiplicative; i.e., if m and n are relatively prime, then m(mn) = m(m)m(n). It is easily seen that the definition of ma implies that this property extends to the Möbius function of order a, giving us the following lemma. Lemma 1. For each complex number a, ma is a multiplicative function. Next, we recall the definition of the Dirichlet product (or convolution) of two arithmetic functions f and g (cf. [1,4]). Definition 1. Given two arithmetic functions f and g, the Dirichlet (convolution) product f £ g is again an arithmetic function which is defined by n n ( f £ g)(n) = ∑ f (d)g = ∑ f g(d); djn d djn d where the summations are taken over all positive divisors d of n. Evidently, the product is commutative: f £ g = g £ f . Using a little algebra one easily shows that the following associative law also holds: ( f £ g) £ h = f £ (g £ h). That is, for all positive integers n, (( f £ g) £ h)(n) = ( f £ (g £ h))(n). Moreover, the convolution f £ g is a multiplicative function whenever f and g are multiplicative functions. Definition 2. Let M = fma : a 2 Cg where C denotes the set of complex numbers. The set M may be called the set of generalized Möbius functions of complex order. 2 Lemma 2. For any given numbers a and b in C, we have £ = ma mb ma+b Proof. It is required to show that for all positive integers n, ( £ )( ) = ( ) n = ( ): ma mb n ∑ ma d mb ma+b n djn d £ Since ma and mb are multiplicative (by Lemma1), the Dirichlet product ma mb is also multiplicative. Thus, it suffices to consider the case n = pk, where p is prime and k is a positive integer. We easily find k k p (m £ m )(pk) = m (d)m = m (pi)m (pk i) a b ∑ a b d ∑ a b djpk i=0 k a b = ( 1)i ( 1)k i ∑ i k i i=0 + k a b k = ( 1) = m + (p ); k a b since the relation (1 + x)a (1 + x)b = (1 + x)a+b implies + k a b = a b : ∑ k i=0 i k i Notice that m0 is the Möbius function of order zero that gives the values ( 0 ( ) 1 n = 1; m (n) = ( 1)¶p n = 0 ∏ ( ) > : pjn ¶p n 0 n 1 Let us denote m0 by d. Since from Lemma2 we have ma £ d = d £ ma = ma for all a, we call it the identity element with respect to the Dirichlet product operation £. We are now ready to show that M is an Abelian group. Theorem 1. (M;£) is an Abelian group with identity element d = m0. Proof. By Lemma2 we see that M is closed with respect to the operation £. Moreover, we also have £ = £ ( ; 2 C); ma mb mb ma a b ( £ ) £ = £ ( £ )( ; ; 2 C); ma mb mg ma mb mg a b g ma £ d = d £ ma = ma ma £ m a = m a £ ma = d (a 2 C); Thus, the theorem is proved. 3 Of course, if G is any additive subgroup of C, then MG = fma : a 2 Gg is a subgroup of M. 3 Corollaries, examples and applications Corollary 1. (Generalized Möbius inversion formulae).) For all a 2 C and arithmetic functions f ;g, " # " # 8 2 N ( ) = n ( ) , 8 2 N ( ) = n ( ) : n f n ∑ ma g d n g n ∑ m a f d djn d djn d Proof. In fact, this is equivalent to the statement f = ma £ g , g = m a £ f ; which follows from f = ma £ g , m a £ f = m a £ ma £ g = d £ g = g: Evidently, Corollary1 with a = 1 implies the classical Möbius inversion formulae ( f = m £g , g = m 1 £ f ), since m1 = m and m 1 1: 1 (¶ (n))! ( ) = ( )¶p(n) = p = : m 1 n ∏ ( ) 1 ∏ ( ( )) 1 pjn ¶p n pjn ¶p n ! Note here that the Möbius m-function and m 1 1 are inverses of each other under convolution. Corollary 2. For all n 2 N and a 2 C, ( ) = ( ): ∑ ma d ma 1 n djn This is equivalent to the statement (m 1 £ ma )(n) = ma 