PROPERTIES OF RATIONAL ARITHMETIC FUNCTIONS

Transcription

1 PROPERTIES OF RATIONAL ARITHMETIC FUNCTIONS VICHIAN LAOHAKOSOL AND NITTIYA PABHAPOTE Received 13 January 2005 and in revised form 20 September 2005 Rational arithmetic functions are arithmetic functions of the form g 1 g r h 1 1 h 1 s,whereg i, 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. 1. 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 f + g)n) = f n)+gn), f g)n) = f i)gj). 1.1) It is well known see, e.g., [1, 5, 13, 19, 21]) that the set,+, ) of all arithmetic functions is a unique factorization domain with the arithmetic function 1 ifn = 1, In) = 0 otherwise, ij=n 1.2) being its convolution identity. A nonzero arithmetic function f is called multiplicative, denoted by f, if f mn) = f m) f n)wheneverm,n) = 1. It is called completely multiplicative, denoted by f,if f mn) = f m) f n)forallm,n N. For nonnegative integers r, s by an r,s)-rational arithmetic function f, denoted by f r,s), we mean an arithmetic function which can be written as f = g 1 g r h 1 1 h 1 s, 1.3) Copyright 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005: ) DOI: /IJMMS

2 3998 Properties of rational arithmetic functions where each g i,h j. 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 1,1) whose elements are known as totients, and 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, f 1) R +, the Rearick logarithm of f see [11, 14, 15]), denoted by Log f, is defined via Log f )1) = log f 1), Log f )n) = 1 n f d) f 1 logd = 1 df f 1 ) n) n>1), logn d logn d n 1.4) where df n) = f n)logn denotes the log derivation of f. The Hsu s generalized Möbius function see [2]) µ r, r R,isdefinedas µ r n) = p n r 1) ν p n) νpn), 1.5) where ν p n) is the highest power of the prime p dividing n. It is known see [8, 12]) that for f, and the converse holds under additional hypotheses. 2. Characterizations f = f r = µ r f, 1.6) 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. Theorem 2.1. Let r, s be nonnegative integers and f. Then, f 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 1 [ a1 p) Log f )n) = α α + + a r p) α b 1 p) α b s p) α] if n = p α, 0 otherwise. 2.1)

3 V. Laohakosol and N. Pabhapote 3999 Proof. f r,s) f = g 1 g r h 1 1 h 1 s gi,h j ) Log f = Logg 1 + +Logg r Logh 1 Logh s. 2.2) The result now follows immediately from Carroll s theorem [3] which states that for F, 1 F LogF)n) = α Fp)α if n = p α, 0 otherwise. 2.3) Taking a 1 p) α = f p α+1 )/fp), b 1 p) = bp) intheorem 2.1, we get the following corollary. Corollary 2.2. Let f,with f p) 0 for each prime p. Then f 1,1) for each prime p and each α N,thereisacomplexnumberbp) such that 1 f p α+1 ) ) bp) Log f )n) = α f p) α if n = p α, 0 otherwise. 2.4) Theorem 2.3. Let r, s be nonnegative integers and f. Then f 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, f p α) = s G α k H k, 2.5) k=0 where G α k = a 1 p) j1 a r p) jr, G 0 = 1, H k = 1) k j 1+ +j r =α k 1 i 1<i 2< <i k s b i1 p) b ik p), H 0 = ) Proof. f r,s) f = g 1 g r h 1 1 h 1 s gi,h j ) f p α) = g 1 p) j1 g r p) jr h 1 1 p k 1 1 ) h s p k s. j 1+ +j r+k 1+ +k s=α 2.7) The result now follows by grouping terms on the right-hand side and using h 1 p k ) = 0 for k 2.

4 4000 Properties of rational arithmetic functions A few known characterizations of two particular classes of functions, namely, those in 1,1), that is, totients see [7]), and those in 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 1,s)to be used later. Corollary 2.4. Let f. Then the following hold. i) f 1,1) for each prime p and each α N, there exists a complex number ap) such that ii) f 2,0) for each prime p and each α 2) N, f p α) = ap) α 1 f p). 2.8) f p α+1) = f p) f p α) + f p α 1)[ f p 2) f p) 2]. 2.9) iii) f 1,s) for each prime p and each α N, there exist complex numbers ap), b 1 p),...,b s p) such that for all α s, f p α) = s gp) α k H k, 2.10) k=0 where H k = 1) k b i1 p) b ik p), H 0 = ) 1 i 1<i 2< <i k s 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. Then f 0,s) for each prime p, f p α ) = 0 for all α>s. Proof. f 0,s) f = h 1 1 h 1 s f p α) = i 1+ +i s=α h 1 1 hi ) p i 1 1 ) h p i s. 2.12) The result now follows by noting that for h, wehaveh 1 p) = hp), 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 X s + f p)x s f p s 1) X + f p s) = ) This indeed renders their existence, which was stated in the result of Theorem 2.3, tobe redundant. s

