KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017
Unitary Convolution and Generalized MӦbius Function
Pussadee Yangklan*, Pattira Ruengsinsub and Vichian Laohakosol
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
Abstract
The concept of unitary convolution in the commutative ring of arithmetic functions, under the two operations of addition and unitary convolution, is investigated. A generalized unitary M bius function is defined and its basic properties, which extend those corresponding classical ones are derived. Of particular interest are certain generalized unitary M bius inversion formula and ӧsome characterizations of multiplicative functions. ӧ Keywords: arithmetic function, M bius inversion formula, multiplicative function, unitary convolution ӧ
1. Introduction
An arithmetic function [1] is a complex-valued function defined on the set of positive integers. The set of arithmetic functions, , is a commutative ring under the usual addition defined by (f+=+ g )() n f () n gn () and the unitary convolution, , defined by (f gn )()= ∑∑ fndgd( /) ()= gndfd( /) (), dn|| dn|| where dn|| denotes the unitary divisor, i.e., those divisors d for which gcd (d,n/d) = 1. The identity element under the unitary convolution is the function 1 ifn = 1 In()= 0 ifn > 1, which is also the identity under the Dirichlet convolution, * , defined by (f * g )( n )= ∑ f ( dgn ) ( / d ). dn∣ For f ∈ such that f (1)≠ 0 , its inverse under the unitary convolution exists and is denoted by −1 f . Notice that when n is squarefree, we have (f gn )()(= f * gn )(). Cohen [2] considered a unitary analogue of the M bius function µ ( n ) = ( − 1) ω () n , (1.1) ӧ where ω()n is the number of distinct prime factors of n withω(1)= 0. Generalizing Cohen's notion, we define a generalized unitary M bius function by
*Corresponding author: Tel.: +66 851421272 E-mail: [email protected]
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ω µα= − ()n α ()n ( ) ,
where α ∈ {0} and µ0 = I, so that µµ1 = . Denote by , the set of multiplicative functions, viz., M:={ f∈= A {0}; fmn ( ) fmfn ( )()whenevergcd(,)mn= 1.}
Evidently, µα is a multiplicative function, and it is easily checked that a unitary convolution of two multiplicative functions is a multiplicative function. The objectives in this work are 1) to collect basic properties associated with unitary convolution; 2) to prove the generalized unitaryMӧbius inversion formula and to derive relations among the generalized unitary Mӧbius function and multiplicative functions; 3) to derive some characterizations of multiplicative function.
2. Preliminaries
2.1 The work of Cohen in 1960 [2] Cohen seemed to be the first person who seriously used the concept of unitary convolution to solve arithmetical problems. First, he defined the unitary Euler's totient, ϕ (n ). For ab, ∈ with b > 0, denote by (,)ab* the greatest divisor of a which is a unitary divisor of b. When (,ab)* =1 the integer a is said to be semiprime to b. The unitary Euler's totient is defined as the number of positive integers in {1, 2,… ,n } that are semiprimeto n, i.e.,
ϕ ()n=∈…{ k {1,2,,};(,) n kn* = 1.} Cohen introduced a trigonometric sum c* (,) mn for the case of unitary divisors, mimicking the Ramanujan's sum cmn(,)in the ordinary case, by c* (,) m n= ∑ e ( mx ,) n (,)xn* = 1 where e( m , n )= exp(2π im / n ).He proved that nif nm | ∑c* (,) md = dn|| 0 ifnm . Furthermore, he defined ϕ = * and µ = * (nc ) (0, n ) (ncn ) (1, ). Cohen gave the following interesting consequences: ∑∑ϕµ(d )= n , ( d )= In ( ), c* ( mn , )= ∑ µ ( d )( n / d ), dn|| dn|| dn|| ω()n ϕ()nn= ζµ1 (), µ () n= (1) − whereζ1()nn= ,and for fg,,∈ the unitary Mӧbius inversion holds, namely, f() n= ∑∑ gd ()← → gn ()= µ () d f ( n / d ). dn|| dn|| He also established a number of analytical estimates. If fn()is defined by fn()= ( g hn )()for gh, ∈ and if gn()is bounded, then for real number x ≥ 2, we have x2 ∞ gn()()ϕ n = + 22 ∑∑f() n 3 Ox( log x ). 2 = n nx n1
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2.2 The work of Cohen in 1961 [3] Let ξη, ∈ denote functions satisfying ∑ ξ(d ) ηδ ( )= In ( ). dnδ = (d ,δ )1= Cohen proved fn()= ∑∑ξ ()() dg δ ←→ gn ()= fd ()(),ηδ dnδδ= dn= (dd ,δδ )1= ( , )1= as well as the generalized unitary inversion formula
f() n= ∑∑ g ()δ← → gn ()= µδ () d f () dnkkδδ= dn= (dd ,δδ )1= ( , )1= where k ∈ . Cohen applied the unitary convolution and the generalized unitary inversion formula to treat some asymptotic problems involving the distribution of certain sets of integers.