1(n). Note that the case a = 1 gives the classical identity of Gauss ( ) = ( ) = ( ): ∑ m d m0 n d n djn Corollary 3. Let f be a completely multiplicative function such that f (mn) = f (n) f (m) for all positive integers m and n, and let r be a positive integer. Then the r-times convolution of mr f with f satisfies (mr f ) £ f £ f £ ¡¡¡ £ f = m0 f ; where (ma f )(n) = ma (n) f (n). This follows easily form Corollary2 and induction on r. Indeed we have (( ) £ )( ) = ( ) ( ) n = ( ) ( ) = ( )( ): mr f f n ∑ mr d f d f f n ∑ mr d mr 1 f n djn d djn 4 Moreover, it may be of interest to note that ( ) = ( £ )( ) = = ( ); m 2 n m 1 m 1 n ∑1 t n djn where t(n) denotes the number of positive divisors of n. Thus, t = m 2. Consequently, from m 2 £ m1 = m 1 and m 2 £ m2 = m0, we may obtain the identities ( ) n = ( ) n = ( ): ∑t d m 1 and ∑t d m2 d n djn d djn d Example 1. Let sr(n) denote the sum of the rth powers of the divisors of n. The well-known identity n nr = m(d)s ∑ r d r can be proved very simply in the following way. Let ir(n) = n . Then, since m 1 1, we have ir £ m 1 = sr, and hence r (m £ sr)(n) = (m £ ir £ m 1)(n) = (ir £ m £ m 1)(n) = (ir £ d)(n) = ir(n) = n : Example 2. Euler’s j-function may be written as j = i1 £ m1. Moreover, using t = m 2 we can easily prove the identity n s(n) = ∑j(d)t : djn d In fact, these statements follow easily from the relations ( ) = n = ( £ )( ) j n ∑dm i1 m1 n djn d and j £ t = (i1 £ m1) £ m 2 = i1 £ m 1 = s (see Example1). Example 3. Fix a positive integer r 1, and define jr = i1 £ mr. Then, if n is ‘r-powerful’, that is, ¶p(n) r for every prime divisor p of n, we have r ( ) = 1 : jr n n∏ 1 pjn p This may be verified as follows: ( ) r j r ( ) = n = mr d = r 1 = 1 : jr n ∑dmr n∑ n∏ ∑ n∏ 1 djn d djn d pjn j=0 j p pjn p Note that, if r = 1, then j1 = j is the classical Euler function. Thus, jr may be called the generalized Euler function of order r. This function has a similar meaning to that of j, in that fr counts the number of integers a, 1 a n, such that a is ‘rth-degree prime to n’.
Load more

Recommended publications
  • 3.4 Multiplicative Functions
    3.4 Multiplicative Functions Question how to identify a multiplicative function? If it is the convolution of two other arithmetic functions we can use Theorem 3.19 If f and g are multiplicative then f ∗ g is multiplicative. Proof Assume gcd ( m, n ) = 1. Then d|mn if, and only if, d = d1d2 with d1|m, d2|n (and thus gcd( d1, d 2) = 1 and the decomposition is unique). Therefore mn f ∗ g(mn ) = f(d) g d dX|mn m n = f(d1d2) g d1 d2 dX1|m Xd2|n m n = f(d1) g f(d2) g d1 d2 dX1|m Xd2|n since f and g are multiplicative = f ∗ g(m) f ∗ g(n) Example 3.20 The divisor functions d = 1 ∗ 1, and dk = 1 ∗ 1 ∗ ... ∗ 1 ν convolution k times, along with σ = 1 ∗j and σν = 1 ∗j are all multiplicative. Note 1 is completely multiplicative but d is not, so convolution does not preserve complete multiplicatively. Question , was it obvious from its definition that σ was multiplicative? I suggest not. Example 3.21 For prime powers pa+1 − 1 d(pa) = a+1 and σ(pa) = . p−1 Hence pa+1 − 1 d(n) = (a+1) and σ(n) = . a a p−1 pYkn pYkn 11 Recall how in Theorem 1.8 we showed that ζ(s) has a Euler Product, so for Re s > 1, 1 −1 ζ(s) = 1 − . (7) ps p Y When convergent, an infinite product is non-zero. Hence (7) gives us (as already seen earlier in the course) Corollary 3.22 ζ(s) =6 0 for Re s > 1.
