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7 Asymptotic properties 19 7.1 Meanvalues...... 19 7.2 Asymptoticdensities ...... 20 7.3 Asymptoticformulas...... 22
8 Acknowledgement 24
1 Introduction
Multiplicative arithmetic functions of a single variable are very well known in the liter- ature. Their various properties were investigated by several authors and they represent an important research topic up to now. Less known are multiplicative arithmetic functions of several variables of which detailed study was carried out by R. Vaidyanathaswamy [74] more than eighty years ago. Since then many, sometimes scattered results for the several variables case were published in papers and monographs, and some authors of them were not aware of the paper [74]. In fact, were are two different notions of multiplicative functions of several variables, used in the last decades, both reducing to the usual multiplicativity in the one variable case. For the other concept we use the term firmly multiplicative function. In this paper we survey general properties of multiplicative arithmetic functions of sev- eral 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. The concept of the convolute of a function with respect to the Dirichlet convolution was introduced by R. Vaidyanathaswamy [74]. We also discuss the multiple Dirichlet series and Bell series. We present certain arithmetic and asymptotic re- sults 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. Further- more, we consider a similar question, namely the asymptotic density of the set of ordered r-tuples with pairwise unitary relatively prime components, that is, the greatest common unitary divisor of each two distinct components is 1. For general properties of (multiplicative) arithmetic functions of a single variable see, e.g., the books of T. M. Apostol [4], G. H. Hardy, E. M. Wright [27], P. J. Mc.Carthy [42], W. Schwarz, J. Spilker [53] and R. Sivaramakrishnan [55]. For algebraic proper- ties of the ring of arithmetic functions of a single variable with the Dirichlet convolution we refer to H. N. Shapiro [54]. Incidence algebras and semilattice algebras concerning arith- metic functions of a single variable were investigated by D. A. Smith [57]. For properties of certain subgroups of the group of multiplicative arithmetic functions of a single variable under the Dirichlet convolution we refer to the papers by T. B. Carroll, A. A. Gioia [7], P.-O. Dehaye [14], J. E. Delany [17], T. MacHenry [41] and R. W. Ryden [48]. Algebraical and topological properties of the ring of arithmetic functions of a single variable with the unitary convolution were given by J. Snellman [58, 59]. See also J. Sandor,´ B. Crstici [50, Sect. 2.2] and H. Scheid [51].
2 Notations
Throughout the paper we use the following notations. General notations:
2 N = 1, 2,... , N = 0, 1, 2,... , • { } 0 { } the prime power factorization of n N is n = pνp(n), the product being over the • ∈ p primes p, where all but a finite number of the exponents νp(n) are zero, d n means that d is a unitary divisor of n, i.e.,Qd n and gcd(d, n/d) = 1, • gcud(|| n ,...,n ) denotes the greatest common unitary| divisor of n ,...,n N, • 1 k 1 k ∈ Zn = Z/nZ is the additive group of residue classes modulo n, • ζ is the Riemann zeta function, • γ is Euler’s constant. Arithmetic• functions of a single variable: δ is the arithmetic function given by δ(1) = 0 and δ(n)=0 for n> 1, • id is the function id(n)= n (n N), • φ is the Jordan function of order∈ k given by φ (n)= nk (1 1/pk) (k C), • k k p|n − ∈ φ = φ1 is Euler’s totient function, • Q ψ is the Dedekind function given by ψ(n)= n p|n(1+1/p), • µ is the M¨obius function, • Q τk is the Piltz divisor function of order k, τk(n) representing the number of ways of expressing• n as a product of k factors, τ(n)= τ2(n) is the number of divisors of n, • σ (n)= dk (k C), • k d|n ∈ σ(n)= σ (n) is the sum of divisors of n, • P1 ω(n) = # p : νp(n) =0 stands for the number of distinct prime divisors of n, • µ×(n) = ({ 1)ω(n) 6 } • Ω(n)= −ν (n) is the number of prime power divisors of n, • p p λ(n) = ( 1)Ω(n) is the Liouville function, • P− ξ(n)= p νp(n)!, • N cn(k) = 1≤q≤n,gcd(q,n)=1 exp(2πiqk/n) (n, k ) is the Ramanujan sum, which can be viewed• asQ a function of two variables. ∈ ArithmeticP functions of several variables: r r is the set of arithmetic functions of r variables (r N), i.e., of functions f : N C, • A(1) ∈ → r = f r : f(1,..., 1) =0 , • A1 is the{ constant∈ A 1 function6 in } , i.e., 1 (n ,...,n ) = 1 for every n ,...,n N, • r Ar r 1 r 1 r ∈ δr(n1,...,nr) = δ(n1) δ(nr), that is δr(1,..., 1) = 1 and δr(n1,...,nr) = 0 for n • n > 1, ··· 1 ··· r for f r the function f 1 is given by f(n)= f(n,...,n) for every n N. Other• notations∈ A will be fixed∈ inside A the paper. ∈
3 Multiplicative functions of several variables
In what follows we discuss the notions of multiplicative, firmly multiplicative and com- pletely multiplicative functions. We point out that properties of firmly and completely multiplicative functions of several variables reduce to those of multiplicative, respectively completely multiplicative functions of a single variable. Multiplicative functions can not be reduced to functions of a single variable. We also present examples of such functions.
3.1 Multiplicative functions A function f is said to be multiplicative if it is not identically zero and ∈ Ar
f(m1n1,...,mrnr)= f(m1,...,mr)f(n1,...,nr) holds for any m ,...,m ,n ,...,n N such that gcd(m m ,n n ) = 1. 1 r 1 r ∈ 1 ··· r 1 ··· r
3 If f is multiplicative, then it is determined by the values f(pν1 ,...,pνr ), where p is prime and ν ,...,ν N . More exactly, f(1,..., 1) = 1 and for any n ,...,n N, 1 r ∈ 0 1 r ∈ νp(n1) νp(nr ) f(n1,...,nr)= f(p ,...,p ). p Y If r = 1, i.e., in the case of functions of a single variable we reobtain the familiar notion of multiplicativity: f 1 is multiplicative if it is not identically zero and f(mn)= f(m)f(n) for every m,n N ∈such A that gcd(m,n) = 1. Let denote∈ the set of multiplicative functions in r variables. Mr 3.2 Firmly multiplicative functions
We call a function f r firmly multiplicative (following P. Haukkanen [28]) if it is not identically zero and ∈ A
f(m1n1,...,mrnr)= f(m1,...,mr)f(n1,...,nr) holds for any m ,...,m ,n ,...,n N such that gcd(m ,n ) = ... = gcd(m ,n ) = 1. 1 r 1 r ∈ 1 1 r r Let r denote the set of firmly multiplicative functions in r variables. AF firmly multiplicative function is completely determined by its values at ν (1,..., 1,p , 1,..., 1), where p runs through the primes and ν N0. More exactly, f(1,..., 1) = 1 and for any n ,...,n N, ∈ 1 r ∈ f(n ,...,n )= f(pνp(n1), 1,..., 1) f(1,..., 1,pνp(nr)) . 1 r ··· p Y If a function f is firmly multiplicative, then it is multiplicative. Also, if f , then ∈ Ar ∈Fr f(n1,...,nr)= f1(n1, 1,..., 1) fr(1,..., 1,nr) for every n1,...,nr N. This immediately gives the following property: ··· ∈
Proposition 1. A function f r is firmly multiplicative if and only if there exist mul- tiplicative functions f ,...,f ∈ A (each of a single variable) such that f(n ,...,n ) = 1 r ∈ M1 1 r f1(n1) fr(nr) for every n1,...,nr N. In this case f1(n) = f(n, 1,..., 1), ..., fr(n) = f(1,...,···1,n) for every n N. ∈ ∈ In the case of functions of a single variable the notion of firmly multiplicative function reduces to that of multiplicative function. For r> 1 the concepts of multiplicative and firmly multiplicative functions are different.
