Unitary convolution for arithmetical functions in several variables

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 ringArðRÞ of arithmetical functions inr variables over an integral domainRwith respect to the unitary convolution. We study a class of norms, and a class of derivations onArðRÞ. 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 commutativeC-algebra with addition and multiplication by scalars, and with the Dirichlet convolution

ðf gÞðnÞ ¼X

djn

fðdÞgðn=dÞ:

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 1_R. Let rb1 be an integer number, and denote ArðRÞ ¼ ff :N^r!Rg. Given f;gAArðRÞ, let us define the unitary convolution flg of f and g by

ðflgÞðn1;. . .;nrÞ ¼ X

d1e1¼n₁ ðd1;e1Þ¼1

. . . X

drer¼nr

ðdr;erÞ¼1

fðd1;. . .;drÞgðe1;. . .;erÞ:

Note thatRhas a natural embedding in the ring ArðRÞ, andArðRÞwith addition and unitary convolution defined above becomes an R-algebra. We define and study a family of norms on A_rðRÞ. Then we show that A_rðRÞ endowed with any of the above norms is complete. A class of derivations on ArðRÞ is then constructed and examined. We also study the logarithmic derivatives of multiplicative arithmetical functions with respect to these derivations.

2. Norms

Let U(R) denote the group of units of R. Let U(ArðRÞ) be the group of units of A_rðRÞ. Thus, UðArðRÞÞ ¼ ff AA_rðRÞ: fð1;. . .;1ÞAUðRÞg. In this sectionRwill denote an integral domain. We start by defining a norm on ArðRÞ. Fix t¼ ðt1;. . .;trÞAR^r witht1;. . .;tr linearly independent overQ, and

t_i>0, ði¼1;2;. . .;rÞ. Given nAN, we define WðnÞ to be the total number

of prime factors of n counting multiplicities, i.e., if n¼ p₁^a1. . .p_k^ak, then WðnÞ ¼ a₁þ þa_k. We now define W_r:N^r!N^r by

Wrðn1;. . .;nrÞ ¼ ðWðn1Þ;. . .;WðnrÞÞ:

Given n¼ ðn1;. . .;nrÞ and m¼ ðm1;. . .;mrÞ in N^r, we denote nm¼ n₁m₁þ þn_rm_r. For f AA_rðRÞ, we define the support of f, suppðfÞ ¼ fnAN^rjfðnÞ00g. We also define for f AArðRÞ,

VtðfÞ ¼

y if f ¼0;

nAminsuppðfÞtWrðnÞ if f00.

(

Note that if f00 then VtðfÞ ¼tWrðnÞ for some nAsuppðfÞ.

Proposition 1. (i) For any f;gAArðRÞ, we have Vtðf þgÞbminfVtðfÞ;VtðgÞg:

(ii) For any f;gAA_rðRÞ, we have

V_tðflgÞbV_tðfÞ þV_tðgÞ:

Proof. (i) Let f;gAA_rðRÞ. If f þg¼0, then clearly V_tðf þgÞb minfVtðfÞ;VtðgÞg. Suppose f þg00. Let nAsuppðf þgÞ. Then either nAsuppðfÞ, or nAsuppðgÞ. If nAsuppðfÞ, then tW_rðnÞbV_tðfÞ, and if nAsuppðgÞ, then tWrðnÞbVtðgÞ. It follows that for all nAsuppðf þgÞ, tWrðnÞbminfVtðfÞ;VtðgÞg. Hence,

Vtðf þgÞbminfVtðfÞ;VtðgÞg:

(ii) Again let f;gAArðRÞ. If flg¼0, then the inequality holds trivially. So assume that flg00, and leta₁;. . .;a_r be positive integers such

that ða1;. . .;arÞAsuppðflgÞ and VtðflgÞ ¼t1Wða1Þ þ þtrWðarÞ. Then

00ðflgÞða1;. . .;a_rÞ ¼ X

d1e1¼a1

ðd1;e1Þ¼1

. . . X

drer¼ar

ðdr;erÞ¼1

fðd1;. . .;d_rÞgðe1;. . .;e_rÞ:

Therefore fðd1;. . .;d_rÞ00 and gðe1;. . .;e_rÞ00 for some d_i, e_i with d_ie_i¼a_i, ðdi;eiÞ ¼1, ði¼1;. . .;rÞ. It follows that

VtðfÞ þVtðgÞat1Wðd1Þ þ þtrWðdrÞ þt1Wðe1Þ þ þtrWðerÞ

¼t₁Wða1Þ þ þt_rWðarÞ

¼V_tðflgÞ:

This completes the proof of the proposition.

Next, we define a family of norms on ArðRÞ. Fix a t as above and a number rAð0;1Þ. Then define a norm k:k ¼ k:k_t :A_rðRÞ !R by

kxk_t¼r^VtðxÞ if x00; and kxk_t¼0 if x¼0:

By the above proposition it follows that kxþykamaxfkxk;kykg, and kxlykakxk kyk for all x;yAArðRÞ. Associated with the normk:kwe have a distance d on A_rðRÞ defined by dðx;yÞ ¼ kxyk_t, for all x;yAA_rðRÞ:

Theorem 1. Let R be an integral domain, and let r be a positive inte- ger. Then A_rðRÞ is complete with respect to each of the norms k:k_t.

Proof. Let ðf_nÞ_nb0 be a Cauchy sequence in A_rðRÞ. Then for each e>0, there exists an NAN depending on e such that kfmfnk<e for all m;nbN. For each kAN, taking e¼r^k, there exists N_kAN such that kfmfnk<r^k for all m;nbNk. Equivalently, VtðfmfnÞ>k for all m;nbNk, i.e., we have that for all m;nbNk,

fmðl1;. . .;lrÞ ¼ fnðl1;. . .;lrÞ

whenever t1Wðl1Þ þ þtrWðlrÞak, l1;. . .;lrAN. We choose inductively for each kAN, the smallest natural number N_k with the above property such that

N1<N2< <Nk <Nkþ1< :

Let us define f :N^r!R as follows. Given l¼ ðl1;. . .;l_rÞAN^r, let k be the smallest positive integer such that k<t1Wðl1Þ þ þtrWðlrÞakþ1. We set fðlÞ ¼ fNkþ1ðlÞ. Then we will have fðlÞ ¼ fnðlÞ, for all nbNkþ1. Since this will hold for all l and k as above, it follows that the sequence ðf_nÞ_nb0 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 AA_rðRÞ multiplicative provided that f is not identically zero and

fðn1m1;. . .;nrmrÞ ¼ fðn1;. . .;nrÞfðm1;. . .;mrÞ

for any n1;. . .;nr;m1;. . .;mrAN satisfying ðn1;m1Þ ¼ ¼ ðnr;mrÞ ¼1. We say that an f AA_rðRÞ is additive provided that

fðn1m₁;. . .;n_rm_rÞ ¼ fðn1;. . .;n_rÞ þfðm1;. . .;m_rÞ

for any n1;. . .;nr;m1;. . .;mrAN satisfying ðn1;m1Þ ¼ ¼ ðnr;mrÞ ¼1.

Note that if f is multiplicative then fð1;. . .;1Þ ¼1, while if f is additive

then fð1;. . .;1Þ ¼0. We now proceed to define a derivation on ArðRÞ. For

any additive function cAA_rðRÞ, define D_c:A_rðRÞ !A_rðRÞ by D_cðfÞðnÞ ¼ fðnÞcðnÞ;

for all f AArðRÞ and nAN^r. For n¼ ðn1;. . .;nrÞ, and m¼ ðm1;. . .;mrÞ in N^r, we write nm¼ ðn1m₁;. . .;n_rm_rÞ. 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 cAA_rðRÞ be additive. Then for all f;gAA_rðRÞ and cAR, (a) Dcðf þgÞ ¼DcðfÞ þDcðgÞ,

(b) DcðflgÞ ¼ flDcðgÞ þglDcðfÞ, (c) D_cðcfÞ ¼cD_cðfÞ.

Consequently, we see that D_c is a derivation on A_rðRÞ over R.