    [Show full text]
  • Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs
    ALGEBRAIC COMBINATORICS ON TRACE MONOIDS: EXTENDING NUMBER THEORY TO WALKS ON GRAPHS P.-L. GISCARD∗ AND P. ROCHETy Abstract. Trace monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and unicity of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including an immanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function from an abelianization procedure. Keywords: Digraph; poset; trace monoid; walks; weighted adjacency matrix; incidence algebra; MacMahon master theorem; Ihara zeta function. MSC: 05C22, 05C38, 06A11, 05E99 1. Introduction. Several school of thoughts have emerged from the literature in graph theory, concerned with studying walks (also known as paths) on graphs as algebraic objects. Among the numerous structures proposed over the years are those based on walk concatenation [3], later refined by nesting [13] or the cycle space [9]. A promising approach by trace monoids consists in viewing the directed edges of a graph as letters forming an alphabet and walks as words on this alphabet. A crucial idea in this approach, proposed by [5], is to define a specific commutation rule on the alphabet: two edges commute if and only if they initiate from different vertices.
    [Show full text]
  • Note on the Theory of Correlation Functions
    Note On The Theory Of Correlation And Autocor- relation of Arithmetic Functions N. A. Carella Abstract: The correlation functions of some arithmetic functions are investigated in this note. The upper bounds for the autocorrelation of the Mobius function of degrees two and three C C n x µ(n)µ(n + 1) x/(log x) , and n x µ(n)µ(n + 1)µ(n + 2) x/(log x) , where C > 0 is a constant,≤ respectively,≪ are computed using≤ elementary methods. These≪ results are summarized asP a 2-level autocorrelation function. Furthermore,P some asymptotic result for the autocorrelation of the vonMangoldt function of degrees two n x Λ(n)Λ(n +2t) x, where t Z, are computed using elementary methods. These results are summarized≤ as a 3-lev≫el autocorrelation∈ function. P Mathematics Subject Classifications: Primary 11N37, Secondary 11L40. Keywords: Autocorrelation function, Correlation function, Multiplicative function, Liouville function, Mobius function, vonMangoldt function. arXiv:1603.06758v8 [math.GM] 2 Mar 2019 Copyright 2019 ii Contents 1 Introduction 1 1.1 Autocorrelations Of Mobius Functions . ..... 1 1.2 Autocorrelations Of vonMangoldt Functions . ....... 2 2 Topics In Mobius Function 5 2.1 Representations of Liouville and Mobius Functions . ...... 5 2.2 Signs And Oscillations . 7 2.3 DyadicRepresentation............................... ... 8 2.4 Problems ......................................... 9 3 SomeAverageOrdersOfArithmeticFunctions 11 3.1 UnconditionalEstimates.............................. ... 11 3.2 Mean Value And Equidistribution . 13 3.3 ConditionalEstimates ................................ .. 14 3.4 DensitiesForSquarefreeIntegers . ....... 15 3.5 Subsets of Squarefree Integers of Zero Densities . ........... 16 3.6 Problems ......................................... 17 4 Mean Values of Arithmetic Functions 19 4.1 SomeDefinitions ..................................... 19 4.2 Some Results For Arithmetic Functions .
    [Show full text]
  • Number Theory, Lecture 3
    Number Theory, Lecture 3 Number Theory, Lecture 3 Jan Snellman Arithmetical functions, Dirichlet convolution, Multiplicative functions, Arithmetical functions M¨obiusinversion Definition Some common arithmetical functions Dirichlet 1 Convolution Jan Snellman Matrix interpretation Order, Norms, 1 Infinite sums Matematiska Institutionen Multiplicative Link¨opingsUniversitet function Definition Euler φ M¨obius inversion Multiplicativity is preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself Link¨oping,spring 2019 Lecture notes availabe at course homepage http://courses.mai.liu.se/GU/TATA54/ Number Summary Theory, Lecture 3 Jan Snellman Arithmetical functions Definition Some common Definition arithmetical functions 1 Arithmetical functions Dirichlet Euler φ Convolution Definition Matrix 3 M¨obiusinversion interpretation Some common arithmetical Order, Norms, Multiplicativity is preserved by Infinite sums functions Multiplicative multiplication function Dirichlet Convolution Definition Matrix verification Euler φ Matrix interpretation Divisor functions M¨obius Order, Norms, Infinite sums inversion Euler φ again Multiplicativity is preserved by multiplication 2 Multiplicative function µ itself Matrix verification Divisor functions Euler φ again µ itself Number Summary Theory, Lecture 3 Jan Snellman Arithmetical functions Definition Some common Definition arithmetical functions 1 Arithmetical functions Dirichlet Euler φ Convolution Definition Matrix 3 M¨obiusinversion interpretation Some common arithmetical Order, Norms,
