ScienceAsia 27 (2001) : 199-202

Arithmetic Functions and Operators

Vichian Laohakosola, Nittiya Pabhapoteb,* and Nalinee Wechwiriyakulb a Department of Mathematics, Kasetsart University, Bangkok 10900, Thailand. b Department of Mathematics, The University of the Thai Chamber of Commerce, Bangkok 10400, Thailand * Corresponding author, E-mail: [email protected]

Received 22 Dec 1999 Accepted 1 May 2001

ABSTRACT Basic results about arithmetic functions and their major operations, namely, valuation, derivation and operators, are collected. The logarithmic and related operators, introduced and applied in 1968 by D Rearick to establish isomorphisms among various groups of real-valued arithmetic functions are extended to complex-valued arithmetic functions along the original lines suggested by him.

KEYWORDS: arithmetic functions, operators.

INTRODUCTION A derivation over A7 is a function D : A → A such that for all f, g ∈ A, and for all a, b ∈ C , we An arithmetic function is a function whose have D(f * g) = Df * g + f * Dg, D(af + bg) = aDf + domain is N and range is a subset of C. bDg. Three typical examples of derivation, which Let f and g be arithmetic functions. The sum (or are often used, are addition) of f and g is an arithmetic function f + g (i) log-derivation : Dl f (n) := f (n) log n , defined by (f + g)(n) = f(n) + g(n). The Dirichlet (ii) p-basic derivation (p prime) : Dp f (n) := product ( or convolution or Dirichlet multiplication) f (np) vp(np), where vp(m) denotes the of f and g is an arithmetic function f * g defined by highest power of p dividing m,

(iii) Dh f (n) := f(n) h(n), where h is a completely additive arithmetic function, ie, n ()()f ∗=gn∑ fdg()( ). h(mn) = h(m) + h(n). dn d In 1968, D. Rearick5, 6 constructed a number of (A, +, ) The set * of all arithmetic functions operators over A analogous to the classical logarithmic, together with addition and convolution is a unique exponential and trigonometric operators. He sub- factorization domain but not a principal ideal sequently used them to show that various groups of 1-3, 7-9 Z(n) = 0 (∀ n ∈ N) domain. The function is an real - valued arithmetic functions are isomorphic. additive identity, while the function That is, (AR, +), (PR, *), (MR, *), (PR, x) and (MR, x)  = are isomorphic, where =  11, n I(n )  is a (Dirichlet) multiplicative →  0, otherwise AR = { f: N R } = set of real-valued arithmetic functions, identity. Indeed, the so-called Möbius inversion ∈ PR = { f AR ; f (1) > 0 }, formula1-3, 7-9 states that for f, g ∈ A, we have ∈ MR = { f PR ; f is multiplicative }, n fn() =⇔=∑∑ gd() gn () fd ()()µ , which dn dn d and x is the unitary product defined by (f x g)(n) = Σ' f (d) g(n/d), with Σ' indicating that the sum is taken f g is equivalent to stating that = * u if and only if over the divisors d such that (d, n/d ) = 1. g = f * µ, where u is the unit function defined for all n ∈ N n µ by u( ) = 1 and the Möbius function. The objectives of this work are : f An arithmetic function is said to be multipli- (i) to extend the definition of Rearick in order to f f (mn) f (m) f (n) cative if (1) = 1 and = for all embrace those arithmetic functions which m, n f ∈ A relatively prime integers . Let . The assume complex values of all but one point, f f -1 (Dirichlet) inverse of is an arithmetic function namely, at n = 1; this is indeed suggested at the f f -1 1-3, 7-9 f -1 for which I = * . It is known that exists if end of Rearick paper,5 and and only if f (1) ≠ 0. 200 ScienceAsia 27 (2001)

(ii)to establish relevant isomorphisms among certain Remark. Proposition 1 and 2 show that the map groups of arithmetic functions considered in (i) f a LCf is an isomorphism of the groups (PC, *) and which encompass those in Rearick.5, 6 (A'C, +).