2.3 The work of Cohen in 1964 [4] A positive integer n is said to be k-free if it has no prime factor of multiplicity ≥ k and k-full if it has no prime factor of multiplicity< k. The set of k-full and k-free integers are denoted by Qk and
Lk respectively. Their characteristic functions are denoted by qnk ()and lk (n ), respectively. Cohen * defined the generalized Mӧbius function, µk , to be the multiplicative function such that −< * e 1, if 1 ek µk ()p = 0, if ek for any prime p and e∈ . Cohen proved that for all k ≥1, **n ∑∑µµk()d= l k () n ←→ kk()dq= In () dn|| dn|| d and for fg,,∈ the following inversion formula holds
nn* gn()= ∑∑ f () dqkk←→ f() n= gd ()µ . dn|| dddn|| * Further, Cohen defined ϕk ()n to be the number of integers m in a complete residue system mod n which are semiprime to the maximal unitary divisor of n contained in Qk . He proved **nn* ϕµk()n= ∑∑ d k← → ϕkk(),dq= n dn|| dddn|| and x2 ∞ µϕ()()nn ϕ *= k + 22 ∑∑k ()n 3 Ox( log x ). nx 0 n=1 n
2.4 The work of Horadam [5] ** Horadam defined generalized integers andϕ () n as follows: suppose there is given a finite or infinite sequence {p}of real numbers (generalized primes) such that 1 {} of all possible p-products; these numbers are called generalized integers. Let { n}be the set ** of generalized integers. Defineϕ () n to be the number of generalized integers contained in { n} which are semiprimeto n . Horadam proved: 3 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 • ∑ ϕ** ()dn= ,for all n ∈; ddδδ=n ,( , ) = 1 • if fh(),()nnare multiplicative, then the unitary convolution of g() n and h() n is also multiplicative; • a unitary inversion formula: if G( n )= ∑ Fd ( ), dδ =n (d ,)1δ = then F( n )= ∑ µδ (dG ) ( ). dδ =n (d ,)1δ = 2.5 The work of Rao [6] Rao (see also [2]) gave the following extension of Cohen's totient. For positive integers nmk, , , k * k let (,nm )k denote the largest unitary divisor of m that divides n and is a k th power. Denote by * k k * ϕk ()m the number of integers n in a complete residue system mod m such that (nm , )k = 1. Rao proved that 1 ϕ* ()m= dkk µ ( md /)= ζµ () m = m 1 − , kk∑∏ kmν p () dm|| pn| p k whereζ k (nn ):= and ν p ()m denotes the highest power of the prime p that divides m.This result * implies at once that ϕk is a multiplicative function. Let dn()denote the number of unitary divisors of n, and let σ k ()n denote the sum of the k th power of the unitary divisors of n (see also [7]), i.e., ω ()nk dn()=∑∑ 1 = 2 ,σζkk ()n= d = Un(). dn|| dn|| Rao also established the identities * **kk * σk()n= ∑ ϕk (/)(), nddd ∑ ϕ (/ ndd11 )= ∑∑ ϕk (/ nd2 ) d 2 , σϕsk+ () d k (/) nd= n σs (). n dn|| d12|| n d|| n dn|| 2.6 The work of Rearick [8] Let be the set of all f ∈ such that f (1) is a positive real number. Rearick proved that the groups { ,*},{ ,*},{ , },{ , } and { ,}+ are isomorphic. 2.7 The work of Hansen and Swanson [9] Hansen and Swanson defined the following concepts. 1. The greatest common unitary divisor (gcud) of positive integers ab, , written as gcud (,)ab , is the integer d ∈ such that d|| ad, | | b, and if c|| ac, | | b, then cd|| . 