5 V. Laohakosol and N. Pabhapote 4001 Invoking upon the fact that f r,0) f 1 0,r), we easily deduce our next result which appears as [13, Problem 1.16, page 48]. Corollary 2.6. Let r be a nonnegative integer and f. Then f r,0) for each prime p, f 1 p α) = 0 α>r. 2.14) Corollary 2.7. Let r be a nonnegative integer and f. Then f r,0) for each prime p,andforallα r, f p α+1) = [ f p α) f 1 p)+ f p α 1) f 1 p 2) + + f p α r+1) f 1 p r)]. 2.15) Proof. This follows by expanding f f 1 = I at the prime powers p α and applying the result of Corollary 2.6. Recall that totients are elements of 1,1). It seems natural to further characterize a particular class of r,s), called here r,s)-totients,definedby f = g r h s, g,h. 2.16) Theorem 2.8. Let r, s be nonnegative integers, f. Then f is an r,s)-totient for each prime p and each α> 2) N, there are complex numbers ap), bp) such that f p α) α = 1) α r s ap) α i i α i bp) i. 2.17) Proof. Using the definition and properties of Hsu s generalized Möbius function mentioned in Section 1,wehave f is an r,s)-totient f = g r h s = µ r g µ s h f p α) α = µ r g p α i) µ s h p i). 2.18) Taking ap) = gp), bp) = hp), the result follows. Another important characterization of r,s) involving recurrence is due to Rutkowski [18] which states that f = g 1 g r h 1 1 h 1 s r,s) for each prime p and each α N, there exist complex numbers c 1 p),...,c r p)suchthat where c 1 p) = f p α) = c 1 p) f p α 1) + + c r p) f p α r) α>s), 2.19) r g i p), c 2 p) = i=1 1 i 1<i 2 r We will have occasion to use Rutkowski s result later. g i1 p)g i2 p),..., c r p) = 1) r+1 g 1 p) g r p). 2.20)

6 4002 Properties of rational arithmetic functions 3. Busche-Ramanujan-type identities It is well known see, e.g., [21, page 62], [7, 10], or [13]) that f 2,0) there exists B such that for all m,n N, wehave f m) f n) = mn f d 2 Bd) d m,n) there exists F such that for all m,n N, wehave f mn) = m n f f Fd), d d d m,n) 3.1) 3.2) and that f 1,1) there exists h such that for all m,n N, wehave f m) f n) = mn f hd)µd) d d m,n) there exists F such that for all m,n N, wehave f mn) = m n f f Fd), d d d m,n) 3.3) 3.4) whenever the greatest common unitary divisor m,n) u = 1, f p 2 ) f p) 2 + Fp), and f p) 0forallprimesp. For the notion of unitary divisor, see [21,page9]. Identities 3.1) and3.2) 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 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 f = g 1 g 2 h 1 2,1), we have f mn) = d m,n) m n g1 g 2 f µd)g 1 d)g 2 d), 3.5) d d whenever γm) γn), where γm) 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 r,s) withr = 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. Definition 3.1. Let s be a nonnegative integer. A pair m,n) N N is said to be s- excessive if for each prime p dividing m,n), either ν p m) ν p n)+s or ν p n) ν p m)+ s,whereν 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 1- excessive pairs m,n) correspond exactly to those with the greatest common unitary divisor m,n) u = 1.

