PII. S0161171204304163 http://ijmms.hindawi.com © Hindawi Publishing Corp.
COMPLETELY MULTIPLICATIVE FUNCTIONS ARISING
FROM SIMPLE OPERATIONS
VICHIAN LAOHAKOSOL and NITTIYA PABHAPOTE Received 13 April 2003
Given two multiplicative arithmetic functions, various conditions for their convolution, pow-ers, and logarithms to be completely multiplicative, based on values at the primes, are de-rived together with their applications.
2000 Mathematics Subject Classification: 11A25.
1. Introduction. By an arithmetic function we mean a complex-valued function whose domain is the set of positive integersN. We define the addition and convolu-tion of two arithmetic funcconvolu-tionsfandg, respectively, by
(f+g)(n)=f (n)+g(n), (f∗g)(n)=
ij=n
f (i)g(j). (1.1)
It is well known (see, e.g., [2,6,15,16]) that the set(Ꮽ,+,∗)of all arithmetic functions is a unique factorization domain with the arithmetic function
I(n)=
1 ifn=1,
0 otherwise, (1.2)
being its convolution identity.
A nonzero arithmetic functionf∈Ꮽis calledmultiplicativeif and only if
f (mn)=f (m)f (n) whenever(m, n)=1. (1.3)
It is calledcompletely multiplicativeif this equality holds for allm, n∈N.
Denote byᏹthe set of all multiplicative functions and byᏯthe set of all completely multiplicative functions. It is easy to see that the elements ofᏹare uniquely determined via their values at all prime powers, while those ofᏯare uniquely determined only via their values at the primes. Unlikeᏹ, which is a group with respect to convolution,Ꮿis closed neither with respect to addition nor convolution.
A natural problem is that of characterizingᏹandᏯby various means and there have appeared a number of such investigations, for example, by Apostol [1,2], Carroll [5], Laohakosol et al. [10,11], and Rearick [12,13].
Though the problem seems elementary, we have not been able to locate its complete solution. As we will see, several of the techniques used to resolve certain cases of this problem are not elementary, that is, making use of logarithmic operators (see [10,11, 13]).
The question about sum is trivial because by looking at the values atn=1 we see immediately that a sum of two multiplicative functions is never multiplicative.
To facilitate later discussions, we recall some basic facts about logarithmic operators onᏭ. Forf∈Ꮽ, its Rearick logarithm, denoted by Logf∈Ꮽ, is defined by
(Logf )(1)=logf (1) assumingf (1) >0,
(Logf )(n)=log1n
d|n
f (d)f−1 n
d
logd
=log1ndf∗f−1(n) (n >1),
(1.4)
wheredf (n)=f (n)logndenotes the log-derivation off.
In 1973, Carlitz and Subbarao [4] defined another logarithmic operator off, denoted byβf∈Ꮽ, as follows:
(βf )(1)=1 assumingf (1)=1,
(βf )(n)= ∞
r=1
(−1)r−1 r
a1···ar=n ai=1(i=1,...,r )
fa1
···far (n >1). (1.5)
As proved in [10], these two logarithms are
(i) essentially identical, that is, both take the same values except only atn=1, (ii) both bijective,
(iii) both transform convolution into sum.
Indeed, as shown by Carlitz and Subbarao [4], the inverse (with respect to convolution) ofβis the operatorγ:ᐁ1→ᐁ1, whereᐁ1is the set of all arithmetic functionsf such thatf (1)=1, defined by
(γg)(1)=1,
(γg)(n)= ∞
r=1 1 r!
a1···ar=n ai=1(i=1,...,r )
ga1
···gar (n >1). (1.6)
2. Convolution. We first look at the case of convolution.
Theorem2.1. Letf , g∈ᏹ. Thenf∗g∈Ꮿeither
gpa=f−1pa+f (p)+g(p)f−1pa−1
or
fpa=g−1pa+f (p)+g(p)g−1pa−1
+···+f (p)+g(p)a−1g−1(p)+f (p)+g(p)a (2.2)
for all primespand alla∈N.