2. The least common unitary divisor (lcud) of positive integers ab, ,written as lcud (,)ab, ab is the integer d defined by d = . gcud(ab , ) 3. Any positive integers are unitarily relatively prime if gcud = 1. ab, (,)ab 4. A positive integer n is a unitary perfect number if the sum of all the unitary divisors of n is 2n . 4 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 2.8 The work of Johnson [10] * Johnson studied the multiplicative properties of snk ( ), defined for fg,,∈ by * sk ( n )= ∑ f ( dgk ) ( / d ). d||( nk , )* He proved that for fg,,∈ * ** 1. smk ( ab )= sm () a s k () b whenever (ak,) =( bm ,) = 1; ** 2. sabsamm( )= () whenever (bm,) = 1; ** 3. smk() a= s m ()() agkwhenever (ak,) = 1 ; * 4. if fg, ∈, then snk ()is multiplicative in k for each fixed n. 2.9 The work of Hsu [11] Hsu extended the Mӧbius in version formula based on the prime factorization of integers. Let x1 xs t1 ts np= 1 ps and dp= 1 ps , with pi being distinct primes, xi and ti being nonnegative integers satisfying 0.≤≤txiiReplacing fn()and gd()in the M bius inversion formula by = … and = … respectively. Hsu proved the following theorems: f(( x )) fx (1 , , xs ) g(( t )) gt (1 , , ts ), ӧ Theorem 2.1 For fg,,∈ we have f(,,) x11…= xss∑ gt (,,) … t 0≤≤txii if and only if gx(,,)1… xs =∑ f ( x1 −… t 1 ,, xss − t )(,,),µ11 t … t s 0≤≤txii where ++ −≤tt1 s ( 1) if all ti 1 µ11(,tt…= ,s ) 0 if there is a ti ≥ 2. Hsu also gave a M bius inversion formula in vector form. Let (xt )−≡ () ( x1 , … , xs )with ≡ … ≡…and ≤≤ i.e., ≤≤for all i. ()(,,),()(,,)x x11 xtssӧ t t (0)()(),tx 0 txii Theorem 2.2 For any given ()rrr≡… (,1 ,s )with ri ∈, we have −1 f(( x )) = ∑ µ()r ((tgx )) (( )− ( t )) (0)()()≤≤tx if and only if gx(( )) = ∑ µ()r ((tf )) (( x )− ( t )) (0)()()≤≤tx −1 where µ()r ((t )) and µ()r ((t )) are defined by ssr tr+−1 i ti −1 ii µµ()rr((tt ))=−=∏∏ ( 1) ,() (( )) . tr−1 ii=11ii= 2.10 The work of Hsu and Wang [12] In 1998, Hsu and Wang stated that Theorem 2.2 is also true when each positive integer rii (= 0, … , s ) is replaced by a real number. They defined a generalized Mӧbius function by 5 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 α ν p ()n µα ()n = ∏(− 1) pn| ν p ()n where α ∈. Note that µ10= µµ,, = I and µα is a multiplicative function. It is not difficult to verify that µµµα* β= αβ+ for complex numbers αβ,. They stated the following generalized Mӧbius inversion formula. Theorem 2.3 For fg,,,∈∈ α we have nn f() n= ∑∑ gd ()µµαα← →gn() = f () d − . dn||dddn They also extended the generalized Mӧbius function by replacing the arguments with some arithmetic functions. For α ∈ , define α()p µ = − vnp () () α () n ∏ ( 1) . (2.1) pn| vnp () Note that µµ()α* () β ()nn= µ (αβ+ ) ()for arithmetic functionsαβµ,,(0) = I and is a multiplicative function. The Mӧbius inversion formula still holds when µα and µ−α are replaced by µ()α and µ()−α , respectively. They also considered the M bius function in two variables. For α ∈∈,,z α(p )+ zvp () n α()p ӧ vnp () µ (,)nz = (− 1) . ()α ∏α + pn| (p ) zvp () n vnp () This reduces to (2.1) when z = 0 . Note that (µµ()α * () β )(,)nz= µ()αβ+ (,) nz for arithmetic functions αβ, and µ()α (,)nz is a multiplicative function. Theorem 2.3 is also true when µα ()n is replaced by µ()α (,)nz , i.e., for fg,,∈ we have fn()= ∑∑µµ()αα ( ndzgd / ,)()←→ gn ()= ()− ( ndzfd / ,) (). dn||dn Moreover, Theorem 2.2 still holds when each positive integer ri is replaced by an arithmetic function αi (is= 1, … , ). Theorem 2.4 If ()αα= (1 , … , αs )with all αi ∈ then f(( x )) = ∑ µ()α ((t ), zg ) (( x )− ( t )) (0)()()≤≤tx if and only if g(( x )) = ∑ µ()−α ((t ), zf ) (( x )− ( t )), (0)()()≤≤tx where µ()α ((tz ), ) is defined by s α αii+ zt i ti µ()α ((tz ), ) = ∏ ( − 1) . (2.2) i=1 αii+ zt ti and µ()−α ((tz ), ) is taken from (2.2) with ()α being replaced by (−αα ) ≡ ( −1 , …− , αs ). 2.11 The work of Schinzel [13] In 1998, Schinzelderived an explicit form of the unitary inverse of f ∈ , with f (1)= 1 , as ω()n k −−11k f() n=∑∑( − 1)∏ fd (i ) ( n >= 1), f (1) 1. k=1 ddn1 k = i=1 (ddij , )= 1, d i > 1 6 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 2.12 The work of Buschman [14] In 2003, Buschman considered the unitary M bius function by defining µ= µµ µ(k ∈ ), k ӧ k factors with µ=IU,, µ = µµµ = µ (k factors), and proved that 0− 1 −−−k 11 − 1 fg= µµ−kk← → gf= , which is an extension of the unitary Mӧbius inversion formula. He also derived the identities µσkk= ζ,, ϕ U = σ ζk σ mm= ζ σ k . 3. Main Results 3.1 Generalized unitary Mӧbius inversion formula In this section, we prove two forms of the generalized unitary M bius inversion formula. Lemma 3.1 For αβ, ∈ , we have µµµα β= αβ+ . ӧ Proof. When n =1, we have 1 ∑ µα()d µ β= µµα(1) β (1) = 1 = µαβ+ (1). d||1 d a1 as In general, when the prime factorization of n is np=1 … ps , we have a1asaa12 aa ssa1 µµαβ ()n = µµ αβ (1)( pp1s )+ µα ( p12 )( µ β pp ss )++µα ( pp1 )( µβ1) ss s s0 s11 s− s0 ss = (− 1) αβ ++ αβ+ αβ =−+=( 1) ( α β ) µαβ+ (n ). 01 s Our first main theorem is: Theorem 3.2 A) The set of all generalized unitary M bius functions is an abelian group under the unitary convolution with the identity element I = µ0 . ӧ B) (Generalized unitary Mӧbius inversion formula) Let fg, ∈ andα ∈ {0} . Then n f() n = ∑ gd () µ−α (3.1) dn|| d if and only if n gn() = ∑ fµ α ( d ), (3.2) dn|| d i.e., fg= µµ−αα← → gf= . Proof. If fg= µ−α , then Lemma 3.1 shows that fµµµµαααα= g − = g −+α = gIg = . Conversely, if gf= µα , then gµ−α= f µµα −− α= f µαα = fIf = . Let the prime factorization of n and its divisor d be xt11xtss np= 11 pss,, dp= p where xi∈, 0 ≤≤ t ii xi ( =… 1, , s ) , and pp12< << ps are primes. Writing (x )≡ ( x1 , … , xs ), ( t ) ≡ ( t1 , … , ts ), (( x ) − ( t )) ≡ ( xt11 −… , , xtss − ), 7 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 The generalized unitary M bius inversion formula becomes s Theorem 3.3 For ()(,αα≡…∈1 , αs )( {0}) , we have ӧ f(( x ))= ∑ gt (( )) µ()−α (( x )− ( t )) (3.3) ()t x if and only if gx(( ))= ∑ f (( x ) − ( t ))µ()α (( t )), (3.4) ()t x s s ω xtii− where (t )=…∈ ( tt , , ) {0, x } and ()pi Equivalently, x1 siµα()−α ((xt )−= ( )) ∏ i . i=1 f(( x ))= ( gµµ()αα )(( x ))← → gx (( )) = ( f()− )(( x )). To prove Theorem 3.3, we need the following lemma. Lemma 3.4 For α ∈∈{0}, x and ux∈{0, }, we have ωωvu−−xv 1 if u= x ∑ αα()pp()−= () vx∈{0, } 0if u = 0. Proof. The result easily follows from ω()p0 ωωvu−−xv α =1 if u = x αα()pp()−= () ∑ ωω()ppxx ()0 ω () p vx∈{0, } ()−αα +== α 0if u 0. We return now to the proof of Theorem 3.3. If then fx(( ))= ( g µ()−α )(( x )), ∑∑f(( x )− ( t ))µµ()αα (( t )) = fx (1 −… t 1 , , xss − t )() ( t 1 , … , ts ) ()tx txii∈{0, } =∑∑gu(,,)1… usµµ()αα (,,) t 1 … ts (− ) ( x 1 −− t 1 u 1 ,, … xss −− t u s ) ∈ ∈− tiiiii{0, x } u {0, xt } ωωvu11−−xv11 ωω()ppvuss−−ss ()xv ()pp11 ()… ααss− =…−∑∑gu(1 , , us ){ αα 11( ) } {∑ ss () } uxii∈∈{0, } vx11{0, } vxs∈{0,s } =gx(,1 … , xs ), by Lemma 3.4.The converse implication is proved by retreating the above steps. Next, we verify two identities related to generalized unitary M bius function. Proposition 3.5 A. Let αβ, ∈ . If f ∈, then ν ӧpn() (µαβf µ )( n )=∏ ( −− βαfp ( )). pn| B. For αγ∈∈, and n∈ , we have γν pn() γ (µζαγ )(n )=−=∏ ( p α ) ( ζγ (nn ) : ). pn| Proof. A. For n =1, we have (µµαβff )(1)= µµ αβ (1) (1)= 1. ∈ a1 as Let gf= µµαβ , so that g .Let np= 1 ps be its prime factorization. Clearly, pai ai i ai aa ii ai gp()i=∑ µα ()() dfd µ β= µ αβα (1)(1)()f µ pi+ µ ()()(1) p ii fp µ β=−− βαfp (),i ai d dp|| i and so s ν ()n ai p gn( )=∏∏ gp (i ) = ( −−βαf ( p )). i=1|pn B. For n =1, we have 8 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 (µζαγ )(1)= µζ αγ (1) (1)= 1. Putting g =µζαγ ∈,and evaluating at prime powers, we get ai p γ ai i aaii ai gp()i=∑ ζγ () d µ α= ζµ γα (1)()pii+ ζ γ ()(1) p µ α =−+ αpi , ai d dp|| i yielding γν gn( )=∏ ( p p ()n −α ). pn| 3.2 Multiplicative function and applications In this section, characterizations of multiplicative functions using unitary products are derived. We start with an auxiliary result. Lemma 3.6 For nn∈>,1, and α ∈ , there are only two values of α (namely 0 and 1) for which the expression (1−αα )nn −−− 1 ( ) can vanish. Proof. Putting F(αα ) := (1 − )nn −−− 1 ( α ) , we see that F(0)= 0 and 0 if n is odd F(1)= 0 −−− 1 ( 1) n = −2.if n is even The asserted result follows from the following claims. Claim 3.7 If α < 0, then F()0α > . Proof of Claim 3.7 If α < 0, then nn 2 nnn F()1α=+−+−++−−−− () αα () ()1()0. αα> 12 n Claim 3.8 For α ∈ {0}, if F()α = 0, then F(1 /α )= 0 . Proof of Claim 3.8 If F()α = 0, then nnn 1 1 1 (− 1) FF=1 − −−− 1 =(α ) = 0. α α αα n Claim 3.9 If α belongs to the open interval (0,1), then F(α )≠ 0. Proof of Claim 3.9 If α ∈(0,1), then F()(1)1()(1)1()α=− αnn −−− α <− α −−− αn =−−− αα () n < 0, because n >1. We now present our characterizations of multiplicative functions. Theorem 3.10 Let α ∈ and f ∈. 1. If ∈ , then f µµααff = −1 f. 2. If ≠∉α and , then ∈ f (1) 0, {0,1} µµααff = −1 f f . Proof. 