PROPERTIES OF RATIONAL ARITHMETIC FUNCTIONS

s ,w heregi, h j are completely multiplicative functions and ∗ denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type identities are derived; and finally, properties of the Kesava Menon norm of such functions are proved.


Introduction
By an arithmetic function we mean a complex-valued function whose domain is the set of positive integers N. We define the addition and the Dirichlet convolution of two arithmetic functions f and g, respectively, by It is well known (see, e.g., [1,5,13,19,21]) that the set (Ꮽ,+, * ) of all arithmetic functions is a unique factorization domain with the arithmetic function being its convolution identity.
A nonzero arithmetic function For nonnegative integers r, s by an (r,s)-rational arithmetic function f , denoted by f ∈ Ꮿ(r,s), we mean an arithmetic function which can be written as where each g i ,h j ∈ Ꮿ.Such functions were first studied by Vaidyanathaswamy [23] in 1931, and later by several authors; see, for example, [4,6,7,9,10,13,16,18,20].Two important classes of rational functions are Ꮿ(1,1) whose elements are known as totients, and Ꮿ(2,0) whose elements are the so-called specially multiplicative functions.Characterizations of these two classes can be found in [7,10], respectively.The present work deals with four aspects of rational arithmetic functions.In the next section, some characterizations of these functions are derived and are then used in the next sections to investigate whether two types of identities, the Busche-Ramanujan identity and the binomial identity, which are known to hold for totients and/or specially multiplicative functions, continue to hold for general rational arithmetic functions.In the last section, the Kesava Menon norm of such functions is studied.
We will find it helpful to make use of two important concepts which we now recall.For f ∈ Ꮽ, f (1) ∈ R + , the Rearick logarithm of f (see [11,14,15]), denoted by Log f ∈ Ꮽ, is defined via where df (n) = f (n)logn denotes the log derivation of f .The Hsu's generalized Möbius function (see [2]) µ r , r ∈ R, is defined as where ν p (n) is the highest power of the prime p dividing n.It is known (see [8,12]) that for f ∈ ᏹ, and the converse holds under additional hypotheses.

Characterizations
In this section, r and s will generally denote nonnegative integers.Should either of them be zero, the sum and/or any other expressions connected with them are taken to be zero.
Corollary 2.2.Let f ∈ ᏹ, with f (p) = 0 for each prime p. Then f ∈ Ꮿ(1,1) ⇐⇒ for each prime p and each α ∈ N, there is a complex number b(p) such that Theorem 2.3.Let r, s be nonnegative integers and f ∈ ᏹ.Then f ∈ Ꮿ(r,s) ⇔ for each prime p and each α ∈ N, there exist complex numbers a 1 (p),...,a r (p), b 1 (p),...,b s (p) such that for all α ≥ s, where (2.6) Proof. (2.7) The result now follows by grouping terms on the right-hand side and using A few known characterizations of two particular classes of functions, namely, those in Ꮿ(1,1), that is, totients (see [7]), and those in Ꮿ(2,0), that is, specially multiplicative functions (see [13,Theorem 1.12]), are immediate consequences of Theorem 2.3, which we record in the following corollary together with a characterizing property of Ꮿ(1,s) to be used later.
(i) f ∈ Ꮿ(1,1) ⇔ for each prime p and each α ∈ N, there exists a complex number a(p) such that (2.9) where Simplified characterizations for rational arithmetic functions belonging to the classes where r is 0 can similarly be obtained as in the next corollaries.
Corollary 2.5.Let s be a nonnegative integer and f ∈ ᏹ.Then f ∈ Ꮿ(0,s) ⇔ for each prime p, f (p α ) = 0 for all α > s. Proof. 12) The result now follows by noting that for h ∈ Ꮿ, we have h −1 (p) = −h(p), h −1 (p i ) = 0 for i ≥ 2, and that the s complex numbers h 1 (p),...,h s (p) are uniquely determined by the s values f (p),..., f (p s ), which are generally arbitrary.In fact, by elementary symmetric functions, we note that h 1 (p),...,h s (p) are just all the s roots of This indeed renders their existence, which was stated in the result of Theorem 2.3, to be redundant.
Recall that totients are elements of Ꮿ(1,1).It seems natural to further characterize a particular class of Ꮿ(r,s), called here (r,s)-totients, defined by (2.16) Theorem 2.8.Let r, s be nonnegative integers, f ∈ ᏹ.Then f is an (r,s)-totient ⇔ for each prime p and each α(> 2) ∈ N, there are complex numbers a(p), b(p) such that (2.17) Proof.Using the definition and properties of Hsu's generalized Möbius function mentioned in Section 1, we have Another important characterization of Ꮿ(r,s) involving recurrence is due to Rutkowski [18] which states that s ∈ Ꮿ(r,s) ⇔ for each prime p and each α ∈ N, there exist complex numbers c 1 (p),...,c r (p) such that where (2.20) We will have occasion to use Rutkowski's result later.