Proof. Writingg=(f∗g)∗f−1and evaluating at prime powers, we get
gpa=f−1pa(f∗g)(1)+f−1pa−1(f∗g)(p)
+···+f−1(1)(f∗g)pa. (2.3)
Iff∗g∈Ꮿ, then(f∗g)(pi)=(f∗g)(p)i=(f (p)+g(p))i,and (2.1) results immedi-ately. Further, (2.2) follows by interchangingfandgin the above arguments.
On the other hand, if (2.1) holds, then comparing (2.1) and (2.3) for successive values ofa=1,2,3, . . ., we conclude that
(f∗g)pa=(f∗g)(p)a, (2.4)
and so
(f∗g)∈Ꮿ. (2.5)
The same result holds similarly for (2.2).
Two remarkable consequences are now in order.
Corollary2.2. Letf∈Ꮿandg∈ᏹ. Then
f∗g∈Ꮿ⇐⇒gpa=g(p)g(p)+f (p)a−1
(2.6)
for all primespand alla∈N.
Proof. This is immediate from (2.1) inTheorem 2.1, noting thatf−1(p)= −f (p) and
f∈Ꮿ⇐⇒f−1pi=0 ∀i≥2. (2.7)
Corollary2.3. Letf , g∈Ꮿ. Then
f∗g∈Ꮿ⇐⇒f (p)g(p)=0 (2.8)
for all primesp.
Proof. FromCorollary 2.2, we have
f∗g∈Ꮿ ⇒gp2=g(p)g(p)+f (p)
while
f (p)g(p)=0⇒gpa=g(p)g(p)+f (p)a−1
⇒f∗g∈Ꮿ. (2.10)
3. Powers. We first recall some of the results in [13]. Forh∈Ꮽ, the (Rearick) expo-nential Exphis defined as the unique element
f∈Ꮽ, f (1) >0, such thath=Logf . (3.1)
Forf∈Ꮽ,f (1) >0, andr∈R, therth power functionfr is defined as
fr=Exp(rLogf ). (3.2)
From [12], we know that iff∈ᏹ,r∈R, thenfr∈ᏹ, which also implies that iffr∈ᏹ, r∈R\{0}, is multiplicative, thenfis also multiplicative.
Recall also the Hsu’s generalized Möbius function (see [3,8,11])
µα(n)=
p|n
α νp(n)
(−1)νp(n), (3.3)
whereα∈Randνp(n)is the highest power of the primepdividingn.
Theorem3.1. Letf∈ᏹandr∈R\{0}. Then
fr∈Ꮿ⇐⇒fpa= −1r
a
(−r )af (p)a (3.4)
for all primespand alla∈N.
Proof. Assumefr ∈Ꮿ. By Haukkanen’s theorem [8] and the property of power function, we have
µ−1/rfr=
fr1/r=f , (3.5)
so that
fpa=µ
−1/rpafrpa=
−r1
a
(−1)afr(p)a. (3.6)
By the result of Carroll [5], see also [10], forfr∈Ꮿ, we have
fr(p)=Logfr(p)=rLogf (p)
=logrpf (1)f−1(p)log 1+f (p)f−1(1)logp=r f (p), (3.7)
Conversely, assuming (3.4), to showfr∈Ꮿ, we use a characterization of Apostol [1], namely,
fr∈Ꮿ⇐⇒f−rpa=0 (3.8)
for all primespand all integersa≥2.
Writingg=f−r, then Logg= −rLogf. We will show thatg(pa)=0 for alla≥0. Since
(Logf )(n)=log1ndf∗f−1(n) (n >1), (3.9)
then
f∗dg= −r (g∗df ). (3.10)
Thus
ij=pa
f (i)dg(j)= −r ij=pa
g(i)df (j). (3.11)
Takinga=1, we getg(p)= −r f (p).
Takinga=2 and usingg(p)= −r f (p), we get
2gp2= −r2fp2−(r+1)f (p)2=0. (3.12)
By induction, using (3.4), fora≥2, we deduce thatg(pa)=0.