1. If f ∈, then n (µαf f )() n = ∑∑ µ α ()() dfdf= fn ()µµαα () d= fn ()−1 (). n dn|| d dn|| 2. For n=1, we have (µµααff )(1)= −1 (1) f (1), aa12 using f (1)≠ 0 and α ∉{0, 1} . For n= pp12with distinct primes pp12, , from aa12 aa 12 aa12 µµαα−112()()()()pp fpp 12= f f pp12 9 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 we get 2 aa12 aa12 a1 a 1 a 2 a2 a 2 a 1 (1)(−=+αfpp12 ) fpp ( 12 ) µµαα ()()() p 1 fp 1 fp 2 + ()()() p 2 fp 2 fp 1 aa12 aa 12 + µα ()().pp12 fpp 12 aa12 a1 a 2 Simplifying and using α ∉{0, 1} , we deduce that fpp(12 )= fp ( 1 )( fp 2 ). Assume that aa11aakk fp(11 pkk )= fp ( ) fp ( ) ( k= 2, 3, …− , s 1; s ≥ 3) for distinct primes pp1,,… k . Let pp1,,… s be distinct and aa1,,…∈s . From ppa1 as a1 as 1 s µµαα−11f( p ps )= ∑ ()()dfdf, a1 as d dp|| 1 ps expanding the first and the last terms on the right hand side, we arrive at a1 as s a1as aa 11 aa ss pp1 s (1−=αµ )fp (1 ps )αα f (1) fp ( 11 p ss ) + µ fp ( p ) f (1) +∑ µαfdf() . a1 as d dp|| 1 ps a1 as d∉{1, pp1 s } Using the induction hypothesis and simplifying, we get ssaa11aassss {(1−αα ) −−− 1 ( ) )} fp (11 pss )={ (1 −αα ) −−− 1 ( ) )} fp ( ) fp ( ). aa11aass Sinceα ∉{0, 1} , Lemma 3.6.shows that fp(11 pss )= fp () fp (). Hence f is multiplicative. The conditions imposed in Theorem 3.10 are analyzed in the next theorem. Theorem 3.11 Let f ∈ , and α ∈ . A. Assume that = , and that f (1) 0 µµααff = −1 f A1. If α ≠ 1, then f ≡ 0. A2. If α = then there are infinitely many satisfying 1, f µµααff = −1 f. B. If = , then f (1) 1 µµ01ff = − f. a Proof. A1.Evaluating at pa,,∈ p prime, we get aa a µµαα−1(p )( f p )= ∑ ()() dfdf( p / d) , dp|| a a a1 as which immediately implies fp( )= 0.Next, evaluating at np= 1 ps for distinct primes pi and ai ∈ , we have ppa1 as a1 as 1 s µµαα−11f( p ps )= ∑ ()()dfdf, a1 as d dp|| 1 ps s a1 as a1 as which by induction shows that (1−=α )fp (1 ps ) 0, and so fp(1 ps )= 0. A2. Let n0 ∈ {1} , a ∈ {0}. Consider f ∈ defined by a if n= n fn()= 0 0.if n≠ n0 Then f (1)= 0. For n∈ , whether n is a unitary divisor of n or not, we find that 0 n (µµf f )( n ) = ∑ fdf ( ) =0 = µ0 fn( ). dn|| d Since there are infinitely many such arithmetic functions f , the desired assertion follows. B. If f (1)= 1 , then µµ−10fUffIfIff= = = = = f f. 10 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 Our next application deals with the concept of discriminative products.For g , h∈ , a unitary product kgh= is said to be discriminative if kn( ) = gnh( ) ( 11) + g( ) hn( ) holds only when ω()n = 1. A unitary product kgh= is said to be semi-discriminative if kn( ) = gnh( ) ( 11) + g( ) hn( ) holds only when ω()n = 1 or 1. Using these two concepts, we have the following characterizations of multiplicative functions. Theorem 3.12 Let f ∈. 1. If f ∈, then f distributes over any unitary discriminative product, i.e., f() g h= fg fh for all gh,.∈ 2. If f (1)≠ 0 and f distributes over some unitary discriminative product k , then f ∈. 3. If f (1)= 1 and f distributes over some unitary semi-discriminative product k , then f ∈. Proof. 