7 V. Laohakosol and N. Pabhapote 4003 Theorem 3.2. Let s be a nonnegative integer and f = g h 1 1 h 1 s 1,s). For each prime p,ifgp) 0,and s k=0 gp) s k H k 0,whereH k = 1) k 1 i 1<i 2< <i k s h i1 p) h ik p), H 0 = 1,thenthereexistsF such that for each s-excessive pair m,n). f mn) = d m,n) m n f f Fd), 3.6) d d Proof. Since f, 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 m = p a1 1 p au u q c1 11 q cv 1v, n = p b1 1 p bu u q d1 21 q dw 2w, 3.7) where p i, q 1j, q 2k are distinct primes; a i, b j, c k, d l are positive integers. By multiplicativity, we can write where Q = f q c1 11 f mn) = Q ) f q c v The right-hand side of the identity becomes d m,n) m n f f Fd) = Q d d u i=1 1v u i=1 f p ai+bi i, 3.8) ) f q d 1 21 ) f q d w 2w). 3.9) mina i,b i) f p ai j i ) b f p i j) j) i F p i. 3.10) 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 satisfying f p a+b) = b f p a j) f p b j) F p j), 3.11) for each prime p.itsuffices to exhibit Fp j ), the values of F at prime powers, independent of a and b,suchthat f a+b = b f a j f b j F j, 3.12) where, for short, we put f p i ) = f i, Fp j ) = F j. Substituting b = 1into3.12), we have f a+1 = f a f 1 + f a 1 F 1 a s +1). 3.13)

8 4004 Properties of rational arithmetic functions Replacing f a+1, f a, f a 1, f 1 using Corollary 2.4iii), we have ) s s s s gp) a+1 k H k = gp) h i p) gp) a k H k + F 1 gp) a 1 k H k, 3.14) k=0 i=1 k=0 yielding F 1 = gp) s i=1 h i p), which is independent of a, provided that gp) and sk=0 gp) a 1 k H k are nonzero. Substituting b = 2into3.12), we get k=0 f a+2 = f a f 2 + f a 1 f 1 F 1 + f a 2 F 2 a s +2). 3.15) Replacing f a+2, f a, f a 1, f 2, f 1, using Corollary 2.4iii) and the value of F 1, we find that [ s ) 2 F 2 = gp) 2 i p) i=1h ] h i1 p)h i2 p), 3.16) 1 i 1<i 2 s independent of a. In general, for fixed j,fromcorollary 2.4iii), with a j s,wehave f a+j = g 2j f a j, f a+j 1 = g 2j 1 f a j,..., f a j+1 = gf a j. 3.17) Substituting these and the previous values of F i i<j)into3.12), 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 gp), f s = s k=0 g s k H k being nonzero, we immediately infer that f a 0for all a s. Theorem 3.3. Let s N.If f = g 1 g 2 h 1 1 h 1 s 2,s), then f mn) = d m,n) for each s 1)-excessive pair m,n) with γm) γn). m n g1 g 2 f µd) g 1 g 2 d), 3.18) d d Proof. Clearly, the identity holds for all m, n with γm) γn) andm,n) = 1. It thus remains to prove this identity when m,n) > 1. For each s 1)-excessive pair m,n)with γm) γn), let their prime factorizations be m = p a1 1 p au u, n = p b1 1 p bu u, 3.19) 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 f mn) = u i=1 f p ai+bi i. 3.20)

9 V. Laohakosol and N. Pabhapote 4005 The right-hand side of the identity becomes d m,n) m n g1 g 2 f µd) g 1 g 2 d) d d = u a i i=1 g1 g 2 ) p a i j i ) b f p i j) i µ p j ) g 1 g 2 p j ). 3.21) The identity will be established if we can show that f p a+b) = a g1 g 2 p a j ) f p b j) µ p j) g 1 g 2 p j ), 3.22) for each prime p and a N {0}, b N with b a + s 1. To this end, it suffices to show that f a+b = g a f b g a 1 f b 1 g 1, 3.23) where f p i ) = f i,g 1 g 2 )p i ) = g i,g 1 g 2 )p j ) = g j. For a = 0, 3.23) trivially holds. When a = 1, b s, from Rutkowski s recurrence, we get f b+1 = c 1 f b + c 2 f b ) 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) holdsuptoa 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 f a+b = c 1 f b+a 1 + c 2 f b+a 2 = c 1 g a 1 f b ga 2 f b 1 g 1) + c2 g a 1 f b 1 ga 2 f b 2 g 1 = c 1 ga 1 f b + ga 2 f b c 2 + c 2 ga 1 f b 1 = ga f b ga 1 f b 1 g 1, 3.25) as required. Theorem 3.3 as stated does not include the case 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 2,0) satisfy the Busche-Ramanujan identity, a natural question to ask is whether a 3,0)-function enjoys such property. A trivial example of the identity function I = I I I = I I, which belongs to both 2,0) and 3,0), shows that the answer is affirmativeincertain cases, while u u u = µ 3 3,0) does not satisfy the Busche-Ramanujan identity. Some necessary conditions for 3,0)-functions to satisfy the Busche-Ramanujan identity are given in the next proposition.