Remark3.2. An alternative proof of the “only if” part ofTheorem 3.1can be done as follows. Assuming (3.4), to showfr ∈Ꮿ, we use a characterization of Carroll [5], namely, forf∈ᏹ,f∈Ꮿif and only if
(Logf )(n)=
f (p)a
a ifn=p
a, pprime, a≥1,
0 otherwise.
(3.13)
From the hypothesis and using induction ona, we easily verify that
f−1pa=
r1
a
(−r )af (p)a (3.14)
for all primespand alla∈N.
Next we want to show thatr (Logf )(pa)=(r f (p))a/a. From the definition of Rearick logarithm Logf, we have
(Logf )pa= 1 a
f (p)f−1pa−1+2fp2f−1pa−2
+···+(a−1)fpa−1f−1(p)+afpa,
and so, using (3.14), we get
(Logf )pa=raaf (p)a
a
i=1 i
−1r
i
1r
a−i
(−1)a
. (3.16)
Using Riordan [14, identity (3d), page 10], we deduce that
(Logf )pa=
r f (p)a
a ·
1
r, (3.17)
and the desired result follows.
Corollary3.3. Letf∈Ꮿandr∈R\{0,1}. Then
fr∈Ꮿ⇐⇒f=I. (3.18)
Proof. For all primespand alla∈N, by Haukkanen’s theorem [8] andTheorem 3.1, we have
fr∈Ꮿ⇒µ
−rfpa=frpa=fr(p)a
⇒
r+a−1 a
fpa=r f (p)a
⇒
r+a−1 a
−ra
f (p)a=0.
(3.19)
Takinga=2, sincer=0,1, we getf (p)=0 for all primesp. Being in Ꮿ, this yields f=I.
Iff=I∈Ꮿ, by Haukkanen’s theorem [8], we get
Ir(n)=µ−rI(n)=µ−r(n)I(n)=
1 ifn=1,
0 otherwise, (3.20)
and soIr∈Ꮿ.
Remark3.4. Corollary 3.3is false if the assumptionf∈Ꮿis dropped as seen from the following two examples.
(1) Takef as the Möbius functionµ=Iandr= −1. We know that (see [2])µ−1is the unit functionu,u(n)=1(n≥1), that is,µ−1∈Ꮿ.
(2) Takef = |µ| =µ2=I andr = −1. Hence, |µ|−1 is the Liouville functionλ∈Ꮿ (see [2]).
4. Logarithms. Since any two of the three known logarithms are the same except at n=1, then we may use any of them whichever appropriate. We start by observing a triv-ial fact that iffis multiplicative, then its Rearick logarithm Logf is not multiplicative because(Logf )(1)=logf (1)=0.
Theorem4.1. Letf∈Ꮽwithf (1)=1. Then its Carlitz-Subbarao logarithmβf ∈
Ꮿfor distinct primesp1, . . . , pkand positive integersn1, . . . , nk, there holds
fpn1 1 ···p
nk
k
=αn1, . . . , nk
fp1
n1···fpknk, (4.1)
where
αn1, . . . , nk= ∞
r=1 1 r!Ir
n1, . . . , nk, (4.2)
Ir(n1, . . . , nk)=∗1;∗ denotes the sum taken over nonnegative integersn11, . . . , nr1; . . .;n1k, . . . , nr ksuch that
n11+···+nr1=n1, . . . , n1k+···+nr k=nk,
n11+···+n1k>0, . . . , nr1+···+nr k>0.
