Hiroshima Math. J.
36 (2006), 113–124
Unitary convolution for arithmetical functions in several variables
Emre Alkan, Alexandru Zaharescu and Mohammad Zaki
(Received May 19, 2005)
Abstract. In this paper we investigate the ring Ar ðRÞ of arithmetical functions in r
variables over an integral domain R with respect to the unitary convolution. We study
a class of norms, and a class of derivations on Ar ðRÞ. We also show that the resulting
metric structure is complete.
1. Introduction
The ring A of complex valued arithmetical functions has a natural
structure of commutative C-algebra with addition and multiplication by scalars,
and with the Dirichlet convolution
X
ð f gÞðnÞ ¼
f ðdÞgðn=dÞ:
djn
In [1], Cashwell and Everett proved that ðA; þ; Þ is a unique factorization
domain. Yokom [5] investigated the prime factorization of arithmetical
functions in a certain subring of the regular convolution ring. He determined
a discrete valuation subring of the unitary ring of arithmetical functions.
Schwab and Silberberg [3] constructed an extension of ðA; þ; Þ which is a
discrete valuation ring. In [4], they showed that A is a quasi-noetherian ring.
In the present paper we study the ring of arithmetical functions in several
variables with respect to the unitary convolution over an arbitrary integral
domain. Let R be an integral domain with identity 1R . Let r b 1 be an
integer number, and denote Ar ðRÞ ¼ f f : N r ! Rg. Given f ; g A Ar ðRÞ, let us
define the unitary convolution f l g of f and g by
X
X
ð f l gÞðn1 ; . . . ; nr Þ ¼
...
f ðd1 ; . . . ; dr Þgðe1 ; . . . ; er Þ:
d1 e1 ¼n1
ðd1 ; e1 Þ¼1
dr er ¼nr
ðdr ; er Þ¼1
Note that R has a natural embedding in the ring Ar ðRÞ, and Ar ðRÞ with addition
and unitary convolution defined above becomes an R-algebra. We define and
study a family of norms on Ar ðRÞ. Then we show that Ar ðRÞ endowed with
any of the above norms is complete. A class of derivations on Ar ðRÞ is then
constructed and examined. We also study the logarithmic derivatives of
multiplicative arithmetical functions with respect to these derivations.
114
E. Alkan, A. Zaharescu and M. Zaki
2. Norms
Let U(R) denote the group of units of R. Let U(Ar ðRÞ) be the group
of units of Ar ðRÞ. Thus, UðAr ðRÞÞ ¼ f f A Ar ðRÞ : f ð1; . . . ; 1Þ A UðRÞg. In
this section R will denote an integral domain. We start by defining a norm on
Ar ðRÞ. Fix t ¼ ðt1 ; . . . ; tr Þ A Rr with t1 ; . . . ; tr linearly independent over Q, and
ti > 0, ði ¼ 1; 2; . . . ; rÞ. Given n A N, we define WðnÞ to be the total number
of prime factors of n counting multiplicities, i.e., if n ¼ p1a1 . . . pkak , then WðnÞ ¼
a1 þ þ ak . We now define Wr : N r ! N r by
Wr ðn1 ; . . . ; nr Þ ¼ ðWðn1 Þ; . . . ; Wðnr ÞÞ:
Given n ¼ ðn1 ; . . . ; nr Þ and m ¼ ðm1 ; . . . ; mr Þ in N r , we denote n m ¼
n1 m1 þ þ nr mr . For f A Ar ðRÞ, we define the support of f , suppð f Þ ¼
fn A N r j f ðnÞ 0 0g. We also define for f A Ar ðRÞ,
(
y
if f ¼ 0;
Vt ð f Þ ¼
min t Wr ðnÞ if f 0 0.
n A suppð f Þ
Note that if f 0 0 then Vt ð f Þ ¼ t Wr ðnÞ for some n A suppð f Þ.
Proposition 1.
