Ramanujan Expansions of Arithmetic Functions of Several Variables

Home , Unitary divisor

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8 σppn1,n2qq cq1 pn1qcq2 pn2q “ ζp3q 3 , (3) pn1,n2q rq1,q2s q1,q2“1 ÿ 8 cq1 pn1qcq2 pn2q τppn1,n2qq “ ζp2q 2 . (4) rq1,q2s q1,q2“1 ÿ Here (3) corresponds to the classical identity

σpnq 8 c pnq “ ζp2q q pn P Nq, (5) n q2 q“1 ÿ due to Ramanujan [13]. However, (4) has not a direct one dimensional analog. The identity of Ramanujan for the divisor function, namely,

8 log q τpnq“´ c pnq pn P Nq q q q“1 ÿ cannot be obtained by the same approach, since the mean value Mpτq does not exist. We remark that prior to Delange’s paper, Cohen [2] pointed out how absolutely convergent expansions (2), including (5) can be deduced for some special classes of multiplicative functions of one variable. It is easy to see that the same arguments of the papers by Delange [3] and Ushiroya [23] lead to the extension of Theorem 1 for Ramanujan expansions of (multiplicative) arithmetic functions of k variables, for any k P N. But in order to obtain k dimensional versions of the identities (3) and (4), the method given in [23] to compute the coeﬃcients is complicated. Recall that d is a unitary divisor (or block divisor) of n if d | n and pd, n{dq “ 1. Various problems concerning functions associated to unitary divisors were discussed by several authors in the literature. Many properties of the functions σ˚pnq and τ ˚pnq “ 2ωpnq, representing the sum, respectively the number of unitary divisors of n are similar to σpnq and τpnq. Analogs of ˚ Ramanujan sums deﬁned by unitary divisors, denoted by cq pnq and called unitary Ramanujan sums, are also known in the literature. However, we are not aware of any paper concerning ˚ expansions of functions with respect to the sums cq pnq. The only such paper we found is by

2 ˚ Subbarao [17], but it deals with a series over the other argument, namely n in cq pnq. See Section 3.2. In this paper we show that certain properties on expansions of functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel. We also present a simple approach to deduce Ramanujan expansions of both types for gppn1,...,nkqq, where g : N Ñ C is a certain function. For example, we have the identities

8 σpnq φ2pqqc˚pnq “ ζp2q q pn P Nq, n q4 q“1 ÿ 8 φ3prq1,q2sqc˚ pn1qc˚ pn2q σppn1,n2qq q1 q2 N “ ζp3q 6 pn1,n2 P q, (6) pn1,n2q rq1,q2s q1,q2“1 ÿ 8 φ2prq1,q2sqc˚ pn1qc˚ pn2q q1 q2 N τppn1,n2qq “ ζp2q 4 pn1,n2 P q, (7) rq1,q2s q1,q2“1 ÿ which can be compared to (5), (3) and (4), respectively. ˚ We point out that there are identities concerning the sums cq pnq, which do not have simple counterparts in the classical case. For example,

τ3ppn1,n2qq (8)

8 2 2 τprq1,q2sqpφ2prq1,q2sqq ˚ ˚ N “ pζp2qq 6 cq1 pn1qcq2 pn2q pn1,n2 P q. rq1,q2s q1,q2“1 ÿ For general accounts on classical Ramanujan sums and Ramanujan expansions of functions of one variable we refer to the book by Schwarz and Spilker [12] and to the survey papers by Lucht [8] and Ram Murty [14]. See Sections 3.1 and 3.2 for properties and references on functions deﬁned by unitary divisors and unitary Ramanujan sums. Section 3.3 includes the needed background on arithmetic functions of several variables. Our main results and their proofs are presented in Section 4.