    [Show full text]
  • Lecture 14: Basic Number Theory
    Lecture 14: Basic number theory Nitin Saxena ? IIT Kanpur 1 Euler's totient function φ The case when n is not a prime is slightly more complicated. We can still do modular arithmetic with division if we only consider numbers coprime to n. For n ≥ 2, let us define the set, ∗ Zn := fk j 0 ≤ k < n; gcd(k; n) = 1g : ∗ The cardinality of this set is known as Euler's totient function φ(n), i.e., φ(n) = jZnj. Also, define φ(1) = 1. Exercise 1. What are φ(5); φ(10); φ(19) ? Exercise 2. Show that φ(n) = 1 iff n 2 [2]. Show that φ(n) = n − 1 iff n is prime. Clearly, for a prime p, φ(p) = p − 1. What about a prime power n = pk? There are pk−1 numbers less than n which are NOT coprime to n (Why?). This implies φ(pk) = pk − pk−1. How about a general number n? We can actually show that φ(n) is an almost multiplicative function. In the context of number theory, it means, Theorem 1 (Multiplicative). If m and n are coprime to each other, then φ(m · n) = φ(m) · φ(n) . ∗ ∗ ∗ ∗ ∗ Proof. Define S := Zm × Zn = f(a; b): a 2 Zm; b 2 Zng. We will show a bijection between Zmn and ∗ ∗ ∗ ∗ ∗ S = Zm × Zn. Then, the theorem follows from the observation that φ(mn) = jZmnj = jSj = jZmjjZnj = φ(m)φ(n). ∗ The bijection : S ! Zmn is given by the map :(a; b) 7! bm + an mod mn. We need to prove that is a bijection.
    [Show full text]
  • Möbius Inversion from the Point of View of Arithmetical Semigroup Flows
    Biblioteca de la Revista Matematica´ Iberoamericana Proceedings of the \Segundas Jornadas de Teor´ıa de Numeros"´ (Madrid, 2007), 63{81 M¨obiusinversion from the point of view of arithmetical semigroup flows Manuel Benito, Luis M. Navas and Juan Luis Varona Abstract Most, if not all, of the formulas and techniques which in number theory fall under the rubric of “M¨obiusinversion" are instances of a single general formula involving the action or flow of an arithmetical semigroup on a suitable space and a convolution-like operator on functions. The aim in this exposition is to briefly present the general formula in its abstract context and then illustrate the above claim using an extensive series of examples which give a flavor for the subject. For simplicity and to emphasize the unifying character of this point of view, these examples are mostly for the traditional number theoretical semigroup N and the spaces R or C. 1. Introduction The “M¨obiusInversion Formula" in elementary number theory most often refers to the formula X X n (1.1) fb(n) = f(d) () f(n) = µ(d)fb ; d djn djn where f is an arithmetical function, that is, a function on N with values typically in Z, R or C; the sum ranges over the positive divisors d of a given 2000 Mathematics Subject Classification: Primary 11A25. Keywords: M¨obiusfunction, M¨obius transform, Dirichlet convolution, inversion formula, arithmetical semigroup, flow. 64 M. Benito, L. M. Navas and J. L. Varona n 2 N, and µ is of course the M¨obiusfunction, given by 8 µ(1) = 1; <> (1.2) µ(n) = 0 if n has a squared factor, > k : µ(p1p2 ··· pk) = (−1) when p1; p2; : : : ; pk are distint primes.
    [Show full text]
  • LECTURE 11: ARITHMETIC FUNCTIONS Recall That a Function F
    LECTURE 11: ARITHMETIC FUNCTIONS Recall that a function f : N ! C is called an arithmetical function. Re- call also that a multiplicative function f satisfies the property that whenever (m; n) = 1, one has f(mn) = f(m)f(n): We have proved that the function X g(n) = f(d) djn is multiplicative whenever f(n) is multiplicative. In this section we discuss various properties of arithmetical functions, many of them multiplicative, and seek to understand what they \look" like. 1. Examples of arithmetical functions • the divisor function τ(n). The function τ(n) is defined as the number of divisors of the number n. It can be written explicitly as X τ(n) = 1: djn • the sum of divisors function σ(n). The function σ(n) is defined as the sum of all divisors of the number n. It can be written explicitly as X σ(n) = d: djn • the Euler totient function φ(n). The function φ(n) is defined as the number of reduced residue classes modulo n. We have shown that it is given by a formula Y φ(n) = (pr − pr−1): prkn • the M¨obiusfunction µ(n) The function φ(n) is defined by ( (−1)`; when n = p ··· p is squarefree, µ(n) = 1 ` 0; otherwise. Here, by a squarefree number, we mean an integer that is not divisible by the square of any prime number. It is easy to check that µ is multiplicative. 1 2 LECTURE 11 Note that, just as with our earlier discussion of the Euler totient, a function that is multiplicative will be relatively easy to evaluate when its argument has a known prime factorisation.
