Expansions of Arithmetic Functions of Several Variables with Respect to Certain Modified Unitary Ramanujan Sums

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8 σppn1,...,nkqq φk`1prq1,...,qksq ˚ ˚ “ ζpk ` 1q c pn1q¨¨¨ c pn q pn1,...,n P Nq, 2pk`1q q1 qk k k pn1,...,nkq rq1,...,qks q1,...,qk“1 ÿ (1.3) for any k P N. See [15, Eq. (30)]. The notations used here (and throughout the paper), which are not explained in the text, are included in Section 2.1. In fact, (1.2) and (1.3) are special cases of the following general result, which can be applied to several other special functions, as well.

Theorem 1 ([15, Th. 4.3]). Let g : N Ñ C be an arithmetic function and let k P N. Assume that 8 |pµ ˚ gqpnq| 2kωpnq ă8. nk n“1 ÿ Then for every n1,...,nk P N,

gppn1,...,nkqq “ aq1,...,qk cq1 pn1q¨¨¨ cqk pnkq, q1,...,q “1 ÿk 8 ˚ ˚ ˚ 1 1 gppn ,...,nkqq “ aq1,...,qk cq1 pn q¨¨¨ cqk pnkq q1,...,q “1 ÿk are absolutely convergent, where

1 8 pµ ˚ gqpmQq a 1 “ , q ,...,qk Qk mk m“1 ÿ 1 8 pµ ˚ gqpmQq a˚ “ , q1,...,qk Qk mk m“1 pm,Qÿq“1 with the notation Q “ rq1,...,qks. 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

2 2 π x 5 3 σ˚pnq“ ` Opxplog xq { q. ζ nďx 12 p3q ÿ

2 In this paper we are looking for unitary analogues of formulas (1.1) and (1.2). Theorem 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: ˚˚ N cq pnq“ expp2πikn{qq pq,n P q, 1ďkďq pk,qÿq˚˚“1 but the function q ÞÑ cqpnq is not multiplicative, and its properties are not parallel to the sums ˚ ˚˚ ˚˚ cqpnq and cq pnq. The function cq 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 cqpnq, also associated with unitary divisors, and show that

˚ 8 σ ppn1,...,nkq˚kq φk`1prq1,...,qksq N r “ ζpk ` 1q 2 1 cq1 pn1q¨¨¨ cqk pnkq pn1,...,nk P q, pn1,...,n q rq1,...,q s pk` q k ˚k q1,...,q “1 k ÿk r r (1.4) where pn1,...,nkq˚k denotes the greatest common unitary divisor of n1,...,nk P N. Now formulas (1.2), (1.3) and (1.4) are of the same shape. In the case k “ 1, identity (1.4) gives 8 σ˚pnq φ2pqq “ ζp2q c pnq pn P Nq, n q4 q q“1 ÿ which may be compared to (1.1). r We also deduce a general result for arbitrary arithmetic functions f of several variables (Theorem 2), which is the analogue of [15, Th. 4.1], concerning the Ramanujan sums cqpnq and ˚ their unitary analogues cq pnq. We point out that in the case k “ 1, Theorem 2 is the analogue of the result of Delange [2], concerning classical Ramanujan sums. As applications, we consider the functions fpn1,...,nkq“ gppn1,...,nkq˚kq, 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). For background material on classical Ramanujan sums and Ramanujan expansions (Ramanujan- Fourier 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 Premiminaries

2.1 Notations ‚ P is the set of (positive) primes,

3 N νppnq ‚ the prime power factorization of n P is n “ pPP p , the product being over the primes p, where all but a ﬁnite number of the exponents νppnq are zero, ś N C ‚ pf ˚ gqpnq“ d|n fpdqgpn{dq is the Dirichlet convolution of the functions f, g : Ñ , s ‚ ids is the function idspnq“ n (n P N,s P R), ř ‚ 1 “ id0 is the constant 1 function, ‚ µ is the M¨obius function, ‚ ωpnq stands for the number of distinct prime divisors of n, s s ‚ φs is the Jordan function of order s given by φspnq“ n p|np1 ´ 1{p q (s P R), ‚ φ “ φ1 is Euler’s totient function, ś ‚ 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˚ “ maxtd : d | k, d k nu, c˚ n πikn q q,n N ‚ q p q“ 1ďkďq,pk,qq˚“1 expp2 { q are the unitary Ramanujan sums ( P ), ‚ pn1,...,nkq k denotes the greatest common unitary divisor of n1,...,nk P N, ř ˚ ‚ pn1,n2q˚˚ “ pn1,n2q˚2, ˚ s ‚ σs pnq“ dkn d (s P R), ˚ ˚ ‚ σ pnq“ σ1 pnq is the sum of unitary divisors of n, ˚ ř˚ ωpnq ‚ τ pnq“ σ0 pnq is the number of unitary divisors of n, which equals 2 .

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˚ppνq“ pν ` 1, φ˚ppνq“ pν ´ 1 for any prime powers pν (ν ě 1). The unitary convolution of the functions f and g is

pf ˆ gqpnq“ fpdqgpn{dq pn P Nq, d n ÿk 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 cqpnq by the formula

q, if q k n, cdpnq“r (2.1) d q #0, if q ∦ n. ÿk r It follows that cqpnq is multiplicative in q,

pν ´ 1, if pν k n, r c ν n p p q“ ν (2.2) #´1, if p ∦ n, r 4 for any prime powers pν (ν ě 1) and

˚ cqpnq“ dµ pq{dq pq,n P Nq. d n,q kpÿq˚˚ r We will need the following result.

Proposition 1. For any q,n P N,

ωpq{pn,qq˚˚q |cdpnq| “ 2 pn,qq˚˚, (2.3) d q ÿk r ωpqq |cdpnq| ď 2 n. (2.4) d q ÿk Proof. If q “ pν (ν ě 1) is a prime power,r then we have by (2.2),

For classical Ramanujan sums the inequality corresponding to (2.4) is crucial in the proof of the theorem of Delange [2], while the identity corresponding to (2.3) was pointed out by Grytczuk [3]. In the case of unitary Ramanujan sums the counterparts of (2.3) and (2.4) were proved by the author [15, Prop. 3.1].

Proposition 2. For any q,n P N,

˚ ˚ φ pqqµ pq{pn,qq˚˚q cqpnq“ ˚ . (2.5) φ pq{pn,qq˚˚q

Proof. Both sides of (2.5) are multiplicativer in q. If q “ pν (ν ě 1) is a prime power, then

˚ ν ˚ 1 ˚ ν ˚ ν ν φ pp qµ p q ν ν φ pp qµ pp {pn,p q˚˚q φ˚p1q “ p ´ 1, if p k n, “ ˚ ν ˚ ν “ c ν pnq, φ˚ppν{pn,pνq q φ pp qµ pp q p ˚˚ # φ˚ppν q “´1, otherwise. by (2.2).

For the Ramanujan sums cqpnq the identity similar to (2.5) is usually attributed to H¨older, ˚ but was proved earlier by Kluyver [4]. In the case of the unitary Ramanujan sums cq pnq the counterpart of (2.5) was deduced by Suryanarayana [12]. Basic properties (including those mentioned above) of the classical Ramanujan sums cqpnq, ˚ their unitary analogues cq pnq and the modiﬁed sums cqpnq can be compared by the next table.