3.3 Completely multiplicative functions A function f is called completely multiplicative if it is not identically zero and ∈ Ar f(m1n1,...,mrnr)= f(m1,...,mr)f(n1,...,nr) holds for any m1,...,mr,n1,...,nr N. Note that R. Vaidyanathaswamy [74] used for such a function the term ’linear function’.∈ Let r denote the set of completely multiplicative functions in r variables. If f r, then it isC determined by its values at (1,..., 1, p, 1,..., 1), where p runs through the primes.∈ C More exactly, f(1,..., 1) = 1 and for any n ,...,n N, 1 r ∈ f(n ,...,n )= f(p, 1,..., 1)νp(n1) f(1,..., 1,p)νp(nr ) . 1 r ··· p Y In the case of functions of a single variable we reobtain the familiar notion of completely multiplicative function: f 1 is completely multiplicative if it is not identically zero and f(mn)= f(m)f(n) for every∈ Am,n N. ∈ It is clear that if a function f r is completely multiplicative, then it is firmly multi- plicative. Also, similar to Proposition∈ A 1:
4 Proposition 2. A function f r is completely multiplicative if and only if there ex- ist completely multiplicative functions∈ A f ,...,f (each of a single variable) such that 1 r ∈ C1 f(n1,...,nr)= f1(n1) fr(nr) for every n1,...,nr N. In this case f1(n)= f(n, 1,..., 1), ..., f (n)= f(1,..., 1,n···) for every n N. ∈ r ∈ 3.4 Examples
The functions (n1,...,nr) gcd(n1,...,nr) and (n1,...,nr) lcm(n1,...,nr) are multiplicative for every r N, but7→ not firmly multiplicative for r 2.7→ ∈ ≥ The functions (n1,...,nr) τ(n1) τ(nr), (n1,n2) τ(n1)σ(n2) are firmly multi- plicative, but not completely multiplicative.7→ ··· 7→ The functions (n1,...,nr) n1 nr, (n1,n2) n1λ(n2) are completely multiplicative. According to Propositions7→1 and···2 firmly multiplicative7→ and completely multiplicative functions reduce to multiplicative, respectively completely multiplicative functions of a single variable. There is no similar characterization for multiplicative functions of several variables. Let h . Then the functions (n ,...,n ) h(gcd(n ,...,n )), (n ,...,n ) ∈M1 1 r 7→ 1 r 1 r 7→ h(lcm(n1,...,nr)) are multiplicative. The product and the quotient of (nonvanishing) mul- tiplicative functions are multiplicative. If f r is multiplicative and we fix one (or more, say s) variables, then the resulting function∈M of r 1 (or r s) variables is not necessary multiplicative. For example, (k,n) − − 7→ cn(k) is multiplicative as a function of two variables, see Section 4.1, but for a fixed n the function k cn(k) is in general not multiplicative (it is multiplicative if and only if µ(n) = 1). 7→ If f r is multiplicative, then the function f of a single variable is multiplicative. Other∈M examples from number theory, group theory and combinatorics:
Example 1. Let Ng1,...,gr (n1,...,nr) denote the number of solutions x (mod n), with n = lcm(n1,...,nr), of the simultaneous congruences g1(x) 0 (mod n1), ..., gr(x) 0 (mod nr), where g ,...,g are polynomials with integer coefficients.≡ Then the function (≡n ,...,n ) 1 r 1 r 7→ Ng1,...,gr (n1,...,nr) is multiplicative. See L. Toth´ [65, Sect. 2] for a proof. Example 2. For a fixed integer r 2 let ≥
1, if n1,...,nr are pairwise relatively prime, ̺(n1,...,nr)= (1) (0, otherwise.
This function is multiplicative, which follows from the definition, and for every n1,...,nr N, ∈ ̺(n ,...,n )= τ(d d )µ(n /d ) µ(n /d ), (2) 1 r 1 ··· r 1 1 ··· r r d1|n1X,...,dr|nr cf. Section 4.1. Example 3. For r 2 let ≥
× 1, if gcud(ni,nj) = 1 for every i = j, ̺ (n1,...,nr)= 6 (3) (0, otherwise.
This is the characteristic function of the set of ordered r-tuples (n ,...,n ) Nr such 1 r ∈ that n1,...,nr are pairwise unitary relatively prime, i.e., such that for every prime p there are no i = j with ν (n )= ν (n ) 1. This function is also multiplicative (by the definition). 6 p i p j ≥ Z Z Example 4. Consider the group G = n1 nr . Let s(n1,...,nr) and c(n1,...,nr) de- note the total number of subgroups of the group×···×G and the number of its cyclic subgroups, re- spectively. Then the functions (n ,...,n ) s(n ,...,n ) and (n ,...,n ) c(n ,...,n ) 1 r 7→ 1 r 1 r 7→ 1 r
5 are multiplicative. For every n ,...,n N, 1 r ∈
φ(d1) φ(dr) c(n1,...,nr)= ··· , (4) φ(lcm(d1,...,dr)) d1|n1X,...,dr|nr see L. Toth´ [65, Th. 3], [66, Th. 1]. In the case r = 2 this gives
c(n1,n2)= φ(gcd(d1, d2)). d1|nX1,d2|n2 Also,
s(n1,n2)= gcd(d1, d2), (5) d1|nX1,d2|n2 for every n1,n2 N. See M. Hampejs, N. Holighaus, L. Toth,´ C. Wiesmeyr [25] and M. Hampejs, L.∈ Toth´ [26]. Example 5. We define the sigma function of r variables by
σ(n1,...,nr)= gcd(d1,...,dr) (6) d1|n1X,...,dr|nr having the representation
σ(n ,...,n )= φ(d)τ(n /d) τ(n /d), (7) 1 r 1 ··· r d|gcd(Xn1,...,nr) valid for every n1,...,nr N. Note that for r = 1 this function reduces to the sum-of-divisors function and in the case r∈= 2 we have σ(m,n)= s(m,n) given in Example 4. r r We call (n1,...,nr) N a perfect r-tuple if σ(n1,...,nr) = 2gcd(n1,...,nr). If2 1= p is a Mersenne prime, then∈ (p,p,...,p) is a perfect r-tuple. For example, (3, 3) is a perfect− pair and (7, 7, 7) is perfect triple. We formulate as an open problem: Which are all the perfect r-tuples? Example 6. The Ramanujan sum (k,n) c (k) having the representation 7→ n
cn(k)= dµ(n/d) (8) d|gcd(Xk,n) is multiplicative as a function of two variables. This property was pointed out by K. R. John- son [34], see also Section 4.1. Example 7. For n ,...,n N let n := lcm(n ,...,n ). The function of r variables 1 r ∈ 1 r n 1 E(n ,...,n )= c (j) c (j) 1 r n n1 ··· nr j=1 X has combinatorial and topological applications, and was investigated in the papers of V.A. Liskovets [40] and L. Toth´ [67]. All values of E(n1,...,nr) are nonnegative integers and the function E is multiplicative. Furthermore, it has the following representation ([67, Prop. 3]): d1µ(n1/d1) drµ(nr/dr) E(n1,...,nr)= ··· , (9) lcm(d1,...,dr) d1|n1X,...,dr|nr valid for every n1,...,nr N. See also L. Toth´ [68, 71] for generalizations of the function E. ∈
6 Example 8. Another multiplicative function, similar to E is
n 1 A(n ,...,n )= gcd(k,n ) gcd(k,n ), 1 r n 1 ··· r Xk=1 where n ,...,n N and n := lcm(n ,...,n ), as above. One has for every n ,...,n N, 1 r ∈ 1 r 1 r ∈ φ(d1) φ(dr) A(n1,...,nr)= ··· , (10) lcm(d1,...,dr) d1|n1X,...,dr|nr see L. Toth´ [64, Eq. (45)], [67, Prop. 12]. See the paper of M. Peter [47] for properties of recurrent multiplicative arithmetical functions of several variables.