2 Notations

‚ N “ t1, 2,...u, N0 “ t0, 1, 2,...u, ‚ P is the set of (positive) primes, 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 ‚ pn1,...,nkq and rn1,...,nks denote the greatest common divisor and the least common multiple, respectively of n1,...,nk P N, N C ‚ pf ˚ gqpnq“ d|n fpdqgpn{dq is the Dirichlet convolution of the functions f, g : Ñ , ‚ ωpnq stands for the number of distinct prime divisors of n, ř ‚ µ is the M¨obius function, s R ‚ σspnq“ d|n d (s P ), ‚ σpnq“ σ1pnq is the sum of divisors of n, ř

3 ‚ τpnq“ σ0pnq is the number of divisors of n, s s R ‚ φs is the Jordan function of order s given by φspnq“ n p|np1 ´ 1{p q (s P ), ‚ φ “ φ1 is Euler’s totient function, ś s s R ‚ ψs is the generalized Dedekind function given by ψspnq“ n p|np1 ` 1{p q (s P ), ‚ ψ “ ψ1 is the Dedekind function, ś N ‚ cqpnq“ 1ďkďq,pk,qq“1 expp2πikn{qq are the Ramanujan sums (q,n P ), ‚ τkpnq is the Piltz divisor function, deﬁned as the number of ways that n can be written as ř a product of k positive integers, Ωpnq ‚ λpnq“p´1q is the Liouville function, where Ωpnq“ p νppnq, ‚ ζ is the Riemann zeta function, ř ‚ d || 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 || n used for prime powers pν.)

3 Preliminaries

3.1 Functions of one variable defined by unitary divisors The study of arithmetic functions defined by unitary divisors goes back to Vaidyanathaswamy [24] and Cohen [1]. The functions σ˚pnq and τ ˚pnq were already mentioned in the Introduction. ˚ ˚ The analog of Euler’s φ function is φ , defined by φ pnq “ #tk P N : 1 ď k ď n, pk,nq˚ “ 1u, where pk,nq˚ “ maxtd : d | k, d || nu. ˚ ˚ ˚ Note that d || pk,nq˚ holds if and only if d | k and d || n. The functions σ , τ , φ are all multiplicative and σ˚ppνq“ pν `1, τ ˚ppνq“ 2, φ˚ppν q“ 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 ÿ|| 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. See, e.g., Derbal [4], McCarthy [10], Sitaramachandrarao and Suryanarayana [15], Snellman [16] for properties and generalizations of functions associated to unitary divisors.

3.2 Unitary Ramanujan sums

˚ The unitary Ramanujan sums cq pnq were deﬁned by Cohen [1] as follows: ˚ N cq pnq“ expp2πikn{qq pq,n P q. 1ďkďq pk,qÿq˚“1 The identities ˚ ˚ N cq pnq“ dµ pq{dq pq,n P q, d n,q ˚ ||pÿ q 4 ˚ q, if q | n, cd pnq“ (9) d q #0, otherwise ÿ|| can be compared to the corresponding ones concerning the Ramanujan sums cqpnq. ˚ N It turns out that cq pnq is multiplicative in q for any ﬁxed n P , and

ν ν ˚ p ´ 1, if p | n, cpν pnq“ (10) #´1, otherwise,

ν ˚ ˚ ˚ ˚ N for any prime powers p (ν ě 1). Furthermore, cq pqq“ φ pqq, cq p1q“ µ pqq (q P ). ˚ If q is squarefree, then the divisors of q coincide with its unitary divisors. Therefore, cq pnq“ cqpnq for any squarefree q and any n P N. However, if q is not squarefree, then there is no direct relation between these two sums.

Proposition 1. For any q,n P N,

˚ ωpq{pn,qq˚q |cd pnq| “ 2 pn,qq˚, (11) d q ÿ|| ˚ ωpqq |cd pnq| ď 2 n. (12) d q ÿ|| Proof. If q “ pν (ν ě 1) is a prime power, then we have by (10),

ν ν ν ˚ ˚ ˚ 1 ` p ´ 1 “ p , if p | n, |cd pnq| “ |c1 pnq| ` |cpν pnq| “ d pν #1 ` 1 “ 2, otherwise. ÿ|| Now (11) follows at once by the multiplicativity in q of the involved functions, while (12) is its immediate consequence.

Remark 1. The corresponding properties for the classical Ramanujan sums are

ωpq{pn,qqq |cdpnq| “ 2 pn,qