Expansions of arithmetic functions of several variables with respect to ...

arXiv:1705.01363v1 [math.NT] 3 May 2017

Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums L´aszl´o T´oth Department of Mathematics, University of P´ecs Ifj´ us´ag u ´ tja 6, 7624 P´ecs, Hungary E-mail: [email protected] Abstract We introduce new analogues of the Ramanujan sums, denoted by r cq pnq, associated to unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums r cq pnq. We apply these results to certain functions associated with σ ˚ pnq and φ˚ pnq, representing the unitary sigma function and unitary phi function, respectively.

2010 Mathematics Subject Classification: 11A25, 11N37 Key Words and Phrases: Ramanujan expansion of arithmetic functions, arithmetic function of several variables, multiplicative function, unitary divisor, sum of unitary divisors, unitary Euler function, unitary Ramanujan sum

1

Introduction

Basic notations, used throughout the paper, are included in Section 2.1. Further notations are explained in the text. Ramanujan’s [7] classical identity 8 ÿ σpnq cq pnq “ ζp2q n q2 q“1

pn P Nq

(1)

can be generalized as 8 ÿ cq1 pn1 q ¨ ¨ ¨ cqk pnk q σppn1 , . . . , nk qq “ ζpk ` 1q pn1 , . . . , nk q rq1 , . . . , qk sk`1 q ,...,q “1 1

pn1 , . . . , nk P Nq,

(2)

k

valid for any k P N. See the author [15, Eq. (28)]. For k “ 2 identity (2) was deduced by Ushiroya [18, Ex. 3.8]. By making use of the unitary Ramanujan sums c˚q pnq, we also have 8 ÿ φk`1 prq1 , . . . , qk sq ˚ σppn1 , . . . , nk qq cq1 pn1 q ¨ ¨ ¨ c˚qk pnk q pn1 , . . . , nk P Nq, (3) “ ζpk ` 1q 2pk`1q pn1 , . . . , nk q rq1 , . . . , qk s q ,...,q “1 1

k

1

for any k P N. See [15, Eq. (30)]. In fact, (2) and (3) are special cases of the following general result, which can be applied to several other special functions, as well. Theorem 1.1. ([15, Th. 4.3]) Let g : N Ñ C be an arithmetic function and let k P N. Assume that 8 ÿ |pµ ˚ gqpnq| ă 8. 2k ωpnq nk n“1 Then for every n1 , . . . , nk P N,

gppn1 , . . . , nk qq “

8 ÿ

aq1 ,...,qk cq1 pn1 q ¨ ¨ ¨ cqk pnk q,

8 ÿ

a˚q1 ,...,qk c˚q1 pn1 q ¨ ¨ ¨ c˚qk pnk q

q1 ,...,qk “1

gppn1 , . . . , nk qq “

q1 ,...,qk “1

are absolutely convergent, where aq1 ,...,qk

8 1 ÿ pµ ˚ gqpmQq “ k , Q m“1 mk

a˚q1 ,...,qk “

1 Qk

8 ÿ

m“1 pm,Qq“1

pµ ˚ gqpmQq , mk

with the notation Q “ rq1 , . . . , qk s. Recall that d is a unitary divisor of n if d | n and pd, n{dq “ 1. Notation d k n. Let σ ˚ pnq, defined as the sum of unitary divisors of n, be the unitary analogue of σpnq. Properties of the function σ ˚ pnq, compared to those of σpnq were investigated by several authors. See, e.g., Cohen [1], McCarthy [6], Sitaramachandrarao and Suryanarayana [10], Sitaramaiah and Subbarao [11], Trudgian [16]. For example, one has ÿ

nďx

σ ˚ pnq “

π 2 x2 ` Opxplog xq5{3 q. 12ζp3q

In this paper we are looking for unitary analogues of formulas (1) and (2). Theorem 1.1 can be applied to the function gpnq “ σ ˚ pnq{n. However, in this case pµ ˚ gqppq “ 1{p, pµ ˚ gqppν q “ p1 ´ pq{pν for any prime p and any ν ě 2. Hence the coefficients of the corresponding expansion can not be expressed by simple special functions, and we consider the obtained identities unsatisfactory. Let pk, nq˚˚ denote the greatest common unitary divisor of k and n. Note that d k pk, nq˚˚ holds true if and only if d k k and d k n. Bi-unitary analogues of the Ramanujan sums may be defined as follows: ÿ c˚˚ expp2πikn{qq pq, n P Nq, q pnq “ 1ďkďq pk,qq˚˚ “1

2

but the function q ÞÑ cq pnq is not multiplicative, and its properties are not parallel to the sums ˚˚ cq pnq and c˚q pnq. The function c˚˚ q pqq “ φ pqq, called bi-unitary Euler function was investigated in our paper [13]. Therefore, we introduce in Section 2.3 new analogues of the Ramanujan sums, denoted by cq pnq, also associated with unitary divisors, and show that r 8 ÿ φk`1 prq1 , . . . , qk sq σ ˚ ppn1 , . . . , nk q˚k q “ ζpk ` 1q r cq pn1 q ¨ ¨ ¨ r cqk pnk q pn1 , . . . , nk P Nq, pn1 , . . . , nk q˚k rq1 , . . . , qk s2pk`1q 1 q ,...,q “1 1

k

(4) where pn1 , . . . , nk q˚k denotes the greatest common unitary divisor of n1 , . . . , nk P N. Now formulas (2), (3) and (4) are of the same shape. In the case k “ 1, identity (4) gives 8 ÿ σ ˚ pnq φ2 pqq “ ζp2q cq pnq pn P Nq, r n q4 q“1

which may be compared to (1). We also deduce a general result for arbitrary arithmetic functions f of several variables (Theorem 3.1), which is the analogue of [15, Th. 4.1], concerning the Ramanujan sums cq pnq and their unitary analogues c˚q pnq. We point out that in the case k “ 1, Theorem 3.1 is the analogue of the result of Delange [2], concerning classical Ramanujan sums. As applications, we consider the functions f pn1 , . . . , nk q “ gppn1 , . . . , nk q˚k q, where g belongs to a large class of functions of one variable, including σ ˚ pnq{n and φ˚ pnq{n, where φ˚ is the unitary Euler function (Theorem 3.2). For background material on classical Ramanujan sums and Ramanujan expansions (RamanujanFourier series) of functions of one variable we refer to the book by Schwarz and Spilker [9] and to the survey papers by Lucht [5] and Ram Murty [8]. Section 2 includes some general properties on arithmetic functions of one and several variables defined by unitary divisors, needed in the present paper.

2 2.1

Premiminaries Basic notations

‚ N “ t1, 2, . . .u, N0 “ t0, 1, 2, . . .u, ‚ P is the set of (positive) primes, ś ‚ the prime power factorization of n P N is n “ pPP pνp pnq , the product being over the primes p, where all but a finite number of the exponents νp pnq are zero, ‚ pn1 , . . . , nk q and rn1 , . . . , nk s denote the greatest common divisor and the least common multiple, respectively ř of n1 , . . . , nk P N, ‚ pf ˚ gqpnq “ d|n f pdqgpn{dq is the Dirichlet convolution of the functions f, g : N Ñ C, ‚ ids is the function ids pnq “ ns (n P N, s P R), ‚ 1 “ id0 is the constant 1 function, ‚ µ is the M¨obius function, ‚ ωpnq stands for the number of distinct prime divisors of n, 3

‚ σpnq is the sum of divisors of n, ś ‚ φs is the Jordan function of order s given by φs pnq “ ns p|n p1 ´ 1{ps q (s P R), ‚ φ “ φ1 isřEuler’s totient function, ‚ cq pnq “ 1ďkďq,pk,qq“1 expp2πikn{qq are the Ramanujan sums (q, n P N), ‚ ζ is the Riemann zeta function, ‚ d k n means that d is a unitary divisor of n, i.e., d | n and pd, n{dq “ 1 (we remark that this is in concordance with the standard notation pν k n used for prime powers pν ), ‚ pk, nq˚ “řmaxtd : d | k, d k nu, ‚ c˚q pnq “ 1ďkďq,pk,qq˚ “1 expp2πikn{qq are the unitary Ramanujan sums (q, n P N), ‚ pn1 , . . . , nk q˚k denotes the greatest common unitary divisor of n1 , . . . , nk P N, ‚ pn1 , n2 q˚˚ř“ pn1 , n2 q˚2 , ‚ σs˚ pnq “ dkn ds (s P R), ‚ σ ˚ pnq “ σ1˚ pnq is the sum of unitary divisors of n, ‚ τ ˚ pnq “ σ0˚ pnq is the number of unitary divisors of n, which equals 2ωpnq .