(i) For any f ; g A Ar ðRÞ, we have
Vt ð f þ gÞ b minfVt ð f Þ; Vt ðgÞg:
(ii)
For any f ; g A Ar ðRÞ, we have
Vt ð f l gÞ b Vt ð f Þ þ Vt ðgÞ:
Proof. (i) Let f ; g A Ar ðRÞ. If f þ g ¼ 0, then clearly Vt ð f þ gÞ b
minfVt ð f Þ; Vt ðgÞg. Suppose f þ g 0 0. Let n A suppð f þ gÞ. Then either
n A suppð f Þ, or n A suppðgÞ. If n A suppð f Þ, then t Wr ðnÞ b Vt ð f Þ, and if
n A suppðgÞ, then t Wr ðnÞ b Vt ðgÞ. It follows that for all n A suppð f þ gÞ,
t Wr ðnÞ b minfVt ð f Þ; Vt ðgÞg. Hence,
Vt ð f þ gÞ b minfVt ð f Þ; Vt ðgÞg:
(ii) Again let f ; g A Ar ðRÞ. If f l g ¼ 0, then the inequality holds
trivially. So assume that f l g 0 0, and let a1 ; . . . ; ar be positive integers such
that ða1 ; . . . ; ar Þ A suppð f l gÞ and Vt ð f l gÞ ¼ t1 Wða1 Þ þ þ tr Wðar Þ. Then
0 0 ð f l gÞða1 ; . . . ; ar Þ ¼
X
d1 e1 ¼a1
ðd1 ; e1 Þ¼1
...
X
f ðd1 ; . . . ; dr Þgðe1 ; . . . ; er Þ:
dr er ¼ar
ðdr ; er Þ¼1
Therefore f ðd1 ; . . . ; dr Þ 0 0 and gðe1 ; . . . ; er Þ 0 0 for some di , ei with di ei ¼ ai ,
ðdi ; ei Þ ¼ 1, ði ¼ 1; . . . ; rÞ. It follows that
Unitary convolution for arithmetical functions in several variables
115
Vt ð f Þ þ Vt ðgÞ a t1 Wðd1 Þ þ þ tr Wðdr Þ þ t1 Wðe1 Þ þ þ tr Wðer Þ
¼ t1 Wða1 Þ þ þ tr Wðar Þ
¼ Vt ð f l gÞ:
This completes the proof of the proposition.
Next, we define a family of norms on Ar ðRÞ. Fix a t as above and a
number r A ð0; 1Þ. Then define a norm k:k ¼ k:kt : Ar ðRÞ ! R by
kxkt ¼ rVt ðxÞ
if x 0 0;
and
kxkt ¼ 0
if x ¼ 0:
By the above proposition it follows that kx þ yk a maxfkxk; k ykg, and
kx l yk a kxk kyk for all x; y A Ar ðRÞ. Associated with the norm k:k we have
a distance d on Ar ðRÞ defined by dðx; yÞ ¼ kx ykt , for all x; y A Ar ðRÞ:
ger.
Theorem 1. Let R be an integral domain, and let r be a positive inteThen Ar ðRÞ is complete with respect to each of the norms k:kt .
Proof. Let ð fn Þnb0 be a Cauchy sequence in Ar ðRÞ. Then for each
e > 0, there exists an N A N depending on e such that k fm fn k < e for all
m; n b N. For each k A N, taking e ¼ r k , there exists Nk A N such that
k fm fn k < r k for all m; n b Nk . Equivalently, Vt ð fm fn Þ > k for all
m; n b Nk , i.e., we have that for all m; n b Nk ,
fm ðl1 ; . . . ; lr Þ ¼ fn ðl1 ; . . . ; lr Þ
whenever t1 Wðl1 Þ þ þ tr Wðlr Þ a k, l1 ; . . . ; lr A N. We choose inductively for
each k A N, the smallest natural number Nk with the above property such that
N1 < N2 < < Nk < Nkþ1 < :
Let us define f : N r ! R as follows. Given l ¼ ðl1 ; . . . ; lr Þ A N r , let k be
the smallest positive integer such that k < t1 Wðl1 Þ þ þ tr Wðlr Þ a k þ 1. We
set f ðlÞ ¼ fNkþ1 ðlÞ. Then we will have f ðlÞ ¼ fn ðlÞ, for all n b Nkþ1 . Since
this will hold for all l and k as above, it follows that the sequence ð fn Þnb0
converges to f . This completes the proof of Theorem 1.
