Internat. J. Math. & Math. Sci. VOL. 16 NO. 2 (1993) 373-384
373
ARITHMETIC
FUNCTIONS
ASSOCIATED
WITH
THE
INFINITARY DIVISORS
OF AN INTEGER
GRAEME L. COHENandPETERHAGIS, JR.
School ofMathematicalSciences
Department
of Mathematics University ofTechnology, Sydney TempleUniversityBroadway,
NSW
2007,Australia Philadelphia,PA
19122(Received November 21, 1991)
ABSTRACT. The infinitarydivisors ofanaturalnumber n arethe productsofits divisorsof the formp
u2,
wherepV
isaprime-powercomponentofnand ya2(where
ya 0or1)
isthe binary representation ofy.
In
thispaper,weinvestigate the infinitaryanalogues of such familiarnumber theoretic functionsas thedivisor sumfunction, Euler’s phi function and the Mbbius function.KEY
WORDS AND PHRASES. Infinitary divisors, arithmeticfunctions, infinitaryconvolutions, asymptotic formulae.1980 MATHEMATICS SUBJECT CLASSIFICATION. 11A25,11N37. i. SOME PRELIMINARIES
Let
I
{p2
pisaprime anda isanonnegativeinteger}.
If n is a natural numbergreaterthan 1 thenitfollows easily from thefundamentaltheorem of arithmetic and the fact that the binary representation ofanatural numberisunique that n canbewritten inexactly
one way
(except
for the order of thefactors)
as aproduct ofdistinct elements from I. Weshall call each dement of
I
in thisproduct an/-component of n, and weshall say that d isan/-divisorofn ifevery/-componentof disalso an/-component ofn.
From
thediscussion in the first foursectionsofCohen[1],
it followsthat thesetof/-divisors ofn isequaltothe setof infmitary divisors ofn.(See
Section 2inHag’is and Cohen[2]
foraconcisedefinition ofiniinitary
divisors.)
If disaninfmitarydivisor(or/-divisor)
of n,wewrited[o
n.For
every naturalnumber n, 1[oo
n.Now
suppose thatnP1P2... Pt,
whereP1
< P2 <
<
Pt
are the/-componentsofn.We shalldenote by
J(n),
sothatJ(n)
isthenumberof/-componentsofn(with
J(1)
0).
DEFINITION 1. The infinitaryMbbiusfunction,#oo,isgivenbyttoo(n)
(-1)
J()(I.I)
and the infinitary phifunction,
oo,
isgiven byoo
(1)
1,
oo
(n)
H
(5
1)
=, 1 ifn>
1.(1.2)
According to Theorem 13 in
[1],
iftoo(n)
andaoo(n)
denote the number and sum, respectively, of theinfinitary divisorsof n, then:too(n)--
2J(’0,
(1.3)
J(-)
aoo(n)
H
(PJ
+
1).
(1.4)
374 G.L. COHEN AND P. HAGIS
REMARK
1. Wehaveao(n)o
(n)
o(n2).
DEFINITION 2. Ifd isthegreatest common infinitary divisorofthe naturalnumbers m
and n,weshallwrite
(m,
n)o
d. If(m, n)o
1,weshallsaythatm andn are/-prime.REMARK
2. Foreverynatural number m,(m,
1)o
1.REMARK
3. If(m,
n)
1, then(m, n)oo
1.REMARK
4. IfdIoo
n, then dandn/d
are/-prime.DEFINITION 3. The arithmetic function
]
(where
]
is not identicallyzero)
issaid tobe I-multiplicativeif(m,n)oo
1implies thatf(mn)= f(m)f(n).
REMARK
5. If jrisI-multiplicative,then]
isalso multiplicative. Thefollowingresultis obvious.TtlEOttEM 1.
Suppose
thatf
andg areI-multiplicativefunctions.
Thenf(1)
1, andf
g andI/I
are I-raultiplicativefunctions.
If
g(n)
#
0for
every n, thenf
/g
isI-multiplicative.From
(I.I), (1.2), (1.3), (1.4),
wehave:THEOREM 2. Each
of
thefunctions
poo,oo,
’oo,aoo
isI-raultiplicative.TItEOREM 3.
Suppose
thatf
isanI-multiplicativefunction.
If
F(n)
Ealoon
f(d) (where
d runs overall
of
the infinitarydivisorsof
n),
thenF
isalso an1-multiplicativefunction.
Proof.
Suppose
that m and n are /-prime. Ifm 1or n I then(since
f(1)
1)
F(m)
1orF(n)
1, anditfollows thatF(mn)
F(m)F(n).
Otherwise,sincethe infinitary divisors ofanatural numberare1 and the set of allproductsof its/-components takenoneat atime, two at atime,etc.,
we seethat disaninfinitary divisor ofmn if andonly if dd d=
whered
Ioo
mandd2
Ioo
n. Of course,d
andd2
are/-prime. Therefore,F(mn)
y
y(d)=
](dld)
dloomn dloo dlo
f(dl)f(d2)=
f(dl)
f(d2)=
F(m)F(n).
dl,rndl.n dl**rn dl.n
T.oas
4.We
have’1
o
(d)
n.Proof.
LetF(n)
EaI
oo
(d).
ItisimmediatethatF(1)
1. Also,ifnP
5I
thenr()
oo
(a)
oo
()
+
oo
(e)
+
(e
-.)
n.al.n
Since
oo
isanI-muliplicaive function,i followsfrom Theorem3 thatF
isI-multiplicive.Therefore,
ifn PP2...Pt,thenF(n)
I-[=
F(Pj)
1-I)=
PJ
n.DEFINITION 4. The arithmetic function i, given by
i(n)=
[] {1,0,
ififn=n>l,l’
iscalledtheidentityfunction.The proofof thefollowingtheorem is very similar to that of Theorem 4 and thereforeis omitted.
’J2OIE
.
We
hve-"1’
I(d)
i(n).
2. Tttp..INFINITRY CONVOLUTION
ARITHMETIC FUNCTIONS ASSOCIATED WITH DIVISORS 375
DEFINITION 5. If
f
and g are arithmeticfunctions, their infinitary convolution, denoted(]
*g)o, isthe arithmetic functiongiven byWnEOaE 6. fo =h
o
],,
,
h=(]
)
(g, y)
(h
)
((]
*)
*)
(]
*(
*))
(
o
).
TEOaE 7. Fo =h
o
TEOaEM 8.
if/
i anarithmetic]uncionuch hat/(1) #
0,herei auniquearithmetic]uncion
(]-),
called heinfinitaryin,erieo/],
such thatREMaE
6. FromTheorems 6, 7, 8, it follows that the set ofwith
](1)
#
0 forms abeligroup withrespecttotheoperationof infinity convolution, the identity elementbeingthe functioni.DEFINITION 6. Theunitfunctionuistheithmeticfunction such that
u(n)
1foreverynatur
numbern.om
Theorem 5 d Definition6,wehave(p
u)
i.Therefore,
om
Theorem8,(.2)
=dTnEOaEM 9.
(The
Mbius in,ersion]oula.)
We
ha,eF(n)
,
](d)
ff
and onlyProo].
Suppose
thatF
(]
u).
Then(F
p)
((],u),p)
(],(u,))
(],i)
].
Convemely, suppose that/
(r,p).
Then(] )
((f
) )
(,
(,,
))
(, )
F.COROLLARY 9.1.
We
have(n)=
,p(d)(n/d)=
nProo].
om
Theorem4,nd],
(d).
Ourresult fonowsom
Whrem 9.TEOaEM 10.
ff
]
andg are each I-mul@licative, thenso is(]
g).
TEOaEM 11.
ff
g and(]
g)
are each I-multiplicaive functions, henf
isao
I-mult@licative.
COROLLARY
..
If
g i$I-multiplicative, o i iinfinita
inverse(g-).
Proof.
Both g d(g,(g-))
e I-multiplicative. Therefore,(g-)
isI-multiplicative.
EMAR
7. It followsom
Theorems6, 7, 8, 10 d Corolly 11.1 that the t ofI-multipHcativefctions isasubgroupof the oup ofthmetic functions
f
su
thatf(1)
0.THEOREM 12.
If f
is I-multiplicative, thenProof.
We
havedln dln
376 G.L. COHEN AND P. HAGIS
f(n)
poo(d)=f(n)i(n)= i(n).
COROLLARY
13.]..If f
i, 1-multiplicative, then(/-1)oo(rl)
f(rl) or--/(rl), accordinff
asJ(n)
is even orodd.3. SOME SUMMATORY FUNCTIONS
Throughout this section, we assume that if the natural number n exceeds 1, then n
PIP2... PJ,
whereP1
<
P2 <
<
PJ
are the/-componentsofn. Also, ewill denoteareal numbersatisfying0<
e<
1.DEFINITION 7. The arithmeticfunction
oo
isdefinedbyJ
P1
ifn>1.oo(1)
1,noo(n)
P1 +
1Itisimmediatethat
noo
isI-multiplicative, and thatnoo(n) _<
1foralln.DEFINITION 8. Ifx
>
0 isarealnumber,oo(x,n)
1.REMARK
8.In
generalit isnot true thatoo(n,n)
oo
(n).
SeeCorollary 14.1.LEMMA
1. I,etP
be anyI-component
of
n, wheren>
1. ThenProof.
We
havem<x ra<z m<z (m,n)--oo=1 (rn,n/P)oo=1 Plom
(/P,-)oo
=
THEOREM 14. Foranypositive real numberx and anynatural number n,
Coo(z,
n)
noo(,)x
+
where the multiplieative constant impliedby the big ohnotation depends onlyone.
Proof.
Theproofwill includeanestimationof theconstant,c
say,impliedby thebigoh notation.Let
Q
bethe smallest dement ofIsuchthat1
log(Q’
1)
<
,,
(3.)
ARITHMETIC FUNCTIONS ASSOCIATED WITH DIVISORS 377
andlet
(For
example, takinge 0.8, 0.5, 0.3 and 0.1,wefind that, respectively,Q
3, 11, 107 and 1876451. Since 2 1_<
1,we seethatQ
_> 3.)
Suppose
firstthat x<
1. ThenCoo(x,
n)
0, andso
(3.1)
holdsin this case, providedc
>_
1.From
nowon,we assumethat x>_
1.Suppose
that n 1. ThenCoo(x,
n)=
Ix],
andI(,-)- oo(-)1
-[]
<
1<
.,
so
(3.1)
holdsin this caseaswell,again providedc
>_
1.Wenow assumefurther that n
>
1, andconsider firstthosenforwhichQ _< P1.
Wewill showthat,foranypositive integer k and allx<
P,
[oo(x,n)- oo(n)x[
<
2pn
.
(3.3)
.The
proofisby inductiononk.When k 1,wehave,since x
<
P,
0<
oo(n)
<
1andp>
1,[Coo(x, n)
oo(n)x[
_<
Coo(x, n)
+
oo(n)x < [x]
+
x_<
2x<
2P
_<
2n< 2pn,
which isthe required resultin thiscase.p(z+l), Forsuchxand
Nowsupposetheresult istruewhen k l,andconsiderthosex <. anyj, 1
<_
j_< J,
wehavex/P.
<
P.
By
repeatedapplication ofLemma 1,oo(,)
oo
,
oo
=o
z,-oo
-oo
,n==
Z,piP=e
-oo
P3’PIP=
-oo -2’
--oo
)
PIP
P-I
P-’
PIP
P-[]-
+
1pj
PJ-1
Pj-P1
378
Since
G.L. COHEN AND P. HAGIS
oo(PJPJ-1PJ-2)
Pj
Pj-Pj
+
1 Pj_+
1,oo(.),
P
J-2Pj 1
Pj
+
1PJ-1
+
1PJ-1 1 Pj Pj_
P2
1Pj+IPj_+I
P2+l
P+I
Pj
+
i Pj_I+
IP+i
P+i
P/I Pj-I
+
I P2/Iwethushave, using theinductionhypothesis,
Thiscompletes theproofof
(3.3).
Now,
for givenxwelet k be the smallest integer such thatz<
Pt.
(Note
that k>
1since x_>
1.)
Thenx>_
P(k-)t
>_
Q(k_),
sothat k 1_<
logx/e
logQ.
Then’-=
Q,-I
-<
Q,-logz/logQ
xlog(Ct’/(C’-l))/logC"
<
x,
ARITHMETIC FUNCTIONS ASSOCIATED WITH DIVISORS 379 Considernext thosenwith
P1
< Q <
P2.