2.2

Functions defined by unitary divisors

The study of arithmetic functions defined by unitary divisors goes back to Vaidyanathaswamy [17] and Cohen [1]. The function σ ˚ pnq was already defined above. The analog of Euler’s φ function is φ˚ , defined by φ˚ pnq “ #tk P N : 1 ď k ď n, pk, nq˚ “ 1u. The functions σ ˚ and φ˚ are multiplicative and σ ˚ ppν q “ pν ` 1, φ˚ ppν q “ pν ´ 1 for any prime powers pν (ν ě 1). The unitary convolution of the functions f and g is ÿ pf ˆ gqpnq “ f pdqgpn{dq pn P Nq, dkn

it preserves the multiplicativity of functions, and the inverse of the constant 1 function under the unitary convolution is µ˚ , where µ˚ pnq “ p´1qωpnq , also multiplicative. The set A of arithmetic functions forms a unital commutative ring with pointwise addition and the unitary convolution, having divisors of zero.

2.3

Modified unitary Ramanujan sums

For q, n P N we introduce the functions r cq pnq by the formula # ÿ q, if q k n, cd pnq “ r 0, if q ∦ n. dkq

It follows that r cq pnq is multiplicative in q, # pν ´ 1, r cpν pnq “ ´1,

if pν k n, if pν ∦ n,

for any prime powers pν (ν ě 1) and r cq pnq “

ÿ

dµ˚ pq{dq

dkpn,qq˚˚

4

pq, n P Nq.

(5)

(6)

We will need the following result. Proposition 2.1. For any q, n P N, ÿ |r cd pnq| “ 2ωpq{pn,qq˚˚ q pn, qq˚˚ ,

(7)

dkq

ÿ

|r cd pnq| ď 2ωpqq n.

(8)

dkq

Proof. If q “ pν (ν ě 1) is a prime power, then we have by (6), # ÿ 1 ` pν ´ 1 “ pν , ˚ ˚ ˚ |cd pnq| “ |c1 pnq| ` |cpν pnq| “ 1 ` 1 “ 2, dkpν

if pν k n, otherwise.

Now (7) follows at once by the multiplicativity in q of the involved functions, while (8) is its immediate consequence. For classical Ramanujan sums the inequality corresponding to (8) is crucial in the proof of the theorem of Delange [2], while the identity corresponding to (7) was pointed out by Grytczuk [3]. In the case of unitary Ramanujan sums the counterparts of (7) and (8) were proved by the author [15, Prop. 3.1]. Proposition 2.2. For any q, n P N, r cq pnq “

φ˚ pqqµ˚ pq{pn, qq˚˚ q . φ˚ pq{pn, qq˚˚ q

(9)

Proof. Both sides of (9) are multiplicative in q. If q “ pν (ν ě 1) is a prime power, then # φ˚ ppν qµ˚ p1q “ pν ´ 1, if pν k n, φ˚ ppν qµ˚ ppν {pn, pν q˚˚ q φ˚ p1q “ “ cpν pnq, ˚ ν ˚ ν φ pp qµ pp q φ˚ ppν {pn, pν q˚˚ q “ ´1, otherwise. φ˚ ppν q by (6). For the Ramanujan sums cq pnq the identity similar to (9) is usually attributed to H¨older, but was proved earlier by Kluyver [4]. In the case of the unitary Ramanujan sums c˚q pnq the counterpart of (9) was deduced by Suryanarayana [12]. Basic properties (including those mentioned above) of the classical Ramanujan sums cq pnq, their unitary analogues c˚q pnq and the modified sums r cq pnq can be compared by the next table.

5

ÿ

cq pnq “

d|pn,qq

cq pnq “

d|q

ÿ d|q

2.4

# q, 0,

if pν | n, if pν´1 k n, if pν´1 ∤ n

cq pnq “ r

˚

φ pqqµ pq{pn, qq˚ q φ˚ pq{pn, qq˚ q

c˚ pν pnq “

# pν ´ 1, ´1,

if q | n, if q ∤ n

|cd pnq| “ 2ωpq{pn,qqq pn, qq

˚

c˚ q pnq “

φpqqµpq{pn, qqq φpq{pn, qq

cd pnq “

dµ˚ pq{dq

d|pn,qq˚

$ ν ν´1 ’ , &p ´ p cpν pnq “ ´pν´1 , ’ % 0, ÿ

ÿ

c˚ q pnq “

dµpq{dq

ÿ

c˚ d pnq “

dkq

ÿ

# q, 0,

if pν | n, if pν ∤ n

cq pnq “ r cpν pnq “ r ÿ

if q | n, if q ∤ n

ωpq{pn,qq˚ q |c˚ pn, qq˚ d pnq| “ 2

dkq

dkq

ÿ

ÿ

dµ˚ pq{dq

dkpn,qq˚˚

φ˚ pqqµ˚ pq{pn, qq˚˚ q φ˚ pq{pn, qq˚˚ q # pν ´ 1, ´1,

cd pnq “ r

# q, 0,

if pν k n, if pν ∦ n

if q k n, if q ∦ n

|r cd pnq| “ 2ωpq{pn,qq˚˚ q pn, qq˚˚

dkq

Table: Properties of cq pnq, c˚q pnq and r cq pnq

Arithmetic functions of several variables

For every fixed k P N the set Ak of arithmetic functions f : Nk Ñ C of k variables is a unital commutative ring with pointwise addition and the unitary convolution defined by ÿ f pd1 , . . . , dk qgpn1 {d1 , . . . , nk {dk q, (10) pf ˆ gqpn1 , . . . , nk q “ d1 kn1 ,...,dk knk

the unity being the function δk , where δk pn1 , . . . , nk q “

#

if n1 “ . . . “ nk “ 1, otherwise.

1, 0,

The inverse of the constant 1 function under (10) is µ˚k , given by µ˚k pn1 , . . . , nk q “ µ˚ pn1 q ¨ ¨ ¨ µ˚ pnk q “ p´1qωpn1 q`¨¨¨`ωpnk q

pn1 , . . . , nk P Nq.

A function f P Ak is said to be multiplicative if it is not identically zero and f pm1 n1 , . . . , mk nk q “ f pm1 , . . . , mk qf pn1 , . . . , nk q holds for any m1 , . . . , mk , n1 , . . . , nk P N such that pm1 ¨ ¨ ¨ mk , n1 ¨ ¨ ¨ nk q “ 1. If f is multiplicative, then it is determined by the values f ppν1 , . . . , pνk q, where p is prime and ν1 , . . . , νk P N0 . More exactly, f p1, . . . , 1q “ 1 and for any n1 , . . . , nk P N, ź f pn1 , . . . , nk q “ f ppνp pn1 q , . . . , pνp pnk q q. pPP

Similar to the one dimensional case, the unitary convolution (10) preserves the multiplicativity of functions. See our paper [14], which is a survey on (multiplicative) arithmetic functions of several variables. 6

3

Main results We first prove the following general result.

Theorem 3.1. Let f : Nk Ñ C be an arithmetic function (k P N). Assume that 8 ÿ

2ωpn1 q`¨¨¨`ωpnk q

n1 ,...,nk “1

|pµ˚k ˆ f qpn1 , . . . , nk q| ă 8. n1 ¨ ¨ ¨ nk

(11)

Then for every n1 , . . . , nk P N, 8 ÿ

f pn1 , . . . , nk q “

q1 ,...,qk “1

where r aq1 ,...,qk “

r aq1 ,...,qk r cq1 pn1 q ¨ ¨ ¨ r cqk pnk q,

8 ÿ

m1 ,...,mk “1 pm1 ,q1 q“1,...,pmk ,qk q“1

(12)

pµ˚k ˆ f qpm1 q1 , . . . , mk qk q , m1 q 1 ¨ ¨ ¨ mk q k

(13)

the series (12) being absolutely convergent. Proof. We have for any n1 , . . . , nk P N, by using property (5), ÿ pµ˚k ˆ f qpd1 , . . . , dk q f pn1 , . . . , nk q “ d1 kn1 ,...,dk knk

“

8 ÿ

d1 ,...,dk

“

8 ÿ

ÿ pµ˚k ˆ f qpd1 , . . . , dk q ÿ r cqk pnk q r cq1 pn1 q ¨ ¨ ¨ d ¨ ¨ ¨ d 1 k “1

q1 ,...,qk “1

qk kdk

q1 kd1

cqk pnk q r cq1 pn1 q ¨ ¨ ¨ r

8 ÿ

d1 ,...,dk “1 q1 kd1 ,...,qk kdk

pµ˚k ˆ f qpd1 , . . . , dk q , d1 ¨ ¨ ¨ dk

leading to expansion (12) with the coefficients (13), by denoting d1 “ m1 q1 , . . . , dk “ mk qk . The rearranging of the terms is justified by the absolute convergence of the multiple series, shown hereinafter: 8 ÿ |r aq1 ,...,qk ||r cq1 pn1 q| ¨ ¨ ¨ |r cqk pnk q| q1 ,...,qk “1

8 ÿ

ď

q1 ,...,qk “1 m1 ,...,mk “1 pm1 ,q1 q“1,...,pmk ,qk q“1

“

8 ÿ

t1 ,...,tk

|pµ˚k ˆ f qpm1 q1 , . . . , mk qk q| |r cq1 pn1 q| ¨ ¨ ¨ |r cqk pnk q| m1 q 1 ¨ ¨ ¨ mk q k

|pµ˚k ˆ f qpt1 , . . . , tk q| t1 ¨ ¨ ¨ tk “1

ÿ

m1 q1 “t1 pm1 ,q1 q“1

7

|r cq1 pn1 q| ¨ ¨ ¨

ÿ

mk qk “tk pmk ,qk q“1

|r cqk pnk q|

8 ÿ

ď n1 ¨ ¨ ¨ nk

2ωpt1 q`¨¨¨`ωptk q

t1 ,...,tk “1

|pµ˚k ˆ f qpt1 , . . . , tk q| ă 8, t1 ¨ ¨ ¨ tk

by using inequality (8) and condition (11). Next we consider the case f pn1 , . . . , nk q “ gppn1 , . . . , nk q˚k q. The following result is the analogue of Theorem 1.1. Theorem 3.2. Let g : N Ñ C be an arithmetic function and let k P N. Assume that 8 ÿ

2k ωpnq

n“1

|pµ˚ ˆ gqpnq| ă 8. nk

Then for every n1 , . . . , nk P N, 8 ÿ

gppn1 , . . . , nk q˚k q “

q1 ,...,qk “1

is absolutely convergent, where r aq1 ,...,qk “

1 Qk

cq1 pn1 q ¨ ¨ ¨ r cqk pnk q, r aq1 ,...,qk r

8 ÿ

m“1 pm,Qq“1

pµ˚ ˆ gqpmQq , mk

(14)

with the notation Q “ rq1 , . . . , qk s. Proof. We apply Theorem 3.1. Taking into account the identity ÿ gppn1 , . . . , nk q˚k q “ pµ˚ ˆ gqpdq dkn1 ,...,dknk

we see that now pµ˚k ˆ f qpn1 , . . . , nk q “

#

pµ˚ ˆ gqpnq, 0,

if n1 “ . . . “ nk “ n, otherwise.

Therefore the coefficients of the expansion are r aq1 ,...,qk “

8 ÿ

n“1 m1 q1 “...“mk qk “n pm1 ,q1 q“1,...,pmk ,qk q“1

“

8 ÿ

n“1 q1 kn,...,qk kn

pµ˚k ˆ f qpm1 q1 , . . . , mk qk q m1 q 1 ¨ ¨ ¨ mk q k

pµ˚ ˆ gqpnq , nk

and we use that q1 k n, . . . , qk k n holds if and only if rq1 , . . . , qk s “ Q k n, that is, n “ mQ with pm, Qq “ 1. 8

Corollary 3.3. For every n1 , . . . , nk P N the following series are absolutely convergent: 8 ÿ φs`k pQqr cq1 pn1 q ¨ ¨ ¨ r cqk pnk q σs˚ ppn1 , . . . , nk q˚k q “ ζps ` kq s 2ps`kq pn1 , . . . , nk q˚k Q q ,...,q “1 1

ps P R, s ` k ą 1q,

(15)

k

8 ÿ

τ ˚ ppn1 , . . . , nk q˚k q “ ζpkq

q1 ,...,qk

φk pQqr cq1 pn1 q ¨ ¨ ¨ r cqk pnk q 2k Q “1

pk ě 2q.