4 Convolutions of arithmetic functions of several vari- ables
In this Section we survey the basic properties of the Dirichlet and unitary convolutions of arithmetic functions of several variables. We also define and discuss the gcd, lcm and binomial convolutions, not given in the literature in the several variables case. We point out that the gcd convolution reduces to the unitary convolution in the one variable case, but they are different for r variables with r > 1. For other convolutions we refer to the papers of J. Sandor,´ A. Bege [49], E. D. Schwab [52] and M. V. Subbarao [61].
4.1 Dirichlet convolution
For every r N the set r of arithmetic functions of r variables is a C-linear space with the usual linear∈ operations.A With the Dirichlet convolution defined by
(f g)(n ,...,n )= f(d ,...,d )g(n /d ,...,n /d ) ∗ 1 r 1 r 1 r r r d1|n1X,...,dr|nr the space forms a unital commutative C-algebra, the unity being the function δ , and Ar r ( r, +, ) is an integral domain. Moreover, ( r, +, ) is a unique factorization domain, as pointedA ∗ out by T. Onozuka [45]. In the caseAr = 1∗ this was proved by E. D. Cashwell, (1) C. J. Everett [9]. The group of invertible functions is r . The inverse of f will be −1∗ A denoted by f . The inverse of the constant 1 function 1r is µr, given by µr(n1,...,nr)= µ(n1) µ(nr) (which is firmly multiplicative, where µ is the classical M¨obius function). The··· Dirichlet convolution preserves the multiplicativity of functions. This property, well known in the one variable case, follows easily from the definitions. Using this fact the multiplicativity of the functions c(n1,...,nr), s(n1,n2), σ(n1,...,nr), E(n1,...,nr) and A(n1,...,nr) is a direct consequence of the convolutional representations (4), (5), (6), (9) and (10), respectively. The multiplicativity of the Ramanujan sum (k,n) c (k) follows in a 7→ n similar manner from (8), showing that cn(k) is the convolution of the multiplicative functions f and g defined by f(m,n)= m for m = n, f(m,n)=0 for m = n and g(m,n)= µ(m) for every m,n N. 6 If f,g ∈ , then ∈Fr (f g)(n ,...,n ) = (f g )(n ) (f g )(n ) ∗ 1 r 1 ∗ 1 1 ··· r ∗ r r and f −1∗(n ,...,n )= f −1∗(n ) f −1∗(n ), 1 r 1 1 ··· r r with the notations of Proposition 1, hence f g and f −1∗ . We deduce ∗ ∈Fr ∈Fr
7 Proposition 3. One has the following subgroup relations:
( , ) ( , ) ( (1), ). Fr ∗ ≤ Mr ∗ ≤ Ar ∗ −1∗ The set r does not form a group under the Dirchlet convolution. If f r, then f = µ f (well knownC in the case r = 1). Note that for every f one has (f ∈Cf)(n ,...,n )= r ∈Cr ∗ 1 r f(n1,...,nr)τ(n1) τ(nr), in particular (1r 1r)(n1,...,nr)= τ(n1) τ(nr). R. Vaidyanathaswamy··· [74] called the Dirichlet∗ convolution ’composition··· of functions’. Other convolutional properties known in the one variable case, for example M¨obius in- version can easily be generalized. As mentioned in Section 3.4, Example 2 the characteristic r function ̺ of the set of ordered r-tuples (n1,...,nr) N such that n1,...,nr N are pairwise relatively prime is multiplicative. One has for∈ every n ,...,n N, ∈ 1 r ∈ ̺(d ,...,d )= τ(n n ), (11) 1 r 1 ··· r d1|n1X,...,dr|nr
ν1 νr since both sides are multiplicative and in the case of prime powers n1 = p ,...,nr = p both sides of (11) are equal to 1+ ν1 + ... + νr. Now, M¨obius inversion gives the formula (2). For further algebraic properties of the C-algebra r and more generally, of the R-algebra r A Ar(R) = f : N R , where R is an integral domain and using the concept of firmly multiplicative{ functions7→ } see E. Alkan, A. Zaharescu, M. Zaki [1]. That paper includes, among others, constructions of a class of derivations and of a family of valuations on Ar(R). See also P. Haukkanen [28] and A. Zaharescu, M. Zaki [75].
4.2 Unitary convolution
The linear space r forms another unital commutative C-algebra with the unitary con- volution defined by A
(f g)(n ,...,n )= f(d ,...,d )g(n /d ,...,n /d ). × 1 r 1 r 1 1 r r d1||n1X,...,dr||nr
Here the unity is the function δr again. Note that ( r, +, ) is not an integral do- A × (1) main, there exist divisors of zero. The group of invertible functions is again r . The × A unitary r variables M¨obius function µr is defined as the inverse of the function 1r. One has × × × ω(n1)+...+ω(nr) µr (n1,...,nr) = µ (n1) µ (nr) = ( 1) (and it is firmly multiplicative). Similar to Proposition 3, ··· − Proposition 4. One has the following subgroup relations:
( , ) ( , ) ( (1), ). Fr × ≤ Mr × ≤ Ar × If f , then its inverse is f −1× = µ×f and ∈Fr r (f f)(n ,...,n )= f(n ,...,n )2ω(n1)+...+ω(nr), × 1 r 1 r
ω(n1)+...+ω(nr) in particular (1r 1r)(n1,...,nr)=2 . R. Vaidyanathaswamy× [74] used the term ’compounding of functions’ for the unitary convolution. r For further algebraic properties of the R-algebra Ar(R) = f : N R , where R is an integral domain with respect to the unitary convolution and{ using the7→ concept} of firmly multiplicative functions see E. Alkan, A. Zaharescu, M. Zaki [2].
8 4.3 Gcd convolution We define a new convolution for functions f,g , we call it gcd convolution, given by ∈ Ar (f g)(n ,...,n )= f(d ,...,d )g(e ,...,e ), (12) ⊙ 1 r 1 r 1 r d1e1=n1,...,drer =nr gcd(d1···dXr,e1···er )=1 which is in concordance with the definition of multiplicative functions. In the case r = 1 the unitary and gcd convolutions are identic, i.e., f g = f g for × ⊙ every f,g 1, but they differ for r> 1. Main properties:∈ A forms a unital commutative C-algebra with the gcd convolution Ar defined by (12). The unity is the function δr and there exist divisors of zero. The group of in- (1) ⊙ vertible functions is again r . Here the inverse of the constant 1 function is µr (n1,...,nr)= 1 A ( 1)ω(n ···nr). More generally, the inverse f −1⊙ of an arbitrary multiplicative function f is − 1 given by f −1⊙(n ,...,n ) = ( 1)ω(n ···nr)f(n ,...,n ). Also, if f , then 1 r − 1 r ∈Mr (f f)(n ,...,n )= f(n ,...,n )2ω(n1···nr). ⊙ 1 r 1 r Proposition 5. One has ( , ) ( (1), ). Mr ⊙ ≤ Ar ⊙ The set r does not form a group under the gcd convolution. To see this note that F 1 (1 1 )(n ,...,n ) = 2ω(n ···nr), but this function is not firmly multiplicative, since r ⊙ r 1 r ω(n1 nr) = ω(n1)+ ... + ω(nr) does not hold for every n1,...,nr N (cf. Proposi- tion 1···). ∈
4.4 Lcm convolution We define the lcm convolution of functions of r variables by
(f g)(n ,...,n )= f(d ,...,d )g(e ,...,e ). ⊕ 1 r 1 r 1 r lcm(d1,e1)=n1X,...,lcm(dr,er )=nr In the case r = 1 this convolution originates by R. D. von Sterneck [60], was investi- gated and generalized by D. H. Lehmer [37, 38, 39], and is also called von Sterneck-Lehmer convolution. See also R. G. Buschman [5]. Note that the lcm convolution can be expressed by the Dirichlet convolution. More exactly, Proposition 6. For every f,g , ∈ Ar f g = (f 1 )(g 1 ) µ . (13) ⊕ ∗ r ∗ r ∗ r Proof. Write (f g)(a ,...,a ) ⊕ 1 r a1|n1X,...,ar|nr
= f(d1,...,dr)g(e1,...,er) lcm(d1,e1)|n1X,...,lcm(dr,er )|nr
= f(d1,...,dr) g(e1,...,er) d1|n1X,...,dr|nr e1|n1X,...,er |nr = (f 1 )(n ,...,n )(g 1 )(n ,...,n ), ∗ r 1 r ∗ r 1 r and by M¨obius inversion we obtain (13).