By
repeatedapplication ofLemma 1,=
z,-
+oo
,n=
,
-o
,
+0o
--..+oo
:F.--.This is a finite series, terminating after
+
1 terms, wherex/P
+1<
1<
x/P.
Since the smallest/-component ofn/P1
is atleastQ,
the workabove shows thatThe extra multiplicative factorintroduced intheerrortermis
1
P
<
P PI-I
2E_lThe
O(1)
term infact addsno morethan1,which isless than 1.nEx,
totheerrorterm. Since 1+
2p/(2
-
1)
< 3p/(2
-
1) (because
pE>
1>_
2-
1),
wemay takece
>_
3p,/(2
-
1).
Theorem 14hasthus been provedforthosen withatmost one/-componentless than
Q.
IfQ
>
3,then thereexist n forwhichP2 <
Q
_< P3.
Forsuch n,n/P1
has at most one/-componentless than
Q,
soprecisely thesameapproachasinthepreceding paragraph,using the resultthere, again shows Theorem14to betrue, providedthatwetakec
>
4p/(2
1)
2.
By
repeatingthisargument, weseethat ifPs
<
Q
<_ Ps+,
forst)mes 1, 2,J
1,then the theoremis
true,
withc
>_
(s
+
2)p/(2
1)
s.
(This
statement alsoholdsfors 0ifwedefine
P0
to be1.)
IfF(Q)
denotes thenumber of elements ofI
whichareless thanQ,
andPs
<
Q
<_
Ps+l
where s+
1_< J,
thenit is easytoseethat s<
F(Q). It
follows that if x>
1 andQ
<_
Pj, then the theoremistrue,providedwetakec
> (F(Q)
+
2)pE/(2
1)
F(a).
We
mustconsiderfinally thosenwithQ
>
Pj. From Lemma 1,andtheresultoftheprecedingparagraph implies that
Coo(x, n)=
(nQ)x
+
O(nQx
)
+
tco(nQ)-
+
O(nEz).
Itnowfollows easilythat(,.)
oo(n)
+
o(),
380 G.L. COHEN AND P. HAGIS
Theproofof Theorem 14 iscomplete. The constant hnplied by the big oh may be taken
equalto
(F(Q)
+
2)(Q’
+
1)p/(2’
1)
F(q).COROLLARY
14.1.We
haveoo(-,-)
_
(-)
oo(
)
+
o(n"),
for
any with 0<
<_
2.Proof.
Use
Remark 1, Theorem 14,andthe fret thatnm(n)
n/am(n).
COROLLARY 14.2.We have,
wherex>
O,
tm(X,
n)def,_
Z
m<z
1/2(.)
+
o(’+’.’).
Proof.
Use
aStieltjes integration withera(x, n)
asintegrator(maintaining
nasaconstantparameter).
THEOREM 15. I,etg be anybounded arithmetic
function
anddefine
the arithmeticfunction
]
(n
nE
g(-ff’)
Then,
for
anyz>
1,z:z
/(")
3
,
+
o(=
TMZo
n<z n=l
Proof.
WehaveusingCorollary14.2.
Wenowhave
d
(d,d’)==1
g(d)=g(d)
d’
(=,=’)== (=’,)==z
d<
2
n
+0
z+
n
+
logz),
Since 0
<
m(n)
_<
1madn>=
1/n2
O(x-1),
thetheoremisproved.ARITHMETIC FUNCTIONS ASSOCIATED WITH DIVISORS 381
COROLLARY 15.1. Foranyx
>
1,COROLLARY
15.2. Foranyz>
I,
Z
o(n)
-"
nn<z n=l
Wealso have
+
O(z
1+"logz).
COROLLARY 15.3. Forany z
>
1,x
nto(n)=’-
po(n)n(n)n2
n<x n=l
+
O(x
TMlogx).
Proof.
Weneedfirstthe resulto(a).
(.)
d
Thiscan beproveddirectly,orbyusing the M6biusinversionformula
(Theorem 9),
asfollows. Sincen
ao(n)
E
d,oo(n)
andsince
noo
isI-multiplicative,wehaven/d
n
Z
poo(d)noo(n/dj
giving
(3.4).
n
dln
oo()
dNow
substituteg(n)=
poo(n)too(n)
in Theorem 15 toobtainCorollary 15.3.Set
A
Z
noo(.)