Proof. Letn¼ ðn1;. . .;n_rÞAN^r. First, from the definition of D_c we see that

Dcðf þgÞðnÞ ¼ ðf þgÞðnÞcðnÞ

¼ fðnÞcðnÞ þgðnÞcðnÞ

¼D_cðfÞ þD_cðgÞ:

Thus, (a) holds. Also from the definition of Dc we have that DcðflgÞðnÞ ¼ ðflgÞðnÞcðnÞ:

So,

DcðflgÞðnÞ ¼cðnÞ X

d1e1¼n1

ðd1;e1Þ¼1

. . . X

drer¼nr

ðd_r;erÞ¼1

fðd1;. . .;drÞgðe1;. . .;erÞ

¼ X

d1e1¼n1

ðd1;e1Þ¼1

. . . X

drer¼nr

ðdr;erÞ¼1

fðd1;. . .;d_rÞgðe1;. . .;e_rÞcðnÞ

¼ X

d1e1¼n1

ðd1;e1Þ¼1

. . . X

drer¼nr

ðdr;erÞ¼1

fðd1;. . .;drÞgðe1;. . .;erÞ

ðcðd1;. . .;drÞ þcðe1;. . .;erÞÞ

¼ X

d1e1¼n1

ðd1;e1Þ¼1

. . . X

drer¼nr

ðdr;erÞ¼1

fðd1;. . .;drÞcðd1;. . .;drÞgðe1;. . .;erÞ

þ X

d1e1¼n1

ðd1;e1Þ¼1

. . . X

drer¼nr

ðdr;erÞ¼1

fðd1;. . .;d_rÞcðe1;. . .;e_rÞgðe1;. . .;e_rÞ

¼ flD_cðgÞ þglD_cðfÞ:

Therefore (b) holds. Also, it is clear that (c) holds, and this proves the proposition.

Lemma 1. Let f;gAA₁ðRÞ. Let p be a prime, and let M_p be the monoid

f1;p;p²;. . .g under multiplication. Suppose that suppðfÞ;suppðgÞJMp. If

fð1Þ ¼0, and gð1Þ ¼1, then f ¼ flg.

Proof. We have that suppðflgÞJM_p since suppðfÞ;suppðgÞJM_p. Thus both f and flg vanish outside the monoid Mp. Let now n be a positive integer. Then

ðflgÞðpⁿÞ ¼ X

de¼pⁿ ðd;eÞ¼1

fðdÞgðeÞ

¼ fð1ÞgðpⁿÞ þfðpⁿÞgð1Þ

¼ fðpⁿÞ:

Thus, f ¼ f lg.

Lemma 2. Let f AA₁ðRÞbe multiplicative and cAA₁ðRÞbe additive. Let p be a prime, and let M_p be the monoid f1;p;p²;. . .g under multiplication as in Lemma 1. Suppose that suppðfÞJMp. Then ^Dc_f^ðfÞ¼DcðfÞ, 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 DcðfÞ and f¹ will be supported on M_p. We have moreover that f¹ð1Þ ¼1 because f is multiplicative. Also since c is additive, cð1Þ ¼0. Applying Lemma 1, we conclude that ^Dcð_f^fÞ¼DcðfÞ.

Theorem 2. Let f AA1ðRÞ be multiplicative and cAA1ðRÞ 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 fp¼ fj_Mp, i.e.,

fpðmÞ ¼ fðmÞ if m¼p^k;kb1 0 else:

Then D_cðfÞ

f ðnÞ ¼ X

pAPn

DcðfpÞðnÞ ¼

cðnÞfðnÞ if n¼p^k for some p prime and kb1;

0 else:

8<

:

Proof. Fix an n and let n¼p₁^s1. . .p_t^st be the prime factorization of n.

Let Mbe the set of allmANsuch that whenever p is a prime and p dividesm, p also dividesn. Note thatMis a monoid under multiplication, generated by the primes p1;. . .;pt. Let g¼ fj_M, i.e., for any mAN,

gðmÞ ¼ fðmÞ if mAM 0 else:

Suppose that m, k are in N, and ðm;kÞ ¼1. If mBM, or kBM, then fj_MðmÞ ¼0, or fj_MðkÞ ¼0, and so, gðmÞgðkÞ ¼ fj_MðmÞfj_MðkÞ ¼0¼ fj_MðmnÞ ¼gðmnÞ since mnBM whenever one of m, or n does not belong to M. If m, n are relatively prime and m;nAM then mnAM and gðmnÞ ¼ fj_MðmnÞ ¼ fðmnÞ ¼ fðmÞfðnÞ ¼ fj_MðmÞfj_MðnÞ ¼gðmÞgðnÞ. Thus g is multi- plicative. We claim that

g¼ Y

pAPn

fp:

Indeed, let us first observe that if h₁;h₂AA₁ðRÞ are such that suppðh1Þ, suppðh2Þ are contained in M, then suppðh1lh2ÞJM. To see this, let mBM. Then there exists a prime p such that pjm, but p does not divide n.

Now

ðh1lh2ÞðmÞ ¼ X

de¼m ðd;eÞ¼1

h1ðdÞh2ðeÞ:

Since either pjd, or pje whenever m¼de, every term in this sum is 0 because suppðhiÞJM ði¼1;2Þ. Thus, ðh1lh₂ÞðmÞ ¼0 for any mBM.

Hence suppðh1lh2ÞJM. Using the above observation and induction, it follows that suppðQ

pAPn f_pÞJM. 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

pAPn fp on M. Let mAM with

m¼p₁^a1. . .p_t^at;

where all ai are nonnegative integers for i¼1;. . .;t. We have that

Y

pAPn

f_pðmÞ ¼ X

d1...dt¼m ðdi;djÞ¼1; ði0jÞ

f_p1ðd1Þ. . .f_ptðdtÞ

¼ X

b1;...;bt

p₁^b1...p_t^bt¼m

fp1ðp₁^b1Þ. . .fptðp_t^btÞ

¼ fp1ðp₁^a1Þ. . .fptðp_t^atÞ

¼ fðmÞ

¼gðmÞ;

where in the above computation b1;. . .;bt are forced to have unique values equal to a₁;. . .;a_t respectively. Hence g¼ Q

pAPn

f_p, as claimed. Next, we claim that

DcðfÞ f

M

¼DcðgÞlg¹: In order to prove this, we first show that

f¹j_M¼g¹: Note that by the previous claim we know that

g¹ ¼ Y

pAPn

f_p

!1

¼ Y

pAPn

f_p¹;

and as a consequence g¹ is supported on M. We now proceed by induction.

First, since fð1Þ ¼gð1Þ ¼1, it follows immediately that f¹j_Mð1Þ ¼g¹ð1Þ ¼1.

Next, letm>1, and assume that for allk<m,g¹ðkÞ ¼ f¹j_MðkÞ. IfmBM, then f¹j_MðmÞ ¼0¼g¹ðmÞ. Now suppose that mAM. Then, using the equalities ðflf¹ÞðmÞ ¼0¼ ðglg¹ÞðmÞ in combination with the induction hypothesis we derive

f¹j_MðmÞ ¼ f¹ðmÞ

¼ 1

fð1Þ X

de¼m ðd;eÞ¼1

e<m

fðdÞf¹ðeÞ

¼ 1

gð1Þ X

de¼m ðd;eÞ¼1

e<m

gðdÞg¹ðeÞ

¼g¹ðmÞ:

Thus,

f¹j_M¼g¹: Further, it is clear that

D_cðfÞj_M¼D_cðfj_MÞ ¼D_cðgÞ:

By the above two relations we conclude that

DcðgÞlg¹ ¼DcðfÞj_Mlf¹j_M:

Therefore in order to prove the claim it remains to show that DcðfÞ

f

M

¼DcðfÞj_Mlf¹j_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 ar- bitrary point mAM. For such an m, any representation of m as a product m¼de forces both d, e to belong to M. Thus

DcðfÞ f

M

ðmÞ ¼DcðfÞ

f ðmÞ ¼ X

ðd;eÞ¼1de¼m

D_cðfÞðdÞf¹ðeÞ

¼ X

ðd;eÞ¼1de¼m

D_cðfÞj_MðdÞf¹j_MðeÞ ¼ ðDcðfÞj_Mlf¹j_MÞðmÞ:

We conclude that ^Dcð_f^fÞ_M¼DcðfÞj_Mlf¹j_M, and hence D_cðfÞ

f

M

¼DcðgÞlg¹;

as claimed. On the other hand, by applying Proposition 2 (b) repeatedly, we obtain

DcðgÞlg¹¼D_cðQ

pAPn f_pÞ Q

pAPn fp

¼ X

pAPn

Dc fp

fp

:

By the above two relations we deduce that DcðfÞ

f

M

¼ X

pAPn

DcðfpÞ fp

:

But by Lemma 2, P

pAPn

DcðfpÞ

fp equalsP

pAPnDcðfpÞ. Therefore, we have that D_cðfÞ

f

_M¼ X

pAPn

DcðfpÞ:

Since n is in M, it follows in particular that DcðfÞ

f ðnÞ ¼ X

pAPn

D_cðf_pÞðnÞ:

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 AArðRÞ be multiplicative and consider the monoids M1¼ fðk;1;. . .;1ÞAN^r:kANg;. . .;Mr¼ fð1;. . .;1;kÞAN^r:kANg:

Let f₁¼ fj_M1;. . .;f_r¼ fj_Mr. Then

f ¼Y^r

i¼1

f_i¼ f₁l lf_r:

Proof. Let m¼ ðm1;m₂;. . .;m_rÞAN^r. We have that Y^r

i¼1

fi

!

ðmÞ ¼ X

d11...d1r¼m1

ðd1i;d1jÞ¼1; ði0jÞ

. . . X

d11...d1r¼m1

ðd1i;d1jÞ¼1;ði0jÞ

Y^r

i¼1

fiðd1i;. . .;driÞ

¼Y^r

i¼1

fið1;. . .;1;mi;1;. . .;1Þ

¼Y^r

i¼1

fð1;. . .;1;mi;1;. . .;1Þ

¼ fðm1;. . .;m_rÞ Hence, f ¼Q^r

i¼1

f_i, and the lemma is proved.

Theorem 3. Let R be an integral domain, and let r be a positive integer.

Then, for any multiplicative function f AA_rðRÞ, any additive function cAA_rðRÞ, and any n¼ ðn1;. . .;nrÞAN^r, we have

D_cðfÞ f ðnÞ ¼

cðnÞfðnÞ if n1¼ ¼ni1¼niþ1¼ ¼nr¼1 and ni¼p^k for some p prime; kb1; and 1aiar;

0 else;

8<

:

where the division on the left side is taken with respect to the unitary con- volution.

Proof. Let f be multiplicative and consider the monoids

M₁¼ fðk;1;. . .;1ÞAN^r:kANg;. . .;M_r¼ fð1;. . .;1;kÞAN^r:kANg as in Lemma 3. Let f₁¼ fj_M1;. . .;f_r¼ fj_Mr. Then by Lemma 3, f ¼ Q^r

i¼1

f_i. Applying Proposition 2 (b) repeatedly, we get

D_cðfÞ

f ¼

D_c Q^r

i¼1

f_i

Q^r

i¼1

fi

¼X^r

i¼1

D_cðfiÞ f_i :

Therefore the desired equality from the statement of Theorem 3 will hold for f provided it holds for each function f_i. On the other hand, each of the functions f_i is supported on a one dimensional monoid isomorphic toN, 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 3 can be used to compute the logarithmic derivative ^Dc_f^ðfÞ. We end this paper with a few very explicit examples. Take R to be the field of com- plex numbers and r¼1. An additive arithmetical function is for instance cðnÞ ¼logn.

1. With R, r and c as above, let f be the Mo¨bius function m, which is a multiplicative function. By its definition, mð1Þ ¼1, and if n>1, n¼

p₁^a1. . .p_k^ak, then

mðnÞ ¼ ð1Þ^k if a₁ ¼ ¼a_k¼1;

0 else:

By Theorem 2 we then have DcðmÞ

m ðnÞ ¼ logp if n¼p for some prime p;

0 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

DcðfÞ

f ðnÞ ¼ kðp^kp^k1Þlogp if n¼p^k for some prime p and kb1;

0 else:

3. With the same R, r and c as before, let f be the sum of divisors function s, given by sðnÞ ¼P

djn

d, which is also a multiplicative arithmetical function. By Theorem 2 we find that

D_cðsÞ s ðnÞ ¼

kðp^kþ11Þlogp

p1 if n¼ p^k for some prime p and kb1;

0 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 ¼X^y

n¼1

tðnÞqⁿ¼qY^y

n¼1

ð1qⁿÞ²⁴; q¼e^2piz;

which is the unique normalized cusp form of weight 12 on SL₂ðZÞ. Ram- anujan 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 convo- lution.

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.

Emre Alkan

Department of Mathematics

University of Illinois at Urbana-Champaign 273Altgeld Hall, MC-382,1409W. Green Street

Urbana, Illinois 61801-2975, USA e-mail: alkan@math.uiuc.edu

Alexandru Zaharescu Department of Mathematics

University of Illinois at Urbana-Champaign 273Altgeld Hall, MC-382,1409W. Green Street

Urbana, Illinois 61801-2975, USA e-mail: zaharesc@math.uiuc.edu

Mohammad Zaki Department of Mathematics

University of Illinois at Urbana-Champaign 273Altgeld Hall, MC-382,1409W. Green Street

Urbana, Illinois 61801-2975, USA e-mail: mzaki@math.uiuc.edu