1. Assume that f ∈. Then for all gh,,∈∈ n , we have n nn f( g h )() n = f () n∑∑ g () d h= g ()() d f d h f= ( fg fh )(. n) dn|| ddn|| dd 2. Assume that f(1)≠= 0, kgh is a unitary discriminative product, and fk= fg fh . For n =1, we have f (1)= 1 .Next, we show that aa11aass fp(11 pss )= fp () fp () (3.5) for s ∈ , all distinct primes pp1,,… s , and aa1,,…∈s . Clearly, (3.5) holds for s =1. Let pp12, , be distinct primes and aa12, ∈ . Then ppaa12 aa12 aa12 12 fk()()()() p12 p= fg fh p12 p= ∑ f g d fh . aa12 d dp|| 12 p Thus, aa12 aa12 a1 a 2 pp12 0= {fpp (12 ) − fp ( 1 ) fp ( 2 )}∑ gdh ( ) aa12 d dp|| 12 p aa12 d∉{1, pp12 } aa12 a1 a 2 aa12 aa12 aa12 =−{()()()}{()((1)()()(1))}.fpp12 fp 1 fp 2 kpp 12 −+ g hpp12 gpp 12 h Since k is unitary discriminative, (3.5) holds for s = 2 .Let s ≥ 3 , and assume that (3.5) holds for all positive integers whose number of distinct prime factors is less than s. Let pp1,,… s be distinct primes and aa1,,…∈s . Then ppa1 as aa11aass1 s fk()()() p11 pss = fg fh p p = ∑ f g () d fh a1 as d dp|| 1 ps a1 as a1 as pp1 s = fp(1 ) fp (s )∑ gdh () a1 as d dp|| 1 ps a1 as dpp∉{1,1 s } aa11aass a1as a 1 a s ++fppgpphghppfpp()()(1)(1)()().11ss 1 s 1 s 11 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 Thus, ppa1 as aa11aass 1 s 0= {fp (11 pss )− fp ( ) fp ( )}∑ gdh ( ) a1 as d dp|| 1 ps a1 as dpp∉{1,1 s } aa11aass =−×{fp (11 pss ) fp ( ) fp ( )} a1as aa 11 aa ss {(ghppghppgpph )(1s )−+ ( (1) ( 11 ss ) ( ) (1))} aa1as 1 aa ss a 1 aa 11 aa s s =−{()()()}{()((1)()()(1))}.fppfpfpkppghppgpph11s ss 1 −+ 11 ss Since k is discriminative, (3.5) holds for s ∈ . Hence, f is multiplicative. The proof of Part 3 is similar to that of Part 2. References [1] Sivaramakrishnan, R., 1989. Classical Theory of Arithmetic Functions. Marcel Dekker. New York-Basel. [2] Cohen, E., 1960. Arithmetical functions associated with the unitary divisors of aninteger. Math. Zeitschr., 74, 66-80. [3] Cohen, E., 1961. Unitary products of arithmetical functions. Acta Arith., 7, 29-38. [4] Cohen, E., 1964. Arithmetical notes. X. A class of to tients. Proc. Amer. Math. Soc., 15(4), 534-539. [5] Horadam, E.M., 1962. Arithmetical function associated with the unitary divisors of a generalized integer. Amer. Math. Monthly, 69, 196-199. [6] Rao, K.N., 1965. On the unitary an alogues of certain totients. Monatsh. Math., 70, 149-154. [7] Vaidyanathaswamy, R., 1931. The theory of multiplicative arithmetic functions. Trans. Amer. Math. Soc., 33, 579-662. [8] Rearick, D., 1968. Operators on algebras of arithmetic functions. Duke. Math. J., 35, 761-766. [9] Hansen, R.T. and Swanson L.G., 1979. Unitary divisors. Math. Magazine, 52, 217-222. [10] Johnson, K.R., 1982. Unitary an alog of generalized Ramanujan sums. Pacific J. Math., 103, 429-432. [11] Hsu, L.C., 1995. A difference-operational approach to the M bius inversion formulas. Fibonacci Quarterly, 33, 169-173. [12] Hsu, L.C. and Wang J., 1998. Some M bius-type functions and ӧinversions constructed via difference operators. Tamkang J. Math., 29, 89-99. [13] Schinzel, A., 1998. A property of unitary convolution.ӧ Colloq. Math., 78, 93-96. [14] Buschman, R.G., 2003. Unitary product again, StudiaUniv. Babes-Bolyai, Mathematica, 48, 29-34. 12