5 ˚ ˚ ˚ cq pnq“ dµpq{dq cq pnq“ dµ pq{dq cq pnq“ dµ pq{dq d|pn,qq d|pn,qq dkpn,qq ÿ ÿ ˚ ÿ ˚˚ φpqqµpq{pn, qqq φ˚pqqµ˚pq{pn, qq q r φ˚pqqµ˚pq{pn, qq q c n c˚ n ˚ c n ˚˚ qp q“ q p q“ ˚ q p q“ ˚ φpq{pn, qq φ pq{pn, qq˚q φ pq{pn, qq˚˚q

ν ν´1 ν r p ´ p , if p | n, ν ν ν ν 1 1 ˚ p ´ 1, if p | n, p ´ 1, if p k n, ν ν´ ν´ ν ν cp pnq“ $´p , if p k n, cp pnq“ ν cp pnq“ ν ν 1 #´1, if p ∤ n #´1, if p ∦ n &’0, if p ´ ∤ n r %’ q, if q | n, ˚ q, if q | n, q, if q k n, cdpnq“ c pnq“ cdpnq“ 0, if q n d 0, if q n 0, if q n d|q # ∤ dkq # ∤ dkq # ∦ ÿ ÿ ÿ ωpq{pn,qqq ˚ ωpq{pn,qq˚ q r ωpq{pn,qq˚˚ q |cdpnq| “ 2 pn, qq |cd pnq| “ 2 pn, qq˚ |cdpnq| “ 2 pn, qq˚˚ d|q dkq dkq ÿ ÿ ÿ ˚ r Table: Properties of cqpnq, cq pnq and cqpnq

2.4 Arithmetic functions of several variables r

k For every ﬁxed k P N the set Ak of arithmetic functions f : N Ñ C of k variables is a unital commutative ring with pointwise addition and the unitary convolution deﬁned by

pf ˆ gqpn1,...,nkq“ fpd1,...,dkqgpn1{d1,...,nk{dkq, (2.6) d1 n1,...,d n k ÿ kk k the unity being the function δk, where

1, if n1 “¨¨¨“ nk “ 1, δkpn1,...,nkq“ #0, otherwise.

˚ The inverse of the constant 1 function under (2.6) is µk, given by

˚ ˚ ˚ ωpn1q`¨¨¨`ωpnkq N µkpn1,...,nkq“ µ pn1q¨¨¨ µ pnkq“p´1q pn1,...,nk P q.

A function f P Ak is said to be multiplicative if it is not identically zero and

fpm1n1,...,mknkq“ fpm1,...,mkqfpn1,...,nkq holds for any m1,...,mk,n1,...,nk P N such that pm1 ¨ ¨ ¨ mk,n1 ¨ ¨ ¨ nkq“ 1. If f is multiplicative, then it is determined by the values fppν1 ,...,pνk q, where p is prime and ν1,...,νk P N Y t0u. More exactly, fp1,..., 1q“ 1 and for any n1,...,nk P N,

νppn1q νppnkq fpn1,...,nkq“ fpp ,...,p q. pPP ź Similar to the one dimensional case, the unitary convolution (2.6) 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 2. Let f : Nk Ñ C be an arithmetic function (k P N). Assume that

8 ˚ |pµ ˆ fqpn1,...,nkq| 2ωpn1q`¨¨¨`ωpnkq k ă8. (3.1) n1 ¨ ¨ ¨ n n1,...,n “1 k ÿk

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

fpn1,...,nkq“ aq1,...,qk cq1 pn1q¨¨¨ cqk pnkq, (3.2) q1,...,q “1 ÿk where r r r 8 ˚ pµk ˆ fqpm1q1,...,mkqkq aq1,...,qk “ , (3.3) m1q1 ¨ ¨ ¨ mkqk m1,...,mk“1 pm1,q1q“1,...,ÿpmk,qkq“1 r the series (3.2) being absolutely convergent.