9 In the case r = 1 Proposition 6 is due to R. D. von Sterneck [60] and D. H. Lehmer [38]. 2 2 Note that (1r 1r)(n1,...,nr)= τ(n1) τ(nr). It turns out that the lcm convolution preserves the multiplicativity⊕ of functions,··· but does not form a group under the lcm Mr convolution. The unity for the lcm convolution is the function δr again. Here ( r, +, ) is a unital commutative ring having divisors of zero. The group of invertible functionsA ⊕ is = f : (f 1 )(n ,...,n ) = 0 for every (n ,...,n ) Nr and we deduce Ar { ∈ Ar ∗ r 1 r 6 1 r ∈ } Proposition 7. i) Let = f : (f 1 )(n ,...,n ) =0 for every (n ,...n ) Nr . e Mr { ∈Mr ∗ r 1 r 6 1 r ∈ } Then r is a subgroup of r with respect to the lcm convolution. M A ⊕ ⊕ ii) The inverse of thef function 1r r is µr = µr 1/(1r 1r). The function µr is ∈ A 1 ∗ ∗ multiplicativef and for everye prime powers pν ,...,pνr , e ( 1)r µ⊕(pν1 ,...,pνr )= − . r ν (ν + 1) ν (ν + 1) 1 1 ··· r r
Proof. Here (ii) follows from (1r 1r)(n1,...,nr) = τ(n1) τ(nr), already mentioned in Section 4.1. ∗ ···
4.5 Binomial convolution We define the binomial convolution of the functions f,g by ∈ Ar (f g)(n ,...,n ) ◦ 1 r ν (n ) ν (n ) = p 1 p r f(d ,...,d )g(n /d ,...,n /d ), ν (d ) ··· ν (d ) 1 r 1 1 r r p p 1 p r ! d1|n1X,...dr|nr Y a where b is the binomial coefficient. It is remarkable that the binomial convolution preserves the complete multiplicativity of arithmetical functions, which is not the case for the Dirichlet convolution and other convolutions. Let ξr be the firmly multiplicative function given by ξ (n ,...,n ) = ξ(n ) ξ(n ), that is ξ (n ,...,n ) = (ν (n )! ν (n )!). Then for r 1 r 1 ··· r r 1 r p p 1 ··· p r every f,g r, ∈ A f g Q f g = ξ , (14) ◦ r ξ ∗ ξ r r leading to the next result.
Proposition 8. The algebras ( r, +, , C) and ( r, +, , C) are isomorphic under the map- f A ◦ A ∗ ping f . 7→ ξr Formula (14) also shows that the binomial convolution preserves the multiplicativity of functions. Furthermore, for any fixed r N the structure ( r, +, ) is an integral domain ∈ (1)A ◦ with unity δr. The group of invertible functions is again r . If f,g r, then with the notations of Proposition 1, A ∈ F
(f g)(n ,...,n ) = (f g )(n ) (f g )(n ) ◦ 1 r 1 ◦ 1 1 ··· r ◦ r r and the inverse of f is
f −1◦(n ,...,n )= f −1◦(n ) f −1◦(n ), 1 r 1 1 ··· r r −1◦ hence f g r and f r. The inverse of the function 1r under the binomial convo- ◦ ∈F ∈F Ω(n1)+...+Ω(nr) lution is the function λr given by λr(n)= λ(n1) λ(nr), i.e., λr(n) = ( 1) . We deduce ··· −
10 Proposition 9. One has the following subgroup relations:
( , ) ( , ) ( , ) ( (1), ). Cr ◦ ≤ Fr ◦ ≤ Mr ◦ ≤ Ar ◦ In the case r = 1 properties of this convolution were discussed in the paper by L. Toth,´ P. Haukkanen [72]. The proofs are similar in the multivariable case.
5 Generating series
As generating series for multiplicative arithmetic functions of r variables we present certain properties of the multiple Dirichlet series, used earlier by several authors and the Bell series, which constituted an important tool of R. Vaidyanathaswamy [74].
5.1 Dirichlet series The multiple Dirichlet series of a function f is given by ∈ Ar ∞ f(n1,...,nr) D(f; z1,...,zr)= z1 zr . n nr n1,...,n =1 1 Xr ···
Similar to the one variable case, if D(f; z1,...,zr) is absolutely convergent in (s1,...,sr) Cr, then it is absolutely convergent in every (z ,...,z ) Cr with z s (1 j r).∈ 1 r ∈ ℜ j ≥ℜ j ≤ ≤ Proposition 10. Let f,g r. If D(f; z1,...,zr) and D(g; z1,...,zr) are absolutely con- vergent, then D(f g; z ,...,z∈ A ) is also absolutely convergent and ∗ 1 r D(f g; z ,...,z )= D(f; z ,...,z )D(g; z ,...,z ). ∗ 1 r 1 r 1 r (1) Also, if f r , then ∈ A −1∗ −1 D(f ; z1,...,zr)= D(f; z1,...,zr) , formally or in the case of absolute convergence. If f r is multiplicative, then its Dirichlet series can be expanded into a (formal) Euler product,∈ M that is,
∞ f(pν1 ,...,pνr ) D(f; z1,...,zr)= , (15) pν1z1+...+νrzr p ν1,...,ν =0 Y Xr the product being over the primes p. More exactly, r Proposition 11. Let f r. For every (z1,...,zr) C the series D(f; z1,...,zr) is absolutely convergent if and∈ M only if ∈
∞ f(pν1 ,...,pνr ) | | < pν1ℜz1+...+νrℜzr ∞ p ν1,...,νr=0 Y ν1+...X+νr ≥1 and in this case the equality (15) holds. Next we give the Dirichlet series representations of certain special functions discussed above. These follow from the convolutional identities given in Section 3.4. For every g 1 we have formally, ∈ A
∞ g(gcd(n ,...,n )) ζ(z ) ζ(z ) ∞ g(n) 1 r = 1 ··· r . (16) z1 zr z1+...+z n nr ζ(z1 + ... + zr) n r n1,...,n =1 1 n=1 Xr ··· X 11 In particular, taking g(n)= n (16) gives for r 2 and z > 1,..., z > 1, ≥ ℜ 1 ℜ r ∞ gcd(n1,...,nr) ζ(z1) ζ(zr)ζ(z1 + ... + zr 1) z1 zr = ··· − n nr ζ(z1 + ... + zr) n1,...,n =1 1 Xr ··· and taking g = δ one obtains for r 2 and z > 1,..., z > 1, ≥ ℜ 1 ℜ r ∞ 1 ζ(z1) ζ(zr) z1 zr = ··· , (17) n1 nr ζ(z1 + ... + zr) n1,...,nr =1 ··· gcd(n1X,...,nr)=1 where the identity (17) is the Dirichlet serries of the characteristic function of the set of points in Nr, which are visible from the origin (cf. T. M. Apostol [4, Page 248, Ex. 15]). The Dirichlet series of the characteristic function ̺ concerning r pairwise relatively prime integers is ∞ ̺(n1,...,nr) z1 zr n nr n1,...,n =1 1 Xr ··· r r r 1 1 1 = ζ(z1) ζ(zr) 1 + 1 (18) ··· − pzj pzj − pzk p j=1 j=1 k=1 Y Y X kY6=j r j−1 1 = ζ(z1) ζ(zr) 1+ ( 1) (j 1) z +...+z , (19) ··· − − p i1 ij p j=2 1 Y X 1≤i <...