    [Show full text]
  • On the Solution of Functional Equations of Wilson's Type On
    Proyecciones Journal of Mathematics Vol. 36, No 4, pp. 641-651, December 2017. Universidad Cat´olica del Norte Antofagasta - Chile On the solution of functional equations of Wilson’s type on monoids Iz-iddine EL-Fassi Ibn Tofail University, Morocco Abdellatif Chahbi Ibn Tofail University, Morocco and Samir Kabbaj Ibn Tofail University, Morocco Received : December 2016. Accepted : September 2017 Abstract Let S be a monoid, C be the set of complex numbers, and let σ, τ Antihom(S, S) satisfy τ τ = σ σ = id. The aim of this paper is to∈ describe the solution f,g :◦S C◦of the functional equation → f(xσ(y)) + f(τ(y)x)=2f(x)g(y),x,y S, ∈ in terms of multiplicative and additive functions. Subjclass [2010] : Primary 39B52. Keywords : Wilson’s functional equation, monoids, multiplicative function. 642 Iz-iddine EL-Fassi, Abdellatif Chahbi and Samir Kabbaj 1. Notation and terminology Throughout the paper we work in the following framework: A monoid is a semi-group S with an identity element that we denote e,thatisanelement such that ex = xe = x for all x S (a semi-group is an algebraic struc- ture consisting of a set together with∈ an associative binary operation) and σ, τ : S S are two anti-homomorphisms (briefly σ, τ Antihom(S, S)) satisfying→τ τ = σ σ = id. ∈ For any◦ function◦f : S C we say that f is σ-even (resp. τ-even) if → ˇ 1 f σ = f (resp. f τ = f), also we use the notation f(x)=f(x− )inthe case◦ S is a group.
    [Show full text]
  • Arithmetic Functions
    Lecture 13 Arithmetic Functions Today - Arithmetic functions, the Mobius¨ function (Definition) Arithmetic Function: An arithmetic function is a function f : N ! C Eg. π(n) = the number of primes ≤ n d(n) = the number of positive divisors of n σ(n) = the sum of the positive divisors of n σk(n) = the sum of the kth powers of n !(n) = the number of distinct primes dividing n Ω(n) = the number of primes dividing n counted with multiplicity Eg. σ(1) = 1 σ(2) = 1 + 2 = 3 σ(3) = 1 + 3 = 4 σ(6) = 1 + 2 + 3 + 6 = 12 (Definition) Perfect Number: A perfect number n is one for which σ(n) = 2n (eg., 6; 28; 496; etc.) Big open conjecture: Every perfect number is even. Note: One can show that if n is an even perfect number, then n = 2m−1(2m −1) where 2m − 1 is a Mersenne prime (Euler) (Definition) Multiplicative: If f is an arithmetic function such that whenever (m; n) = 1 then f(mn) = f(m)f(n), we say f is multiplicative. If f satisfies the stronger property that f(mn) = f(m)f(n) for all m; n (even if not coprime), we say f is completely multiplicative Eg. ( 1 n = 1 f(n) = 0 n < 1 is completely multiplicative. It’s sometimes called 1 (we’ll see why soon). Eg. f(n) = nk for some fixed k 2 N is also completely multiplicative Eg. !(n) is not multiplicative (adds, but 2!(n) is multiplicative) Eg. φ(n ) is multiplicative (by CRT) Note: If f is a multiplicative function, then to know f(n) for all n, it suffices to know f(n) for prime powers n.