Proof. We have for any n1,...,nk P N, by using property (2.1),

˚ fpn1,...,nkq“ pµk ˆ fqpd1,...,dkq d1 n1,...,d n k ÿ kk k 8 ˚ pµk ˆ fqpd1,...,dkq “ cq1 pn1q¨¨¨ cqk pnkq d1 ¨ ¨ ¨ dk d1,...,d “1 q1 d1 q d ÿk ÿk kÿk k 8 8 r ˚ r pµk ˆ fqpd1,...,dkq “ cq1 pn1q¨¨¨ cqk pnkq , d1 ¨ ¨ ¨ dk q1,...,qk“1 d1,...,dk“1 ÿ q1kd1,...,qÿ kkdk r r leading to expansion (3.2) with the coeﬃcients (3.3), by denoting d1 “ m1q1,...,dk “ mkqk. The rearranging of the terms is justiﬁed by the absolute convergence of the multiple series, shown hereinafter: 8

|aq1,...,qk ||cq1 pn1q| ¨ ¨ ¨ |cqk pnkq| q1,...,q “1 ÿk 8 ˚r r r |pµk ˆ fqpm1q1,...,mkqkq| ď |cq1 pn1q| ¨ ¨ ¨ |cqk pnkq| m1q1 ¨ ¨ ¨ mkqk q1,...,qk“1 m1,...,mÿ k“1 pm1,q1q“1,...,pmk,qkq“1 r r 8 ˚ |pµk ˆ fqpt1,...,tkq| “ |cq1 pn1q| ¨ ¨ ¨ |cqk pnkq| t1 ¨ ¨ ¨ tk t1,...,tk“1 m1q1“t1 mkqk“tk ÿ pm1ÿ,q1q“1 pmkÿ,qkq“1 r r

7 8 µ˚ f t1,...,t ωpt1q`¨¨¨`ωptkq |p k ˆ qp kq| ď n1 ¨ ¨ ¨ nk 2 ă8, t1 ¨ ¨ ¨ t t1,...,t “1 k ÿk by using inequality (2.4) and condition (3.1).

Next we consider the case fpn1,...,nkq “ gppn1,...,nkq˚kq. The following result is the analogue of Theorem 1.

Theorem 3. Let g : N Ñ C be an arithmetic function and let k P N. Assume that

8 |pµ˚ ˆ gqpnq| 2kωpnq ă8. nk n“1 ÿ Then for every n1,...,nk P N,

gppn1,...,nkq˚kq“ aq1,...,qk cq1 pn1q¨¨¨ cqk pnkq, q1,...,q “1 ÿk is absolutely convergent, where r r r

1 8 pµ˚ ˆ gqpmQq a 1 “ , (3.4) q ,...,qk Qk mk m“1 pm,Qÿq“1 r with the notation Q “ rq1,...,qks. Proof. We apply Theorem 2. Taking into account the identity

˚ gppn1,...,nkq˚kq“ pµ ˆ gqpdq d n1,...,d n k ÿ k k we see that now

˚ ˚ pµ ˆ gqpnq, if n1 “¨¨¨“ nk “ n, pµk ˆ fqpn1,...,nkq“ #0, otherwise. Therefore the coeﬃcients of the expansion are

8 ˚ pµk ˆ fqpm1q1,...,mkqkq aq1,...,qk “ m1q1 ¨ ¨ ¨ m q n“1 k k m1q1“¨¨¨“ÿmkqk“n 1 1 r pm1,q1q“ ,...,pmk,qkq“

8 pµ˚ ˆ gqpnq “ , nk n“1 q1kn,...,qÿ kkn and we use that q1 k n,...,qk k n holds if and only if rq1,...,qks“ Q k n, that is, n “ mQ with pm,Qq“ 1.

8 Corollary 1. For every n1,...,nk P N the following series are absolutely convergent:

˚ 8 σ ppn1,...,n q q φ pQqc 1 pn1q¨¨¨ c pn q s k ˚k “ ζps ` kq s`k q qk k ps P R,s ` k ą 1q, (3.5) s 2ps`kq pn1,...,nkq Q ˚k q1,...,qk“1 ÿ r r 8 ˚ φkpQqcq1 pn1q¨¨¨ cqk pnkq τ ppn1,...,n q q“ ζpkq pk ě 2q. (3.6) k ˚k Q2k q1,...,qk“1 ÿ r r ˚ s Proof. Apply Theorem 3 to gpnq“ σs pnq{n . Here 1 ˆ id 1 1 µ˚ ˆ g “ µ˚ ˆ s “ pµ˚ ˆ 1qˆ “ , ids ids ids hence pµ˚ ˆ gqpnq“ 1{ns (n P N). We deduce by (3.4) that

8 1 1 φs`kpQq a 1 “ “ ζps ` kq , q ,...,qk Qs`k ms`k Q2ps`kq m“1 pm,Qÿq“1 r which completes the proof.