10 4006 Properties of rational arithmetic functions Proposition 3.4. Let f 3,0).If f satisfies the Busche-Ramanujan identity f mn) = d m,n) m n f f Fd) m,n N), 3.26) d d where F, then for each prime p, there are five possibilities: 1) f p n ) = 0 for all n 1,or 2) f p n ) = f p)) n for all n 1,or 3) f p 2n ) = f p 2 )) n, f p 2n 1 ) = 0 for all n 1,or 4) f p n ) = 1 + n) f p)/2) n for all n 1,or 5) f p n ) = 1/2)1 + f p)/d) f 1 + D)/2) n +1/2)1 f 1 /D) f 1 D)/2) n for all n 1,whereD = 4 f p2 ) 3 f p)) 2 0. 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 such that b f a+b = f a j f b j F j, 3.27) for each prime p and a b, that is, assuming without loss of generality that ν p m) ν p n). Substituting b = 1into3.27), we obtain the main recurrence relation f a+1 = f a f 1 + f a 1 F 1 a 1). 3.28) Putting a = 1, we get F 1 = f 2 f 2 1.From Corollary 2.7, which entails f a f 1 + f a 1 F 1 = f a f 1 + f a 1 f2 f 2 1 ) + fa 2 f3 2 f 1 f 2 + f 3 ), 3.29) 1 f a 1 F 1 = c 2 f a 1 + c 3 f a 2 a 3), 3.30) where c 2 = f 2 f1 2, c 3 = f 3 2 f 1 f 2 + f1 3. Using F 1 = f 2 f1 2 = c 2, this last relation simplifies to c 3 f a 2 = 0a 3), and so either i) f n = 0foralln 1, or ii) 0 = c 3 = f 3 2 f 1 f 2 + f1 3. In the latter situation, we divide into two cases according to c 2 = 0orc 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.

11 V. Laohakosol and N. Pabhapote 4007 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 ) = 0foralln 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 x 2 f 1 x c 2 = 0, with roots 1/2) f 1 ± D), where D = f1 2 +4c 2. The solutions corresponding to D = 0orD 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 2,0)-functions, and some nontrivial -functions. Proposition 3.4 indicates somewhat that 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 3,0) for each prime p and integers e 3, we have f e+1 = f e f 1 + f e 1 A + f e 2 B, 2.31) where f e = f p e ), A = Ap) = f 2 f 2 1, B = Bp) = f 3 2 f 2 f 1 + f 3 1. 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 satisfying f a+b = b f a i f b i F i, 2.32) where a b, F i = Fp i ). Consider the 3,0)-function defined by satisfying 2.31)with f 1) = 1, f 2 a) a 1) 2.33) B2) = f 2 3) 2 f 2 2) f 2 1) + f 2 3) = 0, f p a) = 0 a 1) 2.34) for all other primes p 3. This particular function f 3,0) because it satisfies 2.31) with A2), Ap), Bp) p prime 3) arbitrary but B2) = 0. It satisfies the Busche- Ramanujan identity 2.32) withf2) = A2), F2 i ) = Fp j ) = 0i 2, j 1). The situations for general 3,s)and r,s)withr 3 are analogous. The details are omitted. Another class of identities for functions in 2, 0), called extended Busche-Ramanujan identity, is due to Redmond and Sivaramakrishnan [16] which states that for f, define tn) ifn k, t 0 n) = tn), t k n) = 2.35) 0 otherwise.

12 4008 Properties of rational arithmetic functions Let T 0 = T, T k = µ t k.if f = g 1 g 2 2,0), then d m,n) m n g1 ) f f g 2 d)tk d) = td) ) mn g 1 g 2 d) f d d d d m,n,k) ) Using exactly the same proof as in [16, Theorem 13], together with the result of Theorem 3.3, we have the following theorem. Theorem 3.5. Let s N.If f = g 1 g 2 h 1 1 h 1 s 2, s), then d m,n) m n g1 ) g1 g 2 f g 2 d)tk d) = td) ) mn g 1 g 2 d) f d d d d m,n,k) 2, 3.37) for each s 1)-excessive pair m,n) with γm) γn). 4. Binomial-type identities It is known, see, for example, [16]or[21, Chapter 13], that if f = g 1 g 2 2,0), then f satisfies the so-called binomial identity f p k) k j = 1) j f p) j k 2j g 1 p)g 2 p) ) j, 4.1) where p is a prime, k N. In [6], another form of binomial identity is found, namely, 2 k f p k) = k +1 f p) k 2i[ f p) 2 4g 1 p)g 2 p) ] i. 4.2) The derivation of 4.1)in[16]is by induction,while that of4.2)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)and4.2): f p) k = k +1)f p) k = { k i k +1 2i } k f p i 1 k 2i) g 1 p)g 2 p) ) i, 4.3) d 2i 2 k 2i f p k 2i) f p) 2 4g 1 p)g 2 p) ) i, 4.4) where d 2i is defined as in [17, Section 3.4], namely, via the generating series relation 2x e x e x = x d 2i 2i 2i)!. 4.5)