LOGARITHMIC OPERATORS ∈ Proposition 3. Let f PC. Then f is multiplicative if and only if LCf (n) = 0 whenever n is not a prime The following notation will be standard power. throughout the whole paper: → AC : = { f ; f : N C} = set of all complex-valued Proof. Assume f is multiplicative. Then f (1) = 1 arithmetic functions , and so L f (1) = 0. ∈ ∈ C A'C : = { f AC ; f (1) R } , Let N be a positive integer which is not a prime ∈ PC : = { f A'C ; f (1) > 0 } , power. Then there are positive integers ∈ MC : = { f AC ; f is multiplicative }. m, n both > 1, (m, n) = 1 such that N = mn. Thus

− mn We define the (complex) logarithmic operator L f (N) = ∑ f ()df 1 ( ) log d → C d LC : PC A'C by dmn

L f (1) = log f (1) and C ∑ ∑ -1 m -1 n +   = fd()()f df ( )f ( )(logdd log ) −1 n -1 1 2 d d 1 2 L fn( )= ∑ fd ( ) f   log d = D ff ∗ ( nn ) ( > 1 ). dmdn 12 C  d  1 1 2 dn = LCf (m) I(n) + LCf (n) I(m) = 0. ∈ Proposition 1. For all f, g PC, we have LC (f * g) = Next assume that LCf (n) = 0 whenever n is not a LCf + LCg. prime power. Since LCf (1) = 0, then f (1) = 1. For n > 1, define g ∈ P by g(1) = 1 and g(n) = ∏ fp( v ) L (f g)(1) = log (f g)(1) = log (f (1)g (1)) C Proof. C * * pn| = log f (1) + log g (1) where pv||n. Clearly, g is multiplicative. We now show = LCf (1) + LCg (1). that f = g. Observe that f (n) = g(n) and f -1(n) = g-1(n) -1 Since LCf = f * D1f, evaluating at n > 1, we see that whenever n is a prime power. From the definition -1 LC (f * g)(n) = ((f * g) * Dl (f * g))(n) of L , we thus get L f (n) = L g (n) whenever n is a -1 -1 C C C = (f * g * (f * Dlg + g * Dlf )) (n) ∈ prime power. Since g PC, then the first half of the = LCg (n) + LCf (n). proof yields that LCg (m) = 0 whenever m is not a prime power. Hence, L f (n) = L g (n) for all n ∈ N ∈ C C Proposition 2. For each h A'C, there is a unique and so f = g by the isomorphism L . ∈ C f PC such that h = LCf. Remarks. Proposition 3 implies that the groups Proof. Define f(1) = exp h(1). Let n > 1 and assume (MC, *) and (A"C, +) are isomorphic, where f (k) has been defined for all k < n. The value ∈ A"C : = { h A' C: h(n) = 0 whenever n is not a prime f -1(k) are recursively determined by the relation power }. The group (A"C, +) is also isomorphic to − the group (A , +)via the map h ↔ H where H(n) = ∑ 1 k = C f (df ) ( ) I (k ). This gives us a triangular h(k ) with {k } being the sequence of prime powers dk d n n arranged in ascending order. system which can be solved for the unknowns Consequently, the groups (M , ) and (A , +) -1 -1 C * C f (k). Now given h(n), f(k) and f (k) for all k < n, we are isomorphic. define f(n) by solving for the term corresponding to

− n OTHER OPERATORS d = n in the equation h(n) = ∑ f ()df1 ( ) log d , d dn ∈ Let h A'C. Denote by ECh, call the (complex) ∈ noting that the term containing f -1(n) disappears exponential of h, the unique element f PC , justified because log 1 = 0 and all other terms are known. by Proposition 2, such that h = LCf. It follows easily from the definition and the properties of logarithmic operators that ScienceAsia 27 (2001) 201

∀ ∈ 2 1/2 (i) EC (h1+h2)= EC h1 * EC h2 ( h1, h2 A'C) (i) SC f + ((SC f ) + I) = EC f. ∀ ∈ (ii) LC (ECh)= h ( h A'C) (ii) If SC f = SC g, then EC f = EC g and f = g, ie SC ∀ ∈ (iii)EC (LCf)= f ( f PC) is injective. ∈ ∈ (iv) EC (Z) = I. (iii)For each h A'C, there exists an f A'C such 2 1/2 that SC f = h, viz, f = LC (h + (h + I) ). This ∈ ∈ th For f PC, and r R define the r power shows that SC is surjective. r 2 1/2 arithmetic function by f : = EC (r LC f). (iv) SC (f + g) = SC f * ( (SC g) + I ) + 2 1/2 It is easily checked that SC g * ( (SC f) + I ) .