(4.3)
Proof. Putg=βf. Theng(1)=βf (1)=1. Assumingg=βf ∈Ꮿ, by [4, Theorem 4.28 and equation (1.2)] of Carlitz and Subbarao, we see that form >1,
f (m)= ∞
r=1 1 r!
a1···ar=m
ai=1(i=1,...,r )
ga1
···gar=g(m)c(m), (4.4)
where
c(m)= ∞
r=1 1 r!
a1···ar=m
ai=1(i=1,...,r )
1. (4.5)
Writingmas a unique prime factorization,m=pn1 1 ···p
nk
k , we have
a1···ar=m
ai=1(i=1,...,r )
1= ∗
1. (4.6)
The sum∗ depends only onr,n1, . . . , nk, but is independent ofp1, . . . , pk; we call itIr(n1, . . . , nk). Thus the function
cpn1 1 ···p
nk k = ∞
r=1 1 r!Ir
n1, . . . , nk
(4.7)
depends only onrandn1, . . . , nk, but is independent ofp1, . . . , pk, so that we may rewrite it asα(n1, . . . , nk). Such a functionc(m)is known as prime-independent arithmetic function (see [9, page 33]). Sincef (p)=g(p)c(p)=g(p),then
fpn1 1 ···p
nk k
=αn1, . . . , nkgp1n1···gpknk =αn1, . . . , nkfp1n1···fpknk,
(4.8)
Conversely, fromg=βf, we get (see [4])f=γg; that is,
∞
r=1 1 r!
a1···ar=pn11···pnkk
ai=1(i=1,...,r ) ga1
···gar
=fpn1 1 ···p
nk
k
=αn1, . . . , nk
fp1
n1···fpknk.
(4.9)
Specializingk=1=n1, we getg(p)=f (p). Thus
∞
r=1 1 r!
a1···ar=pn11···pnkk
ai=1(i=1,...,r )
ga1···gar
=αn1, . . . , nk
gp1
n1···gpknk.
(4.10)
We now show thatg(pi)=g(p)ifor each primepandi∈N.
Takingk=1 andn1=2 in (4.10), observing thatα(2)=∞r=1(1/r!)Ir(2), we have
1 1!g
p2+2!1g(p)2=α(2)g(p)2= 1
1!+ 1 2!
g(p)2, (4.11)
that is,g(p2)=g(p)2.
Assumingg(pj)=g(p)jfor allj < i, and using induction in (4.10), we obtain
1 1!g
pi+g(p)i
∞
r=2 1
r!Ir(i)=α(i)g(p)
i. (4.12)
As observed earlier,α(i)is just the sum of the coefficients of theg’s on the left-hand side and this yieldsg(pi)=g(p)ias desired.
Next, we show thatg(piqj)=g(p)ig(q)jfor distinct primesp,qandi, j∈N. Taking k=2,p1=p,p2=q, andn1=n2=1 in (4.10), we get
α(1,1)g(p)g(q)=1!1g(pg)+g(p)g(q). (4.13)
Again sinceα(1,1)is the sum of the coefficients of theg’s on the right-hand side, we deduce thatg(p)g(q)=g(pq). Next, usingg(pi)=g(p)iand induction oni+j, we deduce thatg(piqj)=g(p)ig(q)j. Finally,g(pi1
1 ···p ik
k)=g(p1)i1···g(pk)ikfollows by induction onk.
5. Applications. Our first application is related to one of our earlier results in [11] which states that forf∈ᏹ,α∈R\{0,1}, iffα=µ
−αf andf satisfies condition (NE),
thenf∈Ꮿ. Here condition (NE) reads: ifαis a negative even integer, then
A similar result to the one above but without condition (NE) is the following proposition easily deduced from [11, Theorem 1.2].
Proposition5.1. Letf∈ᏹandα∈R\{0}. Then
f∈Ꮿ⇐⇒fαpa=µ
−α
pa1 αf
α(p)
a
(5.2)
for all primespand alla∈N, whereµαdenotes the Hsu’s generalized Möbius function.
Our second application is related to the notion of totients (see [7]). Recall thatf∈Ꮽis a totient if it is of the form
f=ft∗fv−1, (5.3)
whereft(integral component) andfv(inverse component) are both inᏯ. Those totients belonging toᏯare characterized in the next proposition which is an easy consequence of Corollaries2.2and2.3.
Proposition5.2. (i)Letf=ft∗fv−1be a totient. Then
f∈Ꮿ⇐⇒fv=Iorfv−1(p)+ft(p)=0 (5.4)
for all primesp.