13 V. Laohakosol and N. Pabhapote 4009 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 2,s). Our starting point comes from the observation that 4.1)and4.2) are indeed equivalentthrough a combinatorial identity, which we now elaborate. Starting from 4.2), we have f p k) = = k +1 k +1 ) f p) 2 ) f p) k 2i i 2 ) k 2i [ f p) 2 i g 1 p)g 2 p)] 2 i j ) f p) 2i 2j [ g1 p)g 2 p) ] j 2 = = = i=j k +1 ) i j ) f p) k 2j [ g1 p)g 2 p) ] j 2 1) j f p) k 2j[ g 1 p)g 2 p) ] j 1 2 k 2j 1) j k j j ) f p) k 2j[ g 1 p)g 2 p) ] j, ) i=j k +1 ) i j 4.6) which is 4.1). The last equality follows from the combinatorial identity i=j k +1 ) i j k j = 2 k 2j j 4.7) which appears in Riordan [17, problem 18c)]. Theorem 4.1. Let s N and f = g G h 1 1 h 1 s 2,s). For each prime p and each k>0, 2 k+s f p k+s) = A + B)S k+s p) 2 Bgp)+AGp) ) S k+s 1 p), 4.8) f p k+s) [k+s)/2] k + s j [ = A + B) 1) j ] k+s 2j [ ] j f p)+hp) gp)gp) j Bgp)+AGp) )S k+s 1p) 2 k+s 1, 4.9)

14 4010 Properties of rational arithmetic functions where Hp) = s h i p), i=1 S k+s p) = = [k+s)/2] [k+s)/2] A = f p 1+s) Gp) f p 2+s) g p 1+s) Gp) gp) ), B = f p 2+s ) gp) f p 1+s) G p 1+s) Gp) gp) ), 4.10) k + s +1 [ ] k+s 2i [ ] 2i f p)+hp) gp) Gp) k + s +1 [ ] k+s 2i [ 2 ] i. f p)+hp) f p)+hp) 4gp)Gp) Proof. Since f = g G H 1,withH 1 = h 1 1 h 1,wehave For brevity, we put By Rutkowski s theorem, where s 4.11) f p) = gp)+gp) Hp). 4.12) f k = f p k), g k = g p k), G k = G p k), H = Hp). 4.13) f k = Cf k 1 + Df k 2 k>s), 4.14) C = g 1 + G 1 = f 1 + H, D = g 1 G ) The characteristic polynomial of this recurrence is r 2 Cr D, whose two roots are g 1 and G 1.Let = g 1 G 1. 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, f k+s = Ag k+s 1 + BG k+s 1 k>0). 4.16) f 1+s = Ag 1+s 1 + BG 1+s 1, f 2+s = Ag 2+s 1 + BG 2+s 1, 4.17) A = f 1+sG 1 + f 2+s g 1+s 1, B = f 2+s + g 1 f 1+s G 1+s. 4.18) 1 C + k+s C k+s f k+s = A + B, 4.19) 2 2

15 and so k+s 2 k+s k + s f k+s = A C i k+s i i + B = A + B) +A B) = A + B) [k+s)/2] [k+s 1)/2] [k+s)/2] + A + B 2 = A + B) [k+s)/2] k+s V. Laohakosol and N. Pabhapote 4011 k + s C i k+s i ) i k + s C 2i k+s 2i 2i k + s C k+s 2i 1 2i+1 k + s C 2i k+s 2i 2i Bg1 + AG 1 C k + s +1 2 [k+s 1)/2] Bg 1 + AG 1 ) )) [k+s 1)/2] C k+s 2i 2i k + s C k+s 2i 2i k + s C k+s 2i 1 2i. 4.20) 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 f k+s = A r k+s +k + s)b r k+s k>0). 4.21) f 1+s = A r 1+s +1+s)B r 1+s, f 2+s = A r 2+s +2+s)B r 2+s, 4.22) Therefore, A = 2 + s)rf 1+s 1 + s) f 2+s r 2+s, B = f 2+s rf 1+s r 2+s, r = C ) 2 k+s f k+s = A +k + s)b ) C k+s, 4.24) 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) [k+s)/2] k + s +1 C k+s 2i 2i 2 Bg 1 + AG 1 ) Sk+s 1 p), 4.25)