3. Derivations
We use the same notation as in the previous section.
Definition 1. We call an arithmetical function f A Ar ðRÞ multiplicative
provided that f is not identically zero and
f ðn1 m1 ; . . . ; nr mr Þ ¼ f ðn1 ; . . . ; nr Þ f ðm1 ; . . . ; mr Þ
116
E. Alkan, A. Zaharescu and M. Zaki
for any n1 ; . . . ; nr ; m1 ; . . . ; mr A N satisfying ðn1 ; m1 Þ ¼ ¼ ðnr ; mr Þ ¼ 1.
say that an f A Ar ðRÞ is additive provided that
We
f ðn1 m1 ; . . . ; nr mr Þ ¼ f ðn1 ; . . . ; nr Þ þ f ðm1 ; . . . ; mr Þ
for any n1 ; . . . ; nr ; m1 ; . . . ; mr A N satisfying ðn1 ; m1 Þ ¼ ¼ ðnr ; mr Þ ¼ 1.
Note that if f is multiplicative then f ð1; . . . ; 1Þ ¼ 1, while if f is additive
then f ð1; . . . ; 1Þ ¼ 0. We now proceed to define a derivation on Ar ðRÞ. For
any additive function c A Ar ðRÞ, define Dc : Ar ðRÞ ! Ar ðRÞ by
Dc ð f ÞðnÞ ¼ f ðnÞcðnÞ;
for all f A Ar ðRÞ and n A N r . For n ¼ ðn1 ; . . . ; nr Þ, and m ¼ ðm1 ; . . . ; mr Þ in
N r , we write nm ¼ ðn1 m1 ; . . . ; nr mr Þ. We state some basic properties of the
map Dc in the next proposition.
Proposition 2. Let R be an integral domain, and let r be a positive
integer. Let c A Ar ðRÞ be additive. Then for all f ; g A Ar ðRÞ and c A R,
(a) Dc ð f þ gÞ ¼ Dc ð f Þ þ Dc ðgÞ,
(b) Dc ð f l gÞ ¼ f l Dc ðgÞ þ g l Dc ð f Þ,
(c) Dc ðcf Þ ¼ cDc ð f Þ.
Consequently, we see that Dc is a derivation on Ar ðRÞ over R.
Proof.
Let n ¼ ðn1 ; . . . ; nr Þ A N r .
First, from the definition of Dc we see
that
Dc ð f þ gÞðnÞ ¼ ð f þ gÞðnÞcðnÞ
¼ f ðnÞcðnÞ þ gðnÞcðnÞ
¼ Dc ð f Þ þ Dc ðgÞ:
Thus, (a) holds.
Also from the definition of Dc we have that
Dc ð f l gÞðnÞ ¼ ð f l gÞðnÞcðnÞ:
So,
X
Dc ð f l gÞðnÞ ¼ cðnÞ
d1 e1 ¼n1
ðd1 ; e1 Þ¼1
¼
X
...
d1 e1 ¼n1
ðd1 ; e1 Þ¼1
¼
X
d1 e1 ¼n1
ðd1 ; e1 Þ¼1
X
...
f ðd1 ; . . . ; dr Þgðe1 ; . . . ; er Þ
dr er ¼nr
ðdr ; er Þ¼1
X
f ðd1 ; . . . ; dr Þgðe1 ; . . . ; er ÞcðnÞ
dr er ¼nr
ðdr ; er Þ¼1
...
X
dr er ¼nr
ðdr ; er Þ¼1
f ðd1 ; . . . ; dr Þgðe1 ; . . . ; er Þ
Unitary convolution for arithmetical functions in several variables
117
ðcðd1 ; . . . ; dr Þ þ cðe1 ; . . . ; er ÞÞ
X
X
¼
...
f ðd1 ; . . . ; dr Þcðd1 ; . . . ; dr Þgðe1 ; . . . ; er Þ
d1 e1 ¼n1
ðd1 ; e1 Þ¼1
þ
X
d1 e1 ¼n1
ðd1 ; e1 Þ¼1
dr er ¼nr
ðdr ; er Þ¼1
...
X
f ðd1 ; . . . ; dr Þcðe1 ; . . . ; er Þgðe1 ; . . . ; er Þ
dr er ¼nr
ðdr ; er Þ¼1
¼ f l Dc ðgÞ þ g l Dc ð f Þ:
Therefore (b) holds.
proposition.
Also, it is clear that (c) holds, and this proves the
Lemma 1. Let f ; g A A1 ðRÞ. Let p be a prime, and let Mp be the monoid
f1; p; p 2 ; . . .g under multiplication. Suppose that suppð f Þ; suppðgÞ J Mp . If
f ð1Þ ¼ 0, and gð1Þ ¼ 1, then f ¼ f l g.
Proof. We have that suppð f l gÞ J Mp since suppð f Þ; suppðgÞ J Mp .
Thus both f and f l g vanish outside the monoid Mp . Let now n be a
positive integer. Then
X
ð f l gÞð p n Þ ¼
f ðdÞgðeÞ
de¼p n
ðd; eÞ¼1
¼ f ð1Þgðp n Þ þ f ð p n Þgð1Þ
¼ f ðp n Þ:
Thus, f ¼ f l g.
Lemma 2. Let f A A1 ðRÞ be multiplicative and c A A1 ðRÞ be additive. Let
p be a prime, and let Mp be the monoid f1; p; p 2 ; . . .g under multiplication
D ðfÞ
as in Lemma 1. Suppose that suppð f Þ J Mp . Then cf ¼ Dc ð f Þ, where the
division on the left side is taken with respect to the unitary convolution.
Proof. Note first that since f is supported on Mp , both Dc ð f Þ and f 1
will be supported on Mp . We have moreover that f 1 ð1Þ ¼ 1 because f is
multiplicative. Also since c is additive, cð1Þ ¼ 0. Applying Lemma 1, we
D ðfÞ
conclude that cf ¼ Dc ð f Þ.
Theorem 2. Let f A A1 ðRÞ be multiplicative and c A A1 ðRÞ be additive.
Let n be a positive integer, and let Pn be the set of all prime divisors of n. For
each prime p, let Mp be as in Lemma 1, and let fp ¼ f jMp , i.e.,
118
E. Alkan, A. Zaharescu and M. Zaki
fp ðmÞ ¼
f ðmÞ
0
if m ¼ p k ; k b 1
else:
Then
8
< cðnÞ f ðnÞ
X
Dc ð f Þ
ðnÞ ¼
Dc ð fp ÞðnÞ ¼
:
f
p A Pn
0
if n ¼ p k for some p prime
and k b 1;
else:
Proof. Fix an n and let n ¼ p1s1 . . . ptst be the prime factorization of n.
Let M be the set of all m A N such that whenever p is a prime and p divides m,
p also divides n. Note that M is a monoid under multiplication, generated by
the primes p1 ; . . . ; pt . Let g ¼ f jM , i.e., for any m A N,
f ðmÞ if m A M
gðmÞ ¼
0
else:
Suppose that m, k are in N, and ðm; kÞ ¼ 1. If m B M, or k B M,
then f jM ðmÞ ¼ 0, or f jM ðkÞ ¼ 0, and so, gðmÞgðkÞ ¼ f jM ðmÞ f jM ðkÞ ¼ 0 ¼
f jM ðmnÞ ¼ gðmnÞ since mn B M whenever one of m, or n does not belong to
M. If m, n are relatively prime and m; n A M then mn A M and gðmnÞ ¼
f jM ðmnÞ ¼ f ðmnÞ ¼ f ðmÞf ðnÞ ¼ f jM ðmÞf jM ðnÞ ¼ gðmÞgðnÞ. Thus g is multiplicative. We claim that
Y
g¼
fp :
p A Pn
Indeed, let us first observe that if h1 ; h2 A A1 ðRÞ are such that suppðh1 Þ,
suppðh2 Þ are contained in M, then suppðh1 l h2 Þ J M. To see this, let
m B M. Then there exists a prime p such that pjm, but p does not divide n.
Now
X
ðh1 l h2 ÞðmÞ ¼
h1 ðdÞh2 ðeÞ:
de¼m
ðd; eÞ¼1
Since either pjd, or pje whenever m ¼ de, every term in this sum is 0
because suppðhi Þ J M ði ¼ 1; 2Þ. Thus, ðh1 l h2 ÞðmÞ ¼ 0 for any m B M.
Hence suppðh1 l h2 Þ J M. Using the above observation and induction, it
Q
follows that suppð p A Pn fp Þ J M. Since g is also supported on M, it follows
that in order to prove the above claim it is enough to show that g equals
Q
p A Pn fp on M. Let m A M with
m ¼ p1a1 . . . ptat ;
where all ai are nonnegative integers for i ¼ 1; . . . ; t. We have that
Unitary convolution for arithmetical functions in several variables
Y
X
fp ðmÞ ¼
119
fp1 ðd1 Þ . . . fpt ðdt Þ
d1 ...dt ¼m
ðdi ; dj Þ¼1; ði0jÞ
p A Pn
X
¼
fp1 ð p1b1 Þ . . . fpt ðptbt Þ
b1 ;...; bt
b
b
p1 1 ...pt t ¼m
¼ fp1 ð p1a1 Þ . . . fpt ð ptat Þ
¼ f ðmÞ
¼ gðmÞ;
where in the above computation b1 ; . . . ; bt are forced to have unique values
Q
equal to a1 ; . . . ; at respectively. Hence g ¼
fp , as claimed. Next, we
p A Pn
claim that
Dc ð f Þ ¼ Dc ðgÞ l g1 :
f M
In order to prove this, we first show that
f 1 jM ¼ g1 :
Note that by the previous claim we know that
!1
Y
Y
1
g ¼
fp
¼
fp1 ;
p A Pn
p A Pn
and as a consequence g1 is supported on M. We now proceed by induction.
First, since f ð1Þ ¼ gð1Þ ¼ 1, it follows immediately that f 1 jM ð1Þ ¼ g1 ð1Þ ¼ 1.
Next, let m > 1, and assume that for all k < m, g1 ðkÞ ¼ f 1 jM ðkÞ. If m B M,
then f 1 jM ðmÞ ¼ 0 ¼ g1 ðmÞ. Now suppose that m A M. Then, using the
equalities ð f l f 1 ÞðmÞ ¼ 0 ¼ ðg l g1 ÞðmÞ in combination with the induction
hypothesis we derive
f 1 jM ðmÞ ¼ f 1 ðmÞ
¼
1 X
f ðdÞ f 1 ðeÞ
f ð1Þ de¼m
ðd; eÞ¼1
e<m
¼
1 X
gðdÞg1 ðeÞ
gð1Þ de¼m
ðd; eÞ¼1
e<m
¼ g1 ðmÞ:
120
E. Alkan, A. Zaharescu and M. Zaki
Thus,
f 1 jM ¼ g1 :
Further, it is clear that
Dc ð f ÞjM ¼ Dc ð f jM Þ ¼ Dc ðgÞ:
By the above two relations we conclude that
Dc ðgÞ l g1 ¼ Dc ð f ÞjM l f 1 jM :
Therefore in order to prove the claim it remains to show that
Dc ð f Þ ¼ Dc ð f ÞjM l f 1 jM :
f M
Here the left side is supported on M, while the right side is the unitary
convolution of two arithmetical functions supported on M, so it is also
supported on M. So we only need to check the desired equality at an arbitrary point m A M. For such an m, any representation of m as a product
m ¼ de forces both d, e to belong to M. Thus
X
Dc ð f Þ Dc ð f Þ
ðmÞ
¼
Dc ð f ÞðdÞ f 1 ðeÞ
ðmÞ
¼
f M
f
de¼m
ðd; eÞ¼1
¼
X
Dc ð f ÞjM ðdÞ f 1 jM ðeÞ ¼ ðDc ð f ÞjM l f 1 jM ÞðmÞ:
de¼m
ðd; eÞ¼1
We conclude that
Dc ð f Þ f M
¼ Dc ð f ÞjM l f 1 jM , and hence
Dc ð f Þ f as claimed.