Busche-Ramanujan-type identities
It is well known (see, e.g., [21, page 62], [7,10], or [13]) that ⇐⇒ there exists F ∈ ᏹ such that for all m,n ∈ N, we have and that ⇐⇒ there exists F ∈ ᏹ such that for all m,n ∈ N, we have whenever the greatest common unitary divisor (m,n , and f (p) = 0 for all primes p.For the notion of unitary divisor, see [21, page 9].Identities (3.1) and (3.2) are known as Busche-Ramanujan identities, while (3.4) is called the restricted Busche-Ramanujan identity because of the restrictions on m, n.In this section, we ask whether similar identities hold for functions in general Ꮿ(r,s).An earlier affirmative answer to a particular case of this problem appears in [9,Theorem 4.2] which in our terminology states that for whenever γ(m) | γ(n), where γ(m) denotes the product of all distinct primes factors of m.We will show that there are similar Busche-Ramanujan-type identities for functions in the classes Ꮿ(r,s) with r = 1,2, but are possible for r ≥ 3 with rather artificial flavor.As to be expected, the identities are of restricted form, that is, hold with conditions on m, n.
where ν p (m) denotes the highest power of p appearing in m.
Note that the 0-excessive pairs are trivially all pairs of natural numbers, while the 1excessive pairs (m,n) correspond exactly to those with the greatest common unitary divisor (m,n) u = 1.