16 4012 Properties of rational arithmetic functions we have f p k+s) = A + B) [k+s)/2] Bg 1 + AG 1 )S k+s 1 p) = A + B) [k+s)/2] C ) k + s +1 k+s 2i[C ) 2 i + D] k+s 1 k + s +1 S k+s 1 p) Bg 1 + AG 1 2 k+s 1 [k+s)/2] = A + B) C k+s 2j D j Bg 1 + AG 1 )S k+s 1 p) = A + B) [k+s)/2] 2 k+s 1 k + s j j ) C ) k+s 2i i k+s 2j i j ) [k+s)/2] i=j ) C ) 2i 2j D j 2 k + s +1 ) i j C k+s 2j D j Bg 1 + AG 1 )S k+s 1 p) 2 k+s 1, ) 4.26) where the last equality follows from the identity of Riordan [17, Problem 18c), page 87]. The results of Theorem 4.1 reduce to the identities 4.1)and4.2)whens = 0, because then A + B = 1andBgp)+AGp) = Hp) = 0. It remains to establish inverse forms of the two identities of Theorem 4.1. Theorem 4.2. Let s N and f = g G h 1 1 h 1 s 2,s). For each prime p and each integer k>0, S k+s p) = 2k+s A + B k 1 Bgp)+AGp) i f p k+s i) ) 2 Bgp)+AGp) k + S s, A + B A + B k + s +1 d 2i 2i S k+s 2i p) k + s +1) f p)+hp) ) [k+s)/2] k+s = f p)+hp) ) k+s = 1 A + B [k+s)/2] [ k + s i [ ] k + s i 1 [ f p)+hp) ) 2 4gp)Gp) ] i, f p k+s 2i) + Bgp)+AGp) )S k+s 2i 1p) 2 k+s 2i 1 ] gp)gp) ) i, where S k+s p), Hp), A, B are as defined in Theorem 4.1 and d 2i is as defined in 4.4). 4.27)

17 V. Laohakosol and N. Pabhapote 4013 Proof. The first identity, for S k+s p), comes immediately from solving the first-order nonhomogeneous recurrence 4.8)inTheorem 4.1,wherewedefineg s := f s := S s p)/2 s.the second identity, for k + s +1)f p)+hp)) k+s, follows from the inverse relation, see, for example, [6, page 160], a k = k +1 b k 2i c i k +1)b k = k +1 d 2i 2i a k 2i c i, 4.28) applied to 4.11) oftheorem 4.1. The third identity, for f p)+hp)) k+s,followsfrom theinverserelation,see,forexample,[6, page 159], k i a k = 1) i b i k 2i c i b k = applied to 4.9)of Theorem4.1. [ ] k k a i i 1 k 2i c i, 4.29) We end this section by remarking that it seems unlikely for functions in r,s) with r>2 to have similar binomial-type identities. 5. Kesava Menon norm For f, its Kesava Menon norm Nf is an arithmetic function defined by see [16, 20]) Nfn):= f λf) n 2), 5.1) where λ is the well-known Liouville s function, λn) = 1) Ωn), Ωn) being the total number of prime factors of n counted with multiplicity. Observe that Nf and λ which implies see [23]) that λ f g) = λf λg when f,g. For nonnegative integer m,themth power Kesava Menon) norm of f is inductively defined by N 0 f = f, N 1 f = Nf, N m f = N N m 1 f ). 5.2) It is shown in [20, Theorem 3.3, page 160] that f 2,0) = Nf 2,0), 5.3) and in [16, Theorem 3, page 214] that for nonnegative integer m, f 1, f 2 2,0) = N m f 1 f 2 ) = N m f 1 N m f ) In this section, we prove that both of these properties hold for elements in general r,s). Theorem 5.1. Let r, s be positive integers and f r,s). Then Nf r,s).