obtain
¼ Dc ðgÞ l g1 ;
M
On the other hand, by applying Proposition 2 (b) repeatedly, we
Dc ðgÞ l g
1
Q
X D c fp
D c ð p A P n fp Þ
¼ Q
¼
:
fp
p A Pn fp
pAP
n
By the above two relations we deduce that
X Dc ð fp Þ
Dc ð f Þ ¼
:
f
fp
M
pAP
n
Unitary convolution for arithmetical functions in several variables
But by Lemma 2,
P
p A Pn
Dc ð fp Þ
fp
equals
P
Dc ð fp Þ.
p A Pn
121
Therefore, we have that
X
Dc ð f Þ ¼
Dc ð fp Þ:
f
M
pAP
n
Since n is in M, it follows in particular that
X
Dc ð f Þ
ðnÞ ¼
Dc ð fp ÞðnÞ:
f
pAP
n
This completes the proof of the theorem.
We now proceed to generalize this theorem to the case of arithmetical
functions of several variables.
Lemma 3. Let f A Ar ðRÞ be multiplicative and consider the monoids
M1 ¼ fðk; 1; . . . ; 1Þ A N r : k A Ng; . . . ; Mr ¼ fð1; . . . ; 1; kÞ A N r : k A Ng:
Let f1 ¼ f jM1 ; . . . ; fr ¼ f jMr . Then
f ¼
r
Y
fi ¼ f1 l l fr :
i¼1
Let m ¼ ðm1 ; m2 ; . . . ; mr Þ A N r .
Proof.
r
Y
!
X
fi ðmÞ ¼
...
d11 ...d1r ¼m1
ðd1i ; d1j Þ¼1; ði0jÞ
i¼1
¼
r
Y
We have that
X
r
Y
fi ðd1i ; . . . ; dri Þ
i¼1
d11 ...d1r ¼m1
ðd1i ; d1j Þ¼1; ði0jÞ
fi ð1; . . . ; 1; mi ; 1; . . . ; 1Þ
i¼1
¼
r
Y
f ð1; . . . ; 1; mi ; 1; . . . ; 1Þ
i¼1
¼ f ðm1 ; . . . ; mr Þ
Hence, f ¼
r
Q
fi , and the lemma is proved.
i¼1
Theorem 3. Let R be an integral domain, and let r be a positive integer.
Then, for any multiplicative function f A Ar ðRÞ, any additive function c A Ar ðRÞ,
and any n ¼ ðn1 ; . . . ; nr Þ A N r , we have
122
E. Alkan, A. Zaharescu and M. Zaki
8
< cðnÞ f ðnÞ if n1 ¼ ¼ ni1 ¼ niþ1 ¼ ¼ nr ¼ 1 and ni ¼ p k
Dc ð f Þ
ðnÞ ¼
for some p prime; k b 1; and 1 a i a r;
:
f
0
else;
where the division on the left side is taken with respect to the unitary convolution.
Proof.
Let f be multiplicative and consider the monoids
M1 ¼ fðk; 1; . . . ; 1Þ A N r : k A Ng; . . . ; Mr ¼ fð1; . . . ; 1; kÞ A N r : k A Ng
as in Lemma 3. Let f1 ¼ f jM1 ; . . . ; fr ¼ f jMr . Then by Lemma 3, f ¼
r
Q
fi . Applying Proposition 2 (b) repeatedly, we get
i¼1
Dc ð f Þ
¼
f
Dc
r
Q
i¼1
r
Q
fi
fi
¼
r
X
D c ð fi Þ
i¼1
fi
:
i¼1
Therefore the desired equality from the statement of Theorem 3 will hold for
f provided it holds for each function fi . On the other hand, each of the
functions fi is supported on a one dimensional monoid isomorphic to N, so the
desired equality for each function fi follows directly from Theorem 2. This
completes the proof of Theorem 3.