12 5.2 Bell series If f is a multiplicative function of r variables, then its (formal) Bell series to the base p (p prime) is defined by
∞ f (x ,...,x )= f(pe1 ,...,per )xe1 xer , (p) 1 r 1 ··· r e1,...,e =0 Xr where the constant term is 1. The main property is the following: for every f,g , ∈Mr (f g) (x ,...,x )= f (x ,...,x )g (x ,...,x ). ∗ (p) 1 r (p) 1 r (p) 1 r The connection of Bell series to Dirichlet series and Euler products is given by
−z1 −zr D(f; z1,...,zr)= f(p)(p ,...,p ), (22) p Y valid for every f . For example, the Bell series of the gcd function f(n ,...,n ) = ∈ Mr 1 r gcd(n1,...,nr) is 1 x x f (x ,...,x )= − 1 ··· r . (p) 1 r (1 x ) (1 x )(1 px x ) − 1 ··· − r − 1 ··· r
The Bell series of other multiplicative functions, in particular of c(m,n), s(m,n), σ(n1,...,nr) and cn(k) can be given from their Dirichlet series representations and using the relation (22). Note that in the one variable case the Bell series to a fixed prime of the unitary convolution of two multiplicative functions is the sum of the Bell series of the functions, that is
(f g) (x )= f (x )+ g (x ), × (p) 1 (p) 1 (p) 1 see R. Vaidyanathaswamy [74, Th. XII]. This is not valid in the case of r variables with r> 1.
6 Convolutes of arithmetic functions of several variables
Let f . By choosing n = ... = n = n we obtain the function of a single variable ∈ Ar 1 r n f(n)= f(n,...,n). If f r, then f 1, as already mentioned. Less trivial ways to7→ retrieve from f functions of∈M a single variable∈M is to consider for r> 1,
Ψdir(f)(n)= f(d1,...,dr), (23) d1···Xdr=n
Ψunit(f)(n)= f(d1,...,dr), (24)
d1···dr=n gcd(di,dXj )=1,i6=j
Ψgcd(f)(n)= f(d1,...,dr), (25)
d1···dr=n gcd(d1X,...,dr)=1
Ψlcm(f)(n)= f(d1,...,dr), (26) lcm(d1X,...,dr)=n
νp(n) Ψbinom(f)(n)= f(d1,...,dr), (27) νp(d1),...,νp(dr) 1 p ! d ···Xdr=n Y
13 r where the sums are over all ordered r-tuples (d1,...,dr) N with the given additional conditions, the last one involving multinomial coefficients.∈ For (24) the condition is that d1 dr = n and d1,...,dr are pairwise relatively prime. Note that (24) and (25) are the same··· for r = 2, but they differ in the case r> 2. Assume that there exist functions g ,...,g (each of a single variable) such that 1 r ∈ A1 f(n1,...,nr)= g1(n1) gr(nr) (in particular this holds if f r by Proposition 1). Then (23), (24), (26) and (27···) reduce to the Dirichlet convolution,∈F unitary convolution, lcm con- volution and binomial convolution, respectively, of the functions g1,...,gr. For r = 2 we have the corresponding convolutions of two given functions of a single variable. Note the following special case of (26):
φ(d ) φ(d )= φ (n) (n N), 1 ··· r r ∈ lcm(d1X,...,dr)=n due to R. D. von Sterneck [60]. We remark that (23), with other notation, appears in [74, p. 591-592], where Ψdir(f) is called the ’convolute’ of f (obtained by the convolution of the arguments). We will call Ψdir(f), Ψunit(f), Ψgcd(f), Ψlcm(f)andΨbinom(f) the Dirichlet convolute, unitary convolute, gcd convolute, lcm convolute and binomial convolute, respectively of the function f. Some special cases of convolutes of functions which are not the product of functions of a single variable are the following. Special Dirichlet convolutes are
gr(n)= gcd(d1,...,dr), (28) d1···Xdr=n
ℓr(n)= lcm(d1,...,dr). (29) d1···Xdr=n N(n)= φ(gcd(d, n/d)). (30) Xd|n For r = 2 (28) and (29) are sequences A055155 and A057670, respectively in [56]. The function N(n) given by (30) represents the number of parabolic vertices of γ0(n) (sequence A001616 in [56]), cf. S. Finch [23]. In the case r = 2 the lcm convolute of the gcd function
c(n)= gcd(d, e), (31) lcm(Xd,e)=n represents the number of cyclic subgroups of the group Zn Zn, as shown by A. Pakapong- pun, T. Ward [46, Ex. 2]. That is, c(n) = c(n,n) for every× n N, with the notation of Section 3.4, Example 4 (it is sequence A060648 in [56]). ∈ The Dirichlet, unitary and lcm convolutes of the Ramanujan sums are
a(n)= cd(n/d), (32) Xd|n
b(n)= cd(n/d), (33) Xd||n
h(n)= cd(e). (34) lcm(Xd,e)=n All the functions gr,ℓr,N,c,a,b,h defined above are multiplicative, as functions of a single variable. See Corollary 1.
14 6.1 General results The Dirichlet, unitary, gcd, lcm and binomial convolutions preserve the multiplicativity of functions of a single variable, cf. Section 4. As a generalization of this property, we prove the next result. Proposition 12. Let f be an arbitrary multiplicative function. Then all the functions ∈Mr Ψdir(f), Ψunit(f), Ψgcd(f), Ψlcm(f) and Ψbinom(f) are multiplicative. Note that for the Dirichlet covolute this property was pointed out by R. Vaidyanathaswamy in [74, p. 591-592].
Proof. By the definitions. Let n,m N such that gcd(n,m)=1. If d1 dr = nm, then there exist unique integers a ,b ,...,a∈ ,b N such that a ,...,a n, ···b ,...,b m and 1 1 r r ∈ 1 r | 1 r | d1 = a1b1,...,dr = arbr. Here gcd(a1 ar,b1 br) = 1. Using the multiplicativity of f we obtain ··· ··· Ψdir(f)(nm)= f(a1b1,...,arbr) a1···ar=n b1···Xbr=m
= f(a1,...,ar) f(b1,...,br) a1···a =n 1 Xr b ···Xbr=m =Ψdir(f)(n)Ψdir(f)(m), showing the multiplicativity of Ψdir(f). The proof in the case of the other functions is similar. For the function Ψgcd(f),
Ψgcd(f)(nm)= f(a1b1,...,arbr), a1···ar=n b1···Xbr =m gcd(a1b1,...,arbr )=1 where 1 = gcd(a1b1,...,arbr) = gcd(a1,...,ar) gcd(b1,...,br), since the gcd function in r variables is multiplicative. Hence
Ψgcd(f)(nm)= f(a1,...,ar) f(b1,...,br)
a1···ar =n b1···br =m 1 gcd(a X,...,ar)=1 gcd(b1X,...,br)=1
=Ψgcd(f)(n)Ψgcd(f)(m).