18 4014 Properties of rational arithmetic functions Proof. Let f = g 1 g r h 1 1 h 1 s. By the distributivity of completely multiplicative functions, we get f λf = g 1 g r h 1 1 h 1 ) s λ g1 g r h 1 1 h 1 )) s = g 1 λg 1 gr λg r h 1 1 λh 1 ) 1 h 1 s λh 1 ) s = g 1 u λ) ) g r u λ) ) h 1 1 λh 1 ) 1 h 1 s λh 1 ) s, 5.5) where u is the unit function, un) = 1n N). Using 1 ifn is a perfect square, u λ)n) = 0 otherwise, and for h, p prime, h 1 λh 1) p k) 1 ifk = 0, = hp) 2 if k = 2, 0 otherwise, 5.6) 5.7) we have for each prime p and k N {0} that Nf p k) = f λf) p 2k) = 2k)g 1 u λ) p i1) g r u λ) p ) ir h 1 1 λh 1 ) 1 p j 1 h s 1 λh 1 ) s p j s = k) g 1 g 1 p i 1 ) gr g r p i r ) h1 h 1 ) 1 p j 1 ) hs h s ) 1 p j s ), 5.8) where l) denotes the sum taken over all r + s)-tuples of nonnegative integers i 1,...,i r, j 1,..., j s )suchthati i s + j j s = l. SinceNf and g i g i,h j h j, it follows that Nf r,s). The gist of Theorem 5.1 is that f = g 1 g r h 1 1 h 1 s = Nf = ) g 1 g 1 gr g r h1 h 1 hs h s 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. Corollary 5.2. Let m,r,s N {0}. Then the following hold. i) f = g 1 g r h 1 1 h 1 s r,s) N m f = g 1 ) 2m g r ) 2m h 1 ) 2m ) 1 h s ) 2m ) 1,whereg i ) m = g i g i m times). ii) f 1, f 2 r,s) N m f 1 f 2 ) = N m f 1 N m f 2.

19 V. Laohakosol and N. Pabhapote 4015 The Kesava Menon norm of f 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 2,0), then f n) 2 = n θd) d nnf g 1 g 2 d), 5.10) d n 2 λd) f d) 2 f = n λd)nfd)nf, 5.11) d d d n where θn) = 2 ωn), ωn) being the number of distinct prime factors of n. We next show that similar identities hold for functions in 2,1). Theorem 5.3. Let f = g 1 g 2 h 1 2,1) and let Nf be its Kesava Menon norm.then there exists G such that g1 g 2 f = Nf G g1 g 2, 5.12) g1 g 2 f λ g1 g 2 f = Nf λn f ) G λg) g1 g 2 u λ). d n Further, G is defined on prime powers by Gp e ) = f 1 p 2e )e N),andun) = 1n N). Proof. Define f = f λf.observethat Nf n) ifn is a perfect square, f n) = 5.13) 0 otherwise, and that f n 2) = f λf 1 ) n 2) = i n 2 f i)λf 1 n 2 ) = i j n Nfj) f 1 n 2 j 2 ). 5.14) Define G by G1) = 1, Gp e ) = f 1 p 2e )pprime,e N), and extend it by multiplicativity to all positive integers. Thus, f n 2 ) = Nf G)n). On the other hand, by Theorem 3.2,whens = 1, we have f n 2) = ) n µg1 ) g1 g 2 f g 2 d). 5.15) d d n Thus g 1 g 2 ) f µg 1 g 2 ) = Nf G, and the first identity follows by noting that as g 1 g 2,thenµg 1 g 2 ) 1 = g 1 g 2. To prove the second identity, using the first identity and the distributivity of λ,we have Thus, λ g 1 g 2 ) f = λ Nf G g1 g 2 ) = λn f λg λg1 g ) g1 g 2 ) f λ g1 g 2 ) f = Nf λn f ) G λg) g1 g 2 λg 1 g 2 ), 5.17) and the desired result follows from the distributivity of g 1 g 2.