(i) (f r)s = f rs. ′ Proposition 5. The system (,)A C forms a group (ii) f r+s = f r * f s. which is isomorphic to (,)A′ + , where (iii)(f * g)r = f r * gr. C (iv) If r is a positive interger, then r f g= f∗∗ g2 + 1/2 g f 2 1/2 f = EC (LC f + ... + LC f) = f * ... * f (r factors), ( I) + ( + I) . agreeing with our previous definition of positive integral power function. Proof. That (,)A′ is a group can be directly checked -1 C (v) If r = -1, then f = EC (-LC f), and so -1 using the identities mentioned above. The map f * f = EC (LC f) * EC (-LC f) = EC (LC f - LC f) = I (,)A′ + → (,)A′ f a S f agreeing with the usual meaning of inverse. C C defined via C provides us ∈ ∈ with a desired isomorphism. (vi) If r R - {0} and f PC, it follows that the r ∈ equation g = f is uniquely solvable for g PC; V := { f ∈ A' : -1 < f (1) < 1} ∆ 1 Let C C and let be a V f ∆ g (f + g) indeed, the solution is g = f r , which amounts binary operation defined over C via := * -1 (I + f * g) . It is easily checked that (V , ∆) forms a to saying that every f ∈ P has a unique rth root C C group with the zero function Z acting as the group in P . C identity. Proposition 4. Let r ∈ R. If f ∈ M , then f r ∈ M . C C ∆ ′ + Proposition 6. The groups (VC, ) and (,)A C are ∈ isomorphic. Proof. If f PC, then by Proposition 3 , LCf (n) = 0 whenever n is not a prime power and so is rL f (n). C → Therefore, Proposition 3 again yields that Proof. The hyperbolic tanh map TC : f TC (f) = r S f * (C f)-1 gives a desired isomorphism from A' f = EC (rLC f) is multiplicative. C C C onto VC. Remark. It follows from the last proposition that for r nonzero real r, the map f → f is an automorphism Proposition 7. The groups (AC, +), (AR, +) and (A' , +) are all isomorphic. of the group (PC, *) which sends multiplicative C elements onto themselves. α → Proof. The map : (AR, +) (AC, +) defined for ∈ each positive integer n by Let f A' C. Define the hyperbolic sinh, cosh and tanh as follows: α (f) (n) := f (2n-1) + i f (2n) 1 SEEfff=−−(()), ccc2 yields a desired isomorphism, while the map β : (A , +) → (A' , +) defined by 1 R C CEEfff=+−(()), ccc 2 β(f) (1) = f (1), β(f) (n) = f (2n-2) + i f (2n-1) (n > 1), TSCfff=∗()−1 . ccc yields the other desired isomorphism.

Since this definition mimics the classical one, it is clear that most elementary identities involving hyperbolic and/or trigonometric functions hold. We list some examples here. 202 ScienceAsia 27 (2001)

To sum up, we have the following isomorphisms ACKNOWLEDGEMENTS diagram. This research is partially supported by a 1998-

(PC , * ) (A'c , ) 1999 research grant from the University of the Thai

SC Chamber of Commerce.

LC REFERENCES

(PR , * ) (A'c , +) (VC , ∆) 1. Apostol TM (1976) “Introduction to Analytic Number TC LR Theory,” Springer-Verlag, New York. β (AR , ) 2. Berberian SK (1962) Number-theoretic functions via con- SR volution rings. Math Magazine 65, 75-92. 3. Carlitz L and Subbarao MV (1973) Transformation of α arithmetic functions. Duke Math J 40, 949-58. (V , ∆) (A , +) (A , +) C R C 4. Cashwell ED and Everett CJ (1959) The ring of number- TR theoretic functions. Pacific J Math 9, 975-85. prime power prime power 5. McCarthy PJ (1986) “Introduction to Arithmetical Functions”, Springer-Verlag, New York. A" A" 6. Rearick D (1968) Operators on algebras of arithmetic (MR ,* ) ( R , +) ( c , +) functions. Duke Math J 35, 761-6. LR 7. Rearick D (1968) The trigonometry of numbers. Duke Math J

LC 35, 767-76. 8. Shapiro HN (1983) “Introduction to the Theory of Numbers,” (MC , * ) John Wiley & Sons, New York. 9. Sivaramakrishnan R (1989) “Classical Theory of Arithmetic Functions,” Marcel Dekker, New York.