In the case of the function Ψlcm(f) use that the lcm function in r variables is multiplica- tive, whence nm = lcm(a1b1,...,arbr) = lcm(a1,...,ar) lcm(b1,...,br) and it follows that lcm(a1,...,ar)= n, lcm(b1,...,br)= m.
Remark 1. Alternative proofs for the multiplicativity of Ψunit(f), Ψgcd(f) and Ψlcm(f) can be given as follows. In the first two cases the property can be reduced to that of Ψdir(f). Let ♭ f(n1,...,nr), if gcd(n1,...,nr)=1, f (n1,...,nr)= (0, otherwise. ♭ ♭ Then Ψgcd(f)=Ψdir(f ). If f is multiplicative, then f is also multiplicative and the multiplicativity of Ψgcd(f) follows by the same property of the Dirichlet convolute. Similar for Ψunit(f). Furthermore, Ψ (f) = (f 1 ) µ. (35) lcm ∗ r ∗ Indeed, we have similar to the proof of (13) given above,
Ψlcm(f)(d)= f(d1,...,dr)= f(d1,...,dr) Xd|n lcm(dX1,...,dr)|d d1|n,...,dX r|n
15 = (f 1 )(n,...,n) = (f 1 )(n). ∗ r ∗ r If f is multiplicative, so is f 1 (as a function of r variables). Therefore, f 1 is multi- ∗ r ∗ r plicative (as a function of a single variable) and deduce by (35) that Ψlcm(f) is multiplicative.
Remark 2. Note that if f is completely multiplicative, then Ψbinom(f) is also completely multiplicative.
Corollary 1. The functions gr,ℓr,N,c,a,b,h defined by (28)-(34) are multiplicative.
Proposition 13. The convolutes of the function f = 1r, that is the number of terms of the sums defining the convolutes are the following multiplicative functions: νp(n)+r−1 i) Ψdir(1r)= τr, the Piltz divisor function of order r given by τr(n)= p r−1 , ii) Ψ (1 )= H given by H (n)= rω(n), unit r r r Q νp(n)+r−1 νp(n)−1 iii) Ψ (1 ) = N , where N (n) = r µ(a)τ (b) = gcd r r r a b=n r p r−1 − r−1 ν−1 with r−1 =0 for ν < r, P Q iv) Ψ (1 )= M , where M (n)= µ(a)τ(b)r = ((ν (n)+1)r ν (n)r), lcm r r r ab=n p p − p v) Ψ (1 )= Q , where Q (n)= rΩ(n). binom r r r P Q Proof. i) and ii) are immediate from the definitions. iii) By the property of the M¨obius function,
Ψgcd(1r)(n)= µ(a)= µ(a)= µ(a)τr(b). k r d1···Xdr=n a|gcd(Xd1,...,dr) a b1X···br=n aXb=n
iv) Follows from (35) in the case f = 1r. v) By Remark 2. See also L. Toth,´ P. Haukkanen [72, Cor. 3.2].
Here i) and iv) of Proposition 13 can be generalized as follows.
Proposition 14. Assume that there is a function g 1 (of a single variable) such that f(n ,...,n )= g(gcd(n ,...,n )) for every n ,...,n ∈ AN. Then for every n N, 1 r 1 r 1 r ∈ ∈
Ψdir(f)(n)= g(gcd(d1,...,dr)) = (µ g)(a)τr(b), (36) r ∗ d1···Xdr=n aXb=n Ψ (f)(n)= g(gcd(d ,...,d )) = (g µ µ τ r)(n). (37) lcm 1 r ∗ ∗ ∗ lcm(d1X,...,dr)=n Proof. The identity (36) is given by E. Kratzel,¨ W. G. Nowak, L. Toth´ [35, Prop. 5.1]. We recall its proof, which is simple and similar to that of iii) of Proposition 13: for f(n1,...,nr)= g(gcd(n1,...,nr)),
Ψ (f)(n)= (µ g)(a) dir ∗ d1···Xdr=n a|gcd(Xd1,...,dr)
= (µ g)(a)= (µ g)(a)τr(b). r ∗ r ∗ a b1X···br=n aXb=n Now for (37),
Ψ (f)(n)= (µ g)(a) lcm ∗ lcm(d1X,...,dr)=n a|gcd(Xd1,...,dr) = (µ g)(a) 1= (µ g)(a)(µ τ r)(n/a) ∗ ∗ ∗ Xa|n lcm(b1,...,bXr)=n/a Xa|n = (g µ µ τ r)(n), ∗ ∗ ∗ using iv) of Proposition 13.
16 Proposition 15. For every f,g , ∈ Ar Ψ (f g)=Ψ (f) Ψ (g), dir ∗ dir ∗ dir Ψ (f g)=Ψ (f) Ψ (g), unit × unit × unit Ψ (f g)=Ψ (f) Ψ (g), gcd ⊙ gcd × gcd Ψ (f g)=Ψ (f) Ψ (g), lcm ⊕ lcm ⊕ lcm Ψ (f g)=Ψ (f) Ψ (g). binom ◦ binom ◦ binom
Proof. For the Dirichlet convolute Ψdir,
Ψ (f g)(n)= (f g)(d ,...,d ) dir ∗ ∗ 1 r d1···Xdr=n
= f(a1,...,ar)g(b1,...,br) d1···Xdr=n a1b1=d1X,...,arbr =dr
= f(a1,...,ar)g(b1,...,br) a1b1···Xarbr =n
= f(a1,...,ar) g(b1,...,br) xy=n a1···a =x 1 X Xr b ···Xbr=y = Ψ (f)(x)Ψ (g)(y)=(Ψ (f) Ψ (g))(n), dir dir dir ∗ dir xy=n X and similar for the other ones.
Proposition 16. Let r 2. The following maps are surjective algebra homomorphisms: Ψ : ( , +, , ) ( , ≥+, , ), Ψ : ( , +, , ) ( , +, , ), dir Ar · ∗ → A1 · ∗ unit Ar · × → A1 · × Ψgcd : ( r, +, , ) ( 1, +, , ), Ψlcm : ( r, +, , ) ( 1, +, , ), Ψ :A ( , +· ,⊙, →) A( , +·, ×, ). A · ⊕ → A · ⊕ binom Ar · ◦ → A1 · ◦ Proof. Use Proposition 15. For the surjectivity: for a given f consider F defined ∈ A1 ∈ Ar by F (n, 1,..., 1) = f(n) for every n N and F (n1,...,nr) = 0 otherwise, i.e., for every n ,...,n N with n n > 1. Then∈ Ψ (F )=Ψ (F )=Ψ (F )=Ψ (F ) = 1 r ∈ 2 ··· r dir unit gcd lcm Ψbinom(F )= f.
Corollary 2. Let r 2. i) The maps Ψ ≥: ( , ) ( , ), Ψ : ( , ) ( , ), dir Mr ∗ → M1 ∗ unit Mr × → M1 × Ψgcd : ( r, ) ( 1, ) and Ψbinom : ( r, ) ( 1, ) are surjective group homomor- phisms.M ⊙ → M × M ◦ → M ◦ ii) The maps Ψ : ( , ) ( , ), Ψ : ( , ) ( , ) and dir Fr ∗ → M1 ∗ unit Fr × → M1 × Ψbinom : ( r, ) ( 1, ) are surjective group homomorphisms. iii) TheF map◦ →Ψ M :◦ ( , ) ( , ) is a surjective group homomorphism. binom Cr ◦ → C1 ◦ Proof. Follows from Propositions 12, 15 and from the fact that for every (completely) mul- tiplicative f the function F constructed in the proof of Proposition 16 is (completely) mul- tiplicative. For iii) use also Remark 2.
