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] 1 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 ω()n µαα ()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 2 KMITL Sci. Tech. J. Vol.17 No.1 Jan.-Jun. 2017 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 −<1, if 1 ek * e 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<pp12 < <…. Form the set {} 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 ++ (−≤ 1) tt1 s if all t 1 i µ11(,tt…= ,s ) 0 if there is a ti ≥ 2.