B
E
po(n)noo(.)
CZ
n
’
n2
n2n=l n=l
These may beappromated
usg
the infinityoe
ofEer
products:THEOaEm 16.
If
f(n)
is an I-multiplicative arithmeticnction
such thate
seriesE
y(n)/n"
absolutet
convergent, thenf(n---)= H
1+n=l P6.I
Proof.
Theproofisanalogous to,buteasierthan,thatofTheorem 11.7in[3].
Using thedefinitionsoftoo andpo,wethus have
(
1)
A=
H
I+
P(P+I)
P6.I
(
1)
B=
H
1-p(p+l)
P6_I
(
C=
H
1-(p+l)2
P.I
In
thisway,wehave calculated thefollowingnumericalvaluesfor thecoefficients inCorollaries15.1,15.2 and 15.3,respectively:
-0.73
B
C-
0.33,-
0.37.382 G.L. COHEN AND P. HAGIS
the corresponding definitions and results. See
[3]
forthe results concerning ordinary divisors.]
,,(-)
(2)
+
0(
os
),
(2)
O.S2;r2
E
a*(n)=
1--x
2+
O(x
log
x),
12((3"--’
0.68;()
+
o(
og
),
-
0.30;E
*(n)=
2
+
O(z
log2z),
: :
H
1 --" 0.35.nz
pprimeP(P
""
1)
Wenowproceedtothe evaluation of
,<
r(n).
LEMMA
2. WithC
asabove,
wehave,
.for
allx>
1,(i)
(.)
c +
o(’
os
);
log
z+
C’
+
O(z
-1+-log
x),
shere
C’
i aconstantto be described.Proof.
We
useasanintegratorthe functionK
definedbyK(x)
0 ifz<
1,(See
Corollary
15.3.)
log
z)if
z>
1.(i)
.<=’=(")=f’dK(")--=
K()
+
f,"
---a,,K(=)
:
+
0(’
log
)
+
+
O(u
-+
log)
dCx
+
O(z
logx).
(ii)
n<_xE
’:’(n)n
f=
dK(U)u
2
--.._K(z)z2
+
2f"
K(u)
du CO(z-
+,f(C
(K(u)
C))du
=+
og)+2
+
2
)
+
o(
-+’os
)
+ c
os
+
2g(=)
C
u 2u du
Sce
K(z)/x
3C/2x
O(x
-2+log
x),
the firstARITHMETIC FUNCTIONS ASSOCIATED WITH DIVISORS 383
TttEOREM 17. We have,/oranyx
>
1,Too(n)
Cx logx+
(2C’
C)x
+
O(x+).
Proof.
Having developed Theorem 14 andLemma2, the prooffollowsstandardlines:n(_x n_xdl n_x dd’=n (d,d’)=l
dd’<_x,d<_v/’
(d,d’)oo=1 (d,d’)o=--1
d<_
’
d’<_ d d<,cr
(d’,d)--1 (d’,d)=o--1X
2(C
log+
C’
+
O(
(-1T’) log)) +
O(
+’)
c
o
+
(c’
c)
+
o(+).
Forordinarydivisors,the result correspondingtothatinTheorem17,asoriginallyobtained by Dirichlet,is
z-(n)=
xlogx+
(20’- l)z
+
O(x1/2),
where 7 is Euler’s constant.
(It
is well known that the error term has subsequently been improved. This isthe classicaldivisorproblem,discussedin[3].)
For
unitarydivisors, wehave thefollowingresult(obtained
byGioia and Vaidya[5]):
x(
)
r*(n)=
f-
logx+
27-
1-2(’(2.__.._)
,<x
(2)
+O(x]).
REFERENCES
1.
COHEN,
G. L. On aninteger’s infinitarydivisors, Math.Comp.
54(1990)
395-411. 2.HAGIS,
P., Jr.,
andCOHEN,
G. L.Infinitaryharmonicnumbers’ Bull. Austral. Math. Soc.(0)
-S.3.
APOSTOL,
T. M. Introduction to Analytic NumberTheory, Springer-Verlag, New York,1976.
4.
COHEN,
E.Arithmetical functions associated withtheunitary divisorsofaninteger, Math. Zeit. 74(1960)
66-80.