(ii)Letf=ft∗f−1
v andg=gt∗g−v1be two totients. Then
f∗gis a totient ⇐⇒ft(p)gt(p)=0=fv(p)gv(p) (5.5)
for all primesp.
Another related concept is the following:f∈Ꮽis said to be arational arithmetical function of degree(r , s)(see, e.g., [7]) if it is of the form
f=g1∗g2∗···∗gr∗h−11∗h−21∗···∗h−s1, gi, hi∈Ꮿ. (5.6)
Again by Corollaries2.2and2.3, we have, for example, the following characterization which can be evidently generalized.
Proposition 5.3. (i) Letf = g1∗g2∗h−11 be a rational arithmetical function of
degree(2,1). Then
fis a totient ⇐⇒g1(p)g2(p)=0 (5.7)
for each primep.
(ii)Letf=g1∗h1−1∗h−21be a rational arithmetical function of degree(1,2). Then
f is a totient ⇐⇒h1(p)h2(p)=0 (5.8)
Our last application is related to the concept of specially multiplicative functions. Recall that an arithmetic function F is said to bespecially multiplicative(see [16]) if there is a completely multiplicative functionfAsuch that
F (m)F (n)=
d|(m,n) F
mn
d2
fA(d). (5.9)
It is also known that each specially multiplicative function is a product of two com-pletely multiplicative functions.
A straightforward consequence ofCorollary 2.3is the following result.
Proposition5.4. LetF=f∗gbe a specially multiplicative function withf , g∈Ꮿ.
Then
F∈Ꮿ⇐⇒f (p)g(p)=0 (5.10)
for all primesp.
Acknowledgment. This work was partially supported by The University of the Thai Chamber of Commerce.
References
[1] T. M. Apostol,Some properties of completely multiplicative arithmetical functions, Amer. Math. Monthly78(1971), 266–271.
[2] ,Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976.
[3] T. C. Brown, L. C. Hsu, J. Wang, and P. J.-S. Shiue,On a certain kind of generalized number-theoretical Möbius function, Math. Sci.25(2000), no. 2, 72–77.
[4] L. Carlitz and M. V. Subbarao,Transformation of arithmetic functions, Duke Math. J.40 (1973), 949–958.
[5] T. B. Carroll,A characterization of completely multiplicative arithmetic functions, Amer. Math. Monthly81(1974), 993–995.
[6] E. D. Cashwell and C. J. Everett,The ring of number-theoretic functions, Pacific J. Math.9 (1959), 975–985.
[7] P. Haukkanen,Some characterizations of totients, Int. J. Math. Math. Sci.19(1996), no. 2, 209–217.
[8] ,On the real powers of completely multiplicative arithmetical functions, Nieuw Arch. Wisk. (4)15(1997), no. 1-2, 73–77.
[9] J. Knopfmacher,Abstract Analytic Number Theory, Holland Mathematical Library, vol. 12, North-Holland Publishing, Amsterdam, 1975.
[10] V. Laohakosol, N. Pabhapote, and N. Wechwiriyakul,Logarithmic operators and characteri-zations of completely multiplicative functions, Southeast Asian Bull. Math.25(2001), no. 2, 273–281.
[11] ,Characterizing completely multiplicative functions by generalized Möbius functions, Int. J. Math. Math. Sci.29(2002), no. 11, 633–639.
[12] D. Rearick,Operators on algebras of arithmetic functions, Duke Math. J.35(1968), 761– 766.
[13] ,The trigonometry of numbers, Duke Math. J.35(1968), 767–776. [14] J. Riordan,Combinatorial Identities, John Wiley & Sons, New York, 1968.
[16] R. Sivaramakrishnan,Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 126, Marcel Dekker, New York, 1989.
Vichian Laohakosol: Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
E-mail address:[email protected]
Nittiya Pabhapote: Department of Mathematics, Faculty of Science, The University of the Thai Chamber of Commerce, Bangkok 10400, Thailand