    [Show full text]
  • Multiplicative Arithmetic Functions of Several Variables: a Survey
    Multiplicative Arithmetic Functions of Several Variables: A Survey L´aszl´oT´oth in vol. Mathematics Without Boundaries Surveys in Pure Mathematics T. M. Rassias, P. M. Pardalos (eds.), Springer, 2014, pp. 483–514 Abstract We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory and combinatorics. We give a new proof to obtain the asymptotic density of the set of ordered r-tuples of positive integers with pairwise relatively prime components and consider a similar question related to unitary divisors. 2010 Mathematics Subject Classification: 11A05, 11A25, 11N37 Key Words and Phrases: arithmetic function of several variables, multiplicative function, greatest common divisor, least common multiple, relatively prime integers, unitary divisor, arithmetic convolution, Dirichlet series, mean value, asymptotic density, asymptotic formula Contents 1 Introduction 2 2 Notations 2 arXiv:1310.7053v2 [math.NT] 19 Nov 2014 3 Multiplicative functions of several variables 3 3.1 Multiplicative functions . 3 3.2 Firmly multiplicative functions . 4 3.3 Completely multiplicative functions . 4 3.4 Examples ...................................... 5 4 Convolutions of arithmetic functions of several variables 7 4.1 Dirichlet convolution . 7 4.2 Unitaryconvolution ................................ 8 4.3 Gcdconvolution .................................. 9 4.4 Lcmconvolution .................................. 9 4.5 Binomial convolution .
    [Show full text]
  • Motivic Symbols and Classical Multiplicative Functions
    Motivic Symbols and Classical Multiplicative Functions Ane Espeseth & Torstein Vik Fagerlia videreg˚aendeskole, Alesund˚ f1; pkg ? Submitted to \Norwegian Contest for Young Scientists" 15 February 2016 Contact details: Ane Espeseth: [email protected] Torstein Vik: [email protected] 1 Abstract A fundamental tool in number theory is the notion of an arithmetical function, i.e. a function which takes a natural number as input and gives a complex number as output. Among these functions, the class of multiplicative functions is particularly important. A multiplicative function has the property that all its values are determined by the values at powers of prime numbers. Inside the class of multiplicative functions, we have the smaller class of completely multiplicative functions, for which all values are determined simply by the values at prime numbers. Classical examples of multiplicative functions include the constant func- tion with value 1, the Liouville function, the Euler totient function, the Jordan totient functions, the divisor functions, the M¨obiusfunction, and the Ramanu- jan tau function. In higher number theory, more complicated examples are given by Dirichlet characters, coefficients of modular forms, and more generally by the Fourier coefficients of any motivic or automorphic L-function. It is well-known that the class of all arithmetical functions forms a com- mutative ring if we define addition to be the usual (pointwise) addition of func- tions, and multiplication to be a certain operation called Dirichlet convolution. A commutative ring is a kind of algebraic structure which, just like the integers, has both an addition and a multiplication operation, and these satisfy various axioms such as distributivity: f · (g + h) = f · g + f · h for all f, g and h.
    [Show full text]
  • Multiplicative Functions
    Jay Daigle Occidental College Math 322: Number Theory 5 Multiplicative Functions For further reading on the material in this subsection, consult Rosen 7.1, Stein 2.2. Definition 5.1. An arithmetic function is a function defined for all natural numbers. A function is multiplicative if it has the property that f(mn) = f(m)f(n) whenever (m; n) = 1. It is completely multiplicative if f(mn) = f(m)f(n) for all natural numbers m; n. Example 5.2. The functions f(n) = 1 and g(n) = n are completely multiplicative. We will see that φ(n) is multiplicative but not completely multiplicative. (Example: φ(4) = 2 6= 1 · 1 = φ(2) · φ(2)). Completely multiplicative functions are easy to understand, but we can get a good grasp even of regularly multiplicative functions. a1 a2 as Qs ai Proposition 5.3. If f is multiplicative and n = p1 p2 : : : ps = i=1 pi is the prime factorization of n, then s a1 a2 as Y ai f(n) = f(p1 )f(p2 ) : : : f(ps ) = f(pi ): i=1 Proof. We prove by induction on s, the number of distinct prime factors of n. If s = 1 then a1 a1 n = p1 and then f(n) = f(p1 ) is trivially true. Suppose the proposition is true for all integers with k distinct prime factors, and suppose Qk+1 ai Qk ai ak+1 n has k + 1 distinct prime factors, say n = i=1 pi . We observe that i=1 pi ; pk+1 = 1, and thus by definition of a multiplicative function we know that k+1 ! k ! Y ai Y ai ak+1 f pi = f pi f(pk+1 ): i=1 i=1 And by inductive hypothesis we know that k ! k Y ai Y ai f pi = f(pi ) i=1 i=1 and thus we have k+1 ! k k+1 Y ai Y ai ak+1 Y ai f pi = f(pi )f(pk+1 ) = f(pi ): i=1 i=1 i=1 Thus for any multiplicative function, if we can compute its value for prime powers, we can easily compute its value for any number.
    [Show full text]