17 6.2 Special cases We present identities for the convolutes of some special functions. For Dirichlet convolutes we have the next result. Corollary 3. For every k C, ∈ k (gcd(d1,...,dr)) = φk(a)τr(b). r d1···Xdr=n aXb=n k σk(gcd(d1,...,dr)) = a τr(b). (38) r d1···Xdr=n aXb=n Proof. Follows from the first identity of Proposition 14.
The function (38) is for r = 2 and k = 0 the sequence A124315 in [56], and for r = 2, k = 1 it is sequence A124316 in [56]. See also [35, Sect. 5]. Special cases of lcm convolutes which do not seem to be known are the following. Let β = id λ be the alternating sum-of-divisors function, see L. Toth´ [70]. ∗ Corollary 4. For every n N, ∈ gcd(d ,...,d ) = (φ M )(n), 1 r ∗ r lcm(d1X,...,dr)=n where the function Mr is given in Proposition 13,
r τ(gcd(d1,...,dr)) = τ(n) , lcm(d1X,...,dr)=n φ(gcd(d, e)) = ψ(n), lcm(Xd,e)=n σ(gcd(d, e)) = (τ ψ)(n) = (φ τ 2)(n), (39) ∗ ∗ lcm(Xd,e)=n β(gcd(d, e)) = σ(n), lcm(Xd,e)=n µ(gcd(d, e)) = µ2(n), lcm(Xd,e)=n λ(gcd(d, e))=1. lcm(Xd,e)=n Proof. Follow from the second identity of Proposition 14.
Here (39) is s(n,n), representing the number of all subgroups of the group Zn Zn (sequence A060724 in [56]). × For the convolutes of the Ramanujan sum we have Proposition 17. √n, n perfect square, cd(e)= (40) (0, otherwise, deX=n 1, n squarefull, cd(e)= (41) 0, otherwise, de=n ( gcd(Xd,e)=1 c (e)= φ(n) (n N). (42) d ∈ lcm(Xd,e)=n
18 Proof. According to Corollary 1 all these functions are multiplicative and it is enough to compute their values for prime powers. Alternatively, the formula (8) can be used.
Formulas (40) and (41) are well known, see, e.g., P. J. McCarthy [42, p. 191–192], while (42) seems to be new.
7 Asymptotic properties
We discuss certain results concerning the mean values of arithmetic functions of r vari- ables and the asymptotic densities of some sets in Nr. We also present asymptotic formulas for special multiplicative functions given in the previous sections.
7.1 Mean values Let f . The mean value of f is ∈ Ar 1 M(f) = lim f(n1,...,nr), x1,...,xr→∞ x1 xr ··· n1≤x1X,...,nr≤xr where x1,...,xr tend to infinity independently, provided that this limit exists. As a gener- alization of Wintner’s theorem (valid for the case r = 1), N. Ushiroya [73, Th. 1] proved the next result. Proposition 18. If f (r 1) and ∈ Ar ≥ ∞ (µ f)(n ,...,n ) | r ∗ 1 r | < , n1 nr ∞ n1,...,n =1 Xr ··· then the mean value M(f) exists and
∞ (µ f)(n ,...,n ) M(f)= r ∗ 1 r . n1 nr n1,...,n =1 Xr ··· For multiplicative functions we have the following result due to N. Ushiroya [73, Th. 4] in a slightly different form. Proposition 19. Let f (r 1). Assume that ∈Mr ≥ ∞ (µ f)(pν1 ,...,pνr ) | r ∗ | < . pν1+...+νr ∞ p ν1,...,νr =0 X ν1+...X+νr ≥1
Then the mean value M(f) exists and
r ∞ 1 f(pν1 ,...,pνr ) M(f)= 1 . − p pν1+...+νr p ν1,...,ν =0 Y Xr Corollary 5. (N. Ushiroya [73, Th. 7]) Let g be a multiplicative function and denote ∈M1 by ag the absolute convergence abscissa of the Dirichlet series D(g; z). Then for every r> 1, r>a the main value of the function (n ,...,n ) g(gcd(n ,...,n )) exists and g 1 r 7→ 1 r 1 ∞ g(n) M(f)= . ζ(r) nr n=1 X
19 Proof. Follows from Proposition 18 and the identity (16).
For example, the mean value of the function (n1,...,nr) gcd(n1,...,nr) is ζ(r 1)/ζ(r) 7→ − 2 (r 3), the mean value of the function (n1,...,nr) φ(gcd(n1,...,nr)) is ζ(r 1)/ζ (r) (r ≥ 3). 7→ − ≥The analog of Proposition 18 for the unitary convolution is the next result (see W. Narkiewicz [43] in the case r = 1). Proposition 20. If f (r 1) and ∈ Ar ≥ ∞ (µ× f)(n ,...,n ) | r × 1 r | < , n1 nr ∞ n1,...,n =1 Xr ··· then the mean value M(f) exists and ∞ (µ× f)(n ,...,n )φ(n ) φ(n ) M(f)= r × 1 r 1 ··· r . n2 n2 n1,...,n =1 1 r Xr ··· For further results on the mean values of multiplicative arithmetic functions of several variables and generalizations to the several variables case of results of G. Halasz´ [24] we refer to the papers of O. Casas [8], H. Delange [15, 16], E. Heppner [29, 30] and K- H. Indlekofer [33]. See also E. Alkan, A. Zaharescu, M. Zaki [3].
7.2 Asymptotic densities Let S Nr. The set S possesses the asymptotic density d if the characteristic function ⊂ S χS of S has the mean value M(χS)= dS . In what follows we consider the densities of certain special sets. Let g 1 be a multiplicative function such that g(n) 0, 1 for every n N. Let S = (n ∈,...,n M ) Nr : g(gcd(n ,...,n ))=1 . It follows from∈ { Corollary} 5 that∈ for r 2 g { 1 r ∈ 1 r } ≥ the set Sg has the asymptotic density given by ∞ 1 g(n) d = . Sg ζ(r) nr n=1 X In particular, for r 2 the set of points (n ,...,n ) Nr which are visible from the ≥ 1 r ∈ origin, i.e., such that gcd(n1,...,nr) = 1 holds has the density 1/ζ(r) (the case g = δ). r Another example: the set of points (n1,...,nr) N such that gcd(n1,...,nr) is squarefree has the density 1/ζ(2r) (the case g = µ2). These∈ results are well known. r Now consider the set of points (n1,...,nr) N such that n1,...,nr are pairwise rela- tively prime. The next results was first proved∈ by L. Toth´ [63], giving also an asymptotic formula for ̺(n ,...,n ) and by J.-Y. Cai, E. Bach [6, Th. 3.3]. Here we give n1,...,nr≤x 1 r a simple different proof. P Proposition 21. Let r 2. The asymptotic density of the set of points in Nr with pairwise relatively prime coordinates≥ is 1 r−1 r 1 A = 1 1+ − . (43) r − p p p Y Proof. Apply Proposition 18 for the function f = ̺ defined by (1). Then according to the Dirichlet series representation (18), the density is
∞ (µ ̺)(n ,...,n ) 1 r r 1 r−1 r ∗ 1 r = 1 + 1 , n1 nr − p p − p n1,...,n =1 p ! Xr ··· Y which equals Ar, given by (43).