(16)

Proof. Apply Theorem 3.2 to gpnq “ σs˚ pnq{ns . Here µ˚ ˆ g “ µ˚ ˆ

1 1 1 ˆ ids “ pµ˚ ˆ 1q ˆ “ , ids ids ids

hence pµ˚ ˆ gqpnq “ 1{ns (n P N). We deduce by (14) that r aq1 ,...,qk “

8 ÿ

1 Qs`k

m“1 pm,Qq“1

1 ms`k

“ ζps ` kq

φs`k pQq , Q2ps`kq

which completes the proof.

case s “ 1 identity (15) reduces to (4). Now let consider the function φ˚s pnq “ ś In the sν ˚ ˚ pν kn pp ´ 1q, representing the unitary Jordan function of order s. Herer φs “ µ ˆ ids , and ˚ ˚ φ1 “ φ is the unitary Euler function, already mentioned in Section 2.2. Corollary 3.4. For every n1 , . . . , nk P N the following series are absolutely convergent: ˙ źˆ 2 φ˚s ppn1 , . . . , nk q˚k q “ ζps ` kq 1 ´ s`k ˆ pn1 , . . . , nk qs˚k p pPP 8 ÿ

µ˚ pQqφs`k pQqr cq pn1 q ¨ ¨ ¨ r cqk pnk q ś 1 ps P R, s ` k ą 1q, 2ps`kq s`k Q q p|Q p1 ´ 2{p q1 ,...,qk “1 ˙ ÿ 8 źˆ µ˚ pQqφk`1 pQqr cq pn1 q ¨ ¨ ¨ r cqk pnk q φ˚ ppn1 , . . . , nk q˚k q 2 ś 1 “ ζpk`1q 1 ´ k`1 2pk`1q k`1 q pn1 , . . . , nk q˚k p Q p1 ´ 2{p p|Q q ,...,q “1 pPP ˆ

1

Proof. Apply Theorem 3.2 to gpnq “

k

φ˚s pnq{ns .

µ˚ ˆ g “ µ˚ ˆ

Here

µ˚ ˆ ids µ˚ µ˚ “ pµ˚ ˆ 1q ˆ “ , ids ids ids

that is, pµ˚ ˆ gqpnq “ µ˚ pnq{ns (n P N). We deduce by (14) that r aq1 ,...,qk “

“

1 Qs`k

8 ÿ

m“1 pm,Qq“1

µ˚ pmQq µ˚ pQq “ ms`k Qs`k

8 ÿ

m“1 pm,Qq“1

µ˚ pmq ms`k

˙źˆ ˙ˆ ˙´1 źˆ 1 µ˚ pQq 2 2 1 ´ , ζps ` kq 1 ´ 1 ´ s`k s`k s`k Qs`k p p p pPP p|Q

leading to (17).

(17)

9

pk ě 1q.

For m P N, m ě 2 consider the functionřgpnq “ mωpnq , which is the unitary analogue of the Piltz divisor function τm pnq. Here mωpnq “ dkn pm ´ 1qωpdq for any n P N. We obtain by similar arguments: Corollary 3.5. For every n1 , . . . , nk P N the following series is absolutely convergent: ˙ źˆ m´2 ωppn1 ,...,nk q˚k q ˆ m “ ζpkq 1` pk pPP ˆ

8 ÿ

q1 ,...,qk

φk pQqpm ´ 1qωpQq r cq1 pn1 q ¨ ¨ ¨ r cqk pnk q ś 2k k Q p|Q p1 ` pm ´ 2q{p q “1

(18)

pm, k ě 2q,

For m “ 2 identity (18) reduces to (16). It is possible to formulate the results of Theorem 3.1 in the case of multiplicative functions f of k variables, and Theorem 3.2 in the case of multiplicative functions g of one variable. Note that if g is multiplicative, then f pn1 , . . . , nk q “ gppn1 , . . . , nk q˚k q is multiplicative, viewed as a function of k variables. See also Delange [2] and the author [15]. Furthermore, it is possible to apply the above results to other special (multiplicative) functions. We do not go into more details.

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