20 4016 Properties of rational arithmetic functions Theorem 5.3 yields the following immediate consequences. Corollary 5.4. If f = g 1 g 2 h 1 2,1), Nf is its Kesava Menon norm, and G as definedin Theorem 5.3, then f n) g 1 g 2 n) = Nfi)Gj) g 1 g 2 k), d n ijk=n ) n λ ) g1 g 2 f g1 g 2 f d) = d ijk=n Note that for f = g 1 g 2 2,0), if we interpret G by Nf λn f )i)g λg)j) g 1 g 2 u λ) ) k). 5.18) G1) = 1, Gp) = g 1 g 2 ) p), G p e ) = 0 forprime p and integer e 2, 5.19) then G g 1 g 2 ) = θg 1 g 2 and G λg) g 1 g 2 )u λ) = I, whereθn) = 2 ωn), I is the convolution identity, and so the identities in Corollary 5.4 reduce to 5.10) and5.11), respectively. Added note. Regarding Theorem 5.1, it has been pointed out by one of the referees that Nf for rational arithmetic functions f of order r,s) has already been given in P. Haukkanen s review on [22]. Acknowledgment This work was partially supported by the University of the Thai Chamber of Commerce. References [1] T.M.Apostol,Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer, New York, [2] T.C.Brown,L.C.Hsu,J.Wang,andP.J.-S.Shiue,On a certain kind of generalized numbertheoretical Möbius function, Math. Sci ), no. 2, [3] T. B. Carroll, A characterization of completely multiplicative arithmetic functions, Amer.Math. Monthly ), no. 9, [4] T.B.CarrollandA.A.Gioia,On a subgroup of the group of multiplicative arithmetic functions, J. Austral. Math. Soc. Ser. A ), no. 3, [5] E.D.CashwellandC.J.Everett,The ring of number-theoretic functions, Pacific J. Math ), no. 4, [6] P. Haukkanen, Binomial formulas for specially multiplicative functions, Math. Student ), no. 1 4, [7], Some characterizations of totients, Int. J. Math. Math. Sci ), no. 2, [8], On the real powers of completely multiplicative arithmetical functions, Nieuw Arch. Wiskd. 4) ), no. 1-2, [9], Rational arithmetical functions of order 2,1) with respect to regular convolutions,port. Math ), no. 3, [10], Some characterizations of specially multiplicative functions, Int. J. Math. Math. Sci ), no. 37,

21 V. Laohakosol and N. Pabhapote 4017 [11] V. Laohakosol, N. Pabhapote, and N. Wechwiriyakul, Logarithmic operators and characterizations of completely multiplicative functions, Southeast Asian Bull. Math ), no. 2, [12], Characterizing completely multiplicative functions by generalized Möbius functions, Int. J. Math. Math. Sci ), no. 11, [13] P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, [14] D. Rearick, Operators on algebras of arithmetic functions, Duke Math. J ), no. 4, [15], The trigonometry of numbers, Duke Math. J ), no. 4, [16] D. Redmond and R. Sivaramakrishnan, Some properties of specially multiplicative functions, J. Number Theory ), no. 2, [17] J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, [18] J. Rutkowski, On recurrence characterization of rational arithmetic functions, Funct. Approx. Comment. Math ), [19] H. N. Shapiro, Introduction to the Theory of Numbers, Pure and Applied Mathematics, John Wiley & Sons, New York, [20] R. Sivaramakrishnan, On a class of multiplicative arithmetic functions, J. reineangew. Math ), [21], Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 126, Marcel Dekker, New York, [22] S. Thajoddin and S. Vangipuram, A note on Jordan s totient function, Indian J. Pure Appl. Math ), no. 12, [23] R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc ), no. 2, Vichian Laohakosol: Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand address: fscivil@ku.ac.th Nittiya Pabhapote: Department of Mathematics, School of Science, University of the Thai Chamber of Commerce, Bangkok 10400, Thailand address: nittiya pab@utcc.ac.th

22 Mathematical Problems in Engineering Special Issue on Space Dynamics Call for Papers Space dynamics is a very general title that can accommodate a long list of activities. This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics. It is possible to make a division in two main categories: astronomy and astrodynamics. By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth. Many important topics of research nowadays are related to those subjects. By astrodynamics, we mean topics related to spaceflight dynamics. It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects. Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts. Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control. The main objective of this Special Issue is to publish topics that are under study in one of those lines. The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research. All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays. Before submission authors should carefully read over the journal s Author Guidelines, which are located at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Lead Guest Editor Antonio F. Bertachini A. Prado, Instituto Nacional de Pesquisas Espaciais INPE), São José dos Campos, São Paulo, Brazil; prado@dem.inpe.br Guest Editors Maria Cecilia Zanardi, São Paulo State University UNESP), Guaratinguetá, São Paulo, Brazil; cecilia@feg.unesp.br Tadashi Yokoyama, Universidade Estadual Paulista UNESP), Rio Claro, São Paulo, Brazil; tadashi@rc.unesp.br Silvia Maria Giuliatti Winter, São Paulo State University UNESP), Guaratinguetá, São Paulo, Brazil; silvia@feg.unesp.br Manuscript Due July 1, 2009 First Round of Reviews October 1, 2009 Publication Date January 1, 2010 Hindawi Publishing Corporation