˚ In the case s “ 1 identity (3.5) reduces to (1.4). Now let consider the function φs pnq “ sν ˚ ˚ pν knpp ´ 1q, representing the unitary Jordan function of order s. Here φs “ µ ˆ ids, and ˚ ˚ φ1 “ φ is the unitary Euler function, already mentioned in Section 2.2. ś Corollary 2. For every n1,...,nk P N the following series are absolutely convergent:

φ˚ppn1,...,n q q 2 s k ˚k “ ζps ` kq 1 ´ ˆ (3.7) pn1,...,n qs ps`k k ˚k pPP ź ˆ ˙ 8 ˚ µ pQqφ pQqc 1 pn1q¨¨¨ c pn q ˆ s`k q qk k ps P R,s ` k ą 1q, Q2ps`kq p1 ´ 2{ps`kq q1,...,q “1 p|Q ÿk ˚ r r φ ppn1,...,nkśq˚kq 2 “ ζpk ` 1q 1 ´ 1 ˆ pn1,...,n q pk` k ˚k pPP ź ˆ ˙ 8 ˚ µ pQqφ 1pQqc 1 pn1q¨¨¨ c pn q ˆ k` q qk k pk ě 1q. Q2pk`1q p1 ´ 2{pk`1q q1,...,qk“1 p|Q ÿ r r ˚ s Proof. Apply Theorem 3 to gpnq“ φs pnq{nś. Here µ˚ ˆ id µ˚ µ˚ µ˚ ˆ g “ µ˚ ˆ s “ pµ˚ ˆ 1qˆ “ , ids ids ids that is, pµ˚ ˆ gqpnq“ µ˚pnq{ns (n P N). We deduce by (3.4) that

1 8 µ˚pmQq µ˚pQq 8 µ˚pmq a 1 “ “ q ,...,qk Qs`k ms`k Qs`k ms`k m“1 m“1 pm,Qÿq“1 pm,Qÿq“1 r 9 1 µ˚pQq 2 1 2 ´ “ ζps ` kq 1 ´ 1 ´ 1 ´ , Qs`k ps`k ps`k ps`k pPP p Q ź ˆ ˙ ź| ˆ ˙ˆ ˙ leading to (3.7).

For m P N, m ě 2 consider the function gpnq“ mωpnq, which is the unitary analogue of the ωpnq ωpdq N Piltz divisor function τmpnq. Here m “ dknpm´1q for any n P . We obtain by similar arguments: ř Corollary 3. For every n1,...,nk P N the following series is absolutely convergent: m ´ 2 mωppn1,...,nkq˚kq “ ζpkq 1 ` ˆ (3.8) pk pPP ź ˆ ˙ 8 ωpQq φ pQqpm ´ 1q c 1 pn1q¨¨¨ c pn q ˆ k q qk k pm, k ě 2q, Q2k p1 ` pm ´ 2q{pkq q1,...,qk“1 p|Q ÿ r r For m “ 2 identity (3.8) reduces toś (3.6). Remark 1. It is possible to formulate the results of Theorem 2 in the case of multiplicative functions f of k variables, and Theorem 3 in the case of multiplicative functions g of one variable. Note that if g is multiplicative, then fpn1,...,nkq“ gppn1,...,nkq˚kq 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.

4 Acknowledgement

Supported by the European Union, co-ﬁnanced by the European Social Fund EFOP-3.6.1.- 16-2016-00004.

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