We remark that if f and c are known, then Theorem 2 and Theorem
D ðfÞ
3 can be used to compute the logarithmic derivative cf . We end this
paper with a few very explicit examples. Take R to be the field of complex numbers and r ¼ 1. An additive arithmetical function is for instance
cðnÞ ¼ log n.
1. With R, r and c as above, let f be the Möbius function m, which
is a multiplicative function. By its definition, mð1Þ ¼ 1, and if n > 1, n ¼
p1a1 . . . pkak , then
ð1Þ k if a1 ¼ ¼ ak ¼ 1;
mðnÞ ¼
0
else:
By Theorem 2 we then have
Dc ðmÞ
log p
ðnÞ ¼
m
0
if n ¼ p for some prime p;
else:
2. Take R, r and c as above and choose f to be the Euler totient
function fðnÞ which is multiplicative. By Theorem 2 we see that
Unitary convolution for arithmetical functions in several variables
Dc ðfÞ
ðnÞ ¼
f
kð p k p k1 Þ log p
0
123
if n ¼ p k for some prime p and k b 1;
else:
3. With the same R, r and c as before, let f be the sum of divisors
P
function s, given by sðnÞ ¼
d, which is also a multiplicative arithmetical
djn
function. By Theorem 2 we find that
Dc ðsÞ
ðnÞ ¼
s
(
kð p kþ1 1Þ log p
p1
0
if n ¼ p k for some prime p and k b 1;
else:
One can of course consider many other interesting arithmetical functions.
For instance one can take f to be the number of divisors function, or the sum
of k-th powers of divisors function for some fixed k, which are multiplicative
functions, or one can let f be a Dirichlet character, which is completely
multiplicative. One may also take f to be the Ramanujan tau function tðnÞ
defined in terms of the Delta function
DðzÞ ¼
y
X
n¼1
tðnÞq n ¼ q
y
Y
ð1 q n Þ 24 ;
q ¼ e 2piz ;
n¼1
which is the unique normalized cusp form of weight 12 on SL 2 ðZÞ. Ramanujan first studied many of the beautiful properties of this arithmetical
function (see his collected works [2]). In particular he conjectured that tðnÞ is
multiplicative, a fact that was later proved by Mordell. One can also replace
c by other additive functions, for instance the logarithm of any multiplicative
arithmetical function is additive. Clearly applying Theorems 2 and 3 to
various combinations of such examples is equivalent in some sense to providing
identities for such arithmetical functions with respect to the unitary convolution.
References
[ 1 ] E. D. Cashwell, C. J. Everett, The ring of number-theoretic functions, Pacific J. Math. 9
(1959), 975–985.
[ 2 ] S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
[ 3 ] E. D. Schwab, G. Silberberg, A note on some discrete valuation rings of arithmetical
functions, Arch. Math. (Brno), 36 (2000), 103–109.
[ 4 ] E. D. Schwab, G. Silberberg, The Valuated ring of the Arithmetical Functions as a Power
Series Ring, Arch. Math. (Brno), 37 (2001), 77–80.
[ 5 ] K. L. Yokom, Totally multiplicative functions in regular convolution rings, Canadian
Math. Bulletin 16 (1973), 119–128.
124
E. Alkan, A. Zaharescu and M. Zaki
Emre Alkan
Department of Mathematics
University of Illinois at Urbana-Champaign
273 Altgeld Hall, MC-382, 1409 W. Green Street
Urbana, Illinois 61801-2975, USA
e-mail: alkan@math.uiuc.edu
Alexandru Zaharescu
Department of Mathematics
University of Illinois at Urbana-Champaign
273 Altgeld Hall, MC-382, 1409 W. Green Street
Urbana, Illinois 61801-2975, USA
e-mail: zaharesc@math.uiuc.edu
Mohammad Zaki
Department of Mathematics
University of Illinois at Urbana-Champaign
273 Altgeld Hall, MC-382, 1409 W. Green Street
Urbana, Illinois 61801-2975, USA
e-mail: mzaki@math.uiuc.edu