20 See the quite recent paper by J. Hu [31] for a generalization of Proposition 21. See J. L. Fernandez,´ P. Fernandez´ [19, 20, 21, 22] for various statistical regularity properties concerning mutually relatively prime and pairwise relatively prime integers. Unitary analogs of the problems of above are the following. Proposition 22. Let r 2. The asymptotic density of the set of points (n ,...,n ) Nr ≥ 1 r ∈ such that gcud(n1,...,nr)=1 is (p 1)r 1 − . − pr(pr 1) p Y − Proof. The characteristic function of this set is given by
× δ(gcud(n1,...,nr)) = µ (d)= G(d1,...,dr), d||gcud(Xn1,...,nr) d1||nrX,...,dr||nr where × µ (n), if n1 = ... = nr = n, G(n1,...,nr)= (0, otherwise. We deduce from Proposition 20 that the density in question is
∞ ∞ G(n ,...,n )φ(n ) φ(n ) µ×(n)φ(n)r 1 r 1 ··· r = n2 n2 n2r n1,...,n =1 1 r n=1 Xr ··· X (p 1)r = 1 − . − pr(pr 1) p Y −
Proposition 22 was proved by L. Toth´ [62] using different arguments. See [62] for other related densities and asymptotic formulas. Corollary 6. (r=2) The set of points (m,n) N2 such that gcud(m,n)=1 has the density ∈ p 1 1 − . − p2(p + 1) p Y Proposition 23. Let r 2. The asymptotic density of the set of points in Nr with pairwise unitary relatively prime≥ coordinates is
∞ ν1 νr ν1 νr × Q(p ,...,p )φ(p ) φ(p ) Ar = ··· , p2ν1+...+2νr p ν1,...,ν =0 Y Xr where Q is the multiplicative function of r variables given as follows: Let pν1 ,...,pνr be arbitrary powers of the prime p with ν ,...,ν N , ν + ... + ν 1. Assume that 1 r ∈ 0 1 r ≥ the exponents ν1,...,νr have q (1 q r) distinct positive values, taken t1,...,tq times (1 t + ... + t r). Then ≤ ≤ ≤ 1 q ≤ Q(pν1 ,...,pνr ) = ( 1)r(1 t ) (1 t ). − − 1 ··· − q Proof. We use Proposition 20 for the function ̺× defined by (3). The density is
∞ Q(n ,...,n )φ(n ) φ(n ) A× = 1 r 1 ··· r , r n2 n2 n1,...,n =1 1 r Xr ··· × × where Qr = µr ̺ and the Euler product formula can be used by the multiplicativity of the involved functions.×
21 Note that for r = 2, 1, ν1 = ν2 =0, Q(pν1 ,pν2 )= 1, ν = ν 1, − 1 2 ≥ 0, otherwise and reobtain Corollary 6. Corollary 7. (r =3, 4)
4 7 9 8 2 3 2 A× = ζ(2)ζ(3) 1 + + + . 3 − p2 p3 − p4 p5 − p6 − p7 p8 p Y 8 3 27 24 14 3 A× = ζ2(2)ζ(3)ζ(4) 1 + + 4 − p2 p3 p4 − p5 − p6 − p7 p Y 37 30 42 33 41 78 44 9 + + + + . p8 − p9 p10 − p11 − p12 p13 − p14 p15 Proof. According to Proposition 23,
1, ν1 = ν2 = ν3 =0, 1, ν = ν 1,ν = 0 and symmetric cases, Q(pν1 ,pν2 ,pν3 )= − 1 2 ≥ 3 2, ν1 = ν2 = ν3 1, ≥ 0, otherwise and
1, ν1 = ν2 = ν3 =0, 1, ν = ν 1,ν = ν = 0 and symmetric cases, − 1 2 ≥ 3 4 2, ν = ν = ν 1,ν = 0 and symmetric cases, ν1 ν2 ν3 ν4 1 2 3 4 Q(p ,p ,p ,p )= − ≥ 3, ν1 = ν2 = ν3 = ν4 1, − ≥ 1, ν1 = ν2 >ν3 = ν4 1 and symmetric cases, ≥ 0, otherwise, and direct computations lead to the given infinite products.
We refer to the papers by J. Christopher [12], H. Delange [15] and N. Ushiroya [73] for related density results.
7.3 Asymptotic formulas One has x2 1 ζ(2) ζ′(2) gcd(m,n)= log x +2γ + O(x1+θ+ε), (44) ζ(2) − 2 − 2 − ζ(2) m,nX≤x for every ε> 0, where θ is the exponent appearing in Dirichlet’s divisor problem, that is
τ(n)= x log x + (2γ 1)x + O(xθ+ε). (45) − nX≤x It is known that 1/4 θ 131/416 0.3149, where the upper bound, the best up to date, is the result of M. N.≤ Huxley≤ [32].≈
22 If r 3, then ≥ ζ(r 1) gcd(n ,...,n )= − xr + O(R (x)), (46) 1 r ζ(r) r n1,...,nXr≤x where R (x)= x2 log x and R (x)= xr−1 for r 4, which follows from the representation 3 r ≥ r gcd(n1,...,nr)= φ(d)[x/d] . n1,...,nXr≤x dX≤x Furthermore, ζ(3) lcm(m,n)= x4 + O(x3 log x). (47) 4ζ(2) m,nX≤x The formulas (44), (46), (47) can be deduced by elementary arguments and go back to the work of E. Cesaro` [10], E. Cohen [13] and P. Diaconis, P. Erdos˝ [18]. See also L. Toth´ [64, Eq. (25)]. A formula similar to (44), with the same error term holds for g(gcd(m,n)), where g = h id, h is bounded, including the cases g = φ, σ, ψ. m,n≤x ∗ ∈ A1 See L. Toth´ [64, p. 7]. See J. L. Fernandez,´ P. Fernandez´ [19, 20, 21] for statistical P regularity properties of the gcd’s and lcm’s of positive integers.
Consider next the function g2(n)= d|n gcd(d, n/d), which is the Dirichlet convolute of the gcd function for r = 2. P Proposition 24. 3 g (n)= x(log2 x + c log x + c )+ R(x), 2 2π2 1 2 nX≤x ′ θ θ 547 ′ 26947 where c1,c2 are constants and R(x)= O(x (log x) ) with θ = 832 =0.65745 ... , θ = 8320 . This was proved using analytic tools (Huxley’s method) by E. Kratzel,¨ W. G. Nowak, L. Toth´ [35, Th. 3.5]. See M. Kuhleitner,¨ W. G. Nowak [36] for omega estimates on the function g2(n). The papers [35] and [36] contain also results for the function gr(n) (r 3), defined by (28) and related functions. ≥ For the function ℓ2(n) = d|n lcm(d, n/d), representing the Dirichlet convolute of the lcm function for r = 2 one can deduce the next asymptotics. P Proposition 25. ζ(3) 2ζ′(2) 2ζ′(3) ℓ (n)/n = x log x +2γ 1 + + O(xθ+ε), 2 ζ(2) − − ζ(2) ζ(3) nX≤x where θ is given by (45).
A similar formula can be given for the function ℓr(n) (r 3) defined by (29). For the functions s(m,n) and c(m,n) defined in Section 3.4≥ , Example 4, W. G. Nowak, L. Toth´ [44] proved the following asymptotic formulas. Proposition 26. For every fixed ε> 0,
2 1117 s(m,n)= x2(log3 x + a log2 x + a log x + a )+ O x 701 +ε , π2 1 2 3 m,nX≤x 12 1117 c(m,n)= x2(log3 x + b log2 x + b log x + b )+ O x 701 +ε , π4 1 2 3 m,nX≤x where 1117/701 1.5934 and a ,a ,a ,b ,b ,b are explicit constants. ≈ 1 2 3 1 2 3 See the recent paper by T. H. Chan, A. V. Kumchev [11] concerning asymptotic formulas for n≤x,k≤y cn(k). P 23 8 Acknowledgement
The author gratefully acknowledges support from the Austrian Science Fund (FWF) under the project Nr. M1376-N18.
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L. T´oth Institute of Mathematics, Universit¨at f¨ur Bodenkultur Gregor Mendel-Straße 33, A-1180 Vienna, Austria and Department of Mathematics, University of P´ecs Ifj´us´ag u. 6, H-7624 P´ecs, Hungary E-mail: [email protected]
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