Theorem 3.2. Let s be a nonnegative integer and f
for each s-excessive pair (m,n).
Proof.Since f ∈ ᏹ, the identity holds for all m, n with (m,n) = 1.It thus remains to prove this identity when (m,n) > 1.For such s-excessive pair (m,n), let their prime factorizations be where p i , q 1 j , q 2k are distinct primes; a i , b j , c k , d l are positive integers.By multiplicativity, we can write where (3.9) The right-hand side of the identity becomes Assuming without loss of generality that ν p (m) ≥ ν p (n) + s, that is, a ≥ b + s, the identity will be established if we can find F ∈ ᏹ satisfying for each prime p.It suffices to exhibit F(p j ), the values of F at prime powers, independent of a and b, such that where, for short, we put f 12), we have Replacing f a+1 , f a , f a−1 , f 1 using Corollary 2.4(iii), we have yielding F 1 = g(p) s i=1 h i (p), which is independent of a, provided that g(p) and s k=0 g(p) a−1−k H k are nonzero.Substituting b = 2 into (3.12),we get Replacing f a+2 , f a , f a−1 , f 2 , f 1 , using Corollary 2.4(iii) and the value of F 1 , we find that independent of a.In general, for fixed j, from Corollary 2.4(iii), with a − j ≥ s, we have Substituting these and the previous values of F i (i < j) into (3.12), and dividing by f a− j , we uniquely determine F j independent of a.Note that the division by f a− j is legitimate because from g(p), f s = s k=0 g s−k H k being nonzero, we immediately infer that f a = 0 for all a ≥ s. ) Proof.Clearly, the identity holds for all m, n with γ(m) | γ(n) and (m,n) = 1.It thus remains to prove this identity when (m,n) > 1.For each (s − 1)-excessive pair (m,n) with γ(m) | γ(n), let their prime factorizations be where p i are distinct primes, a i nonnegative integers, and b i positive integers, a i ≤ b i (i = 1,2,3,...,u).By multiplicativity, we can write The right-hand side of the identity becomes (3.21) The identity will be established if we can show that for each prime p and a To this end, it suffices to show that where (3.24) Noting that c 1 = g * 1 , c 2 = −g 1 , (3.23) follows in this case.Now proceed by induction on a. Assume that (3.23) holds up to a − 1. Again by Rutkowski's recurrence, when b + a ≥ s − 1, noting also that f and g 1 * g 2 satisfy the same recurrence, we have as required.
Theorem 3.3 as stated does not include the case Ꮿ(2, 0) because (−1)-excessive pair is not defined.However, going through the above proof, we see that in this case, we simply get the result of Haukkanen referred to in (3.5) above.Since functions in Ꮿ(2,0) satisfy the Busche-Ramanujan identity, a natural question to ask is whether a Ꮿ(3,0)-function enjoys such property.A trivial example of the identity function I = I * I * I = I * I, which belongs to both Ꮿ(2,0) and Ꮿ(3,0), shows that the answer is affirmative in certain cases, while u * u * u = µ −3 ∈ Ꮿ(3,0) does not satisfy the Busche-Ramanujan identity.Some necessary conditions for Ꮿ(3,0)-functions to satisfy the Busche-Ramanujan identity are given in the next proposition.
where F ∈ ᏹ, then for each prime p, there are five possibilities: Proof.Proceeding as in the proof of Theorem 3.2, we are looking for necessary conditions for f to satisfy the Busche-Ramanujan identity and this amounts to finding F ∈ ᏹ such that for each prime p and a ≥ b, that is, assuming without loss of generality that ν p (m) ≥ ν p (n). Substituting b = 1 into (3.27),we obtain the main recurrence relation From Corollary 2.7, which entails where this last relation simplifies to c 3 f a−2 = 0 (a ≥ 3), and so either (i) In the latter situation, we divide into two cases according to c 2 = 0 or c 2 = 0.
Case 1 (c 2 = 0).In this case, it easily follows from the main recurrence relation that f a = f a 1 for all a ≥ 1.
Case 2 (c 2 = 0).In this case, we further subdivide into two subcases according to f 1 = 0 or not.Subcase 2.1 ( f 1 = 0, and so f 2 = c 2 = 0).Using the main recurrence relation, it is easily checked that f (p 2n ) = ( f (p 2 )) n , and f (p 2n−1 ) = 0 for all n ≥ 1. Subcase 2.2 ( f 1 = 0).In this case, the main recurrence relation is a second-order recurrence with constant coefficients whose characteristic equation is The solutions corresponding to D = 0 or D = 0 are listed as ( 4) and ( 5), respectively, in the statement of the proposition.
In the proof of Proposition 3.4, Case 1 contains, as a special case, the identity function, while other cases contain some nontrivial Ꮿ(2,0)-functions, and some nontrivial Ꮿ-functions.Proposition 3.4 indicates somewhat that Ꮿ(3,0)-functions which satisfy reasonable Busche-Ramanujan-type identity can be artificially constructed from those satisfying conditions in any of the five cases.We now give an example to substantiate this claim.Recall from Corollary 2.7 that f ∈ Ꮿ(3,0) ⇔ for each prime p and integers e ≥ 3, we have where Should there be a Busche-Ramanujan-type identity, subject to certain conditions on m, n, proceeding as in the proof of Theorem 3.2, we deduce that there must exist F ∈ ᏹ satisfying where a ≥ b, F i = F(p i ).Consider the Ꮿ(3,0)-function defined by ).The situations for general Ꮿ(3,s) and Ꮿ(r,s) with r ≥ 3 are analogous.The details are omitted.Another class of identities for functions in Ꮿ(2, 0), called extended Busche-Ramanujan identity, is due to Redmond and Sivaramakrishnan [16] which states that for f ∈ Ꮽ, define (2.36) Using exactly the same proof as in [16,Theorem 13], together with the result of Theorem 3.3, we have the following theorem.

Binomial-type identities
It is known, see, for example, [16] or [21,Chapter 13], that if f = g 1 * g 2 ∈ Ꮿ(2,0), then f satisfies the so-called binomial identity where p is a prime, k ∈ N. In [6], another form of binomial identity is found, namely, The derivation of (4.1) in [16] is by induction, while that of (4.2) in [6] is based on solving second-order recurrence relation.Making use of certain Chebyshev-type identities, Haukkanen also derived the following inverse forms of (4.1) and (4.2):Our objective in this section is to use Rutkowski's recurrence to derive binomial-type identities and their inverse forms similar to (4.1)-(4.4)for elements in Ꮿ(2,s).Our starting point comes from the observation that (4.1) and (4.2) are indeed equivalent through a combinatorial identity, which we now elaborate.
Starting from (4.2), we have which is (4.1).The last equality follows from the combinatorial identity which appears in Riordan [17, problem 18(c)].
For each prime p and each k > 0, ) where (4.11) For brevity, we put By Rutkowski's theorem, where The characteristic polynomial of this recurrence is r 2 − Cr − D, whose two roots are g 1 and If g 1 = G 1 , then ∆ = 0, and the general solution of this recurrence is of the form Using the two initial values we get Thus, V. Laohakosol and N. Pabhapote 4011 and so If g 1 = G 1 , then ∆ = 0. Without loss of generality, we may assume that g 1 = G 1 := r = 0, for otherwise the desired result is trivial.The general solution of our recurrence now takes the shape Using the initial conditions we get Therefore, which agrees with (4.20) under the limit ∆ → 0, and the first identity is established.To establish the second identity, we proceed to use the combinatorial identity alluded to above.Since 2 k+s f p k+s = (A + B) we have applied to (4.9) of Theorem 4.1.
We end this section by remarking that it seems unlikely for functions in Ꮿ(r,s) with r > 2 to have similar binomial-type identities.

Kesava Menon norm
For f ∈ ᏹ, its Kesava Menon norm N f is an arithmetic function defined by (see [16,20]) where λ is the well-known Liouville's function, λ(n) = (−1) Ω(n) , Ω(n) being the total number of prime factors of n counted with multiplicity.Observe that N f ∈ ᏹ and λ ∈ Ꮿ which implies (see [23]) that λ( f * g) = λ f * λg when f ,g ∈ ᏹ.For nonnegative integer m, the mth power (Kesava Menon) norm of f ∈ ᏹ is inductively defined by and in [16, Theorem 3, page 214] that for nonnegative integer m, (5.4) In this section, we prove that both of these properties hold for elements in general Ꮿ(r,s).
. By the distributivity of completely multiplicative functions, we get where and for h ∈ Ꮿ, p prime, we have for each prime p and k ∈ N ∪ {0} that where (l) denotes the sum taken over all (r + s)-tuples of nonnegative integers (i 1 ,...,i r , j 1 ,..., j s ) such that i The gist of Theorem 5.1 is that (5.9) Theorem 5.1 remains valid when r and/or s is 0 for we can always, if needed, convolute by I or I −1 .Immediate from these remarks is the following corollary.
The Kesava Menon norm of f ∈ ᏹ is closely related to its (ordinary) square ( f ) 2 as seen from the following two identities of Sivaramakrishnan [20].If f = g 1 * g 2 ∈ Ꮿ(2,0), then where θ(n) = 2 ω(n) , ω(n) being the number of distinct prime factors of n.We next show that similar identities hold for functions in Ꮿ(2,1).
To prove the second identity, using the first identity and the distributivity of λ ∈ Ꮿ, we have λ g 1 * g 2 f = λ N f * G * g 1 g 2 = λN f * λG * λg 1 g 2 .
Added note.Regarding Theorem 5.1, it has been pointed out by one of the referees that N f for rational arithmetic functions f of order (r,s) has already been given in P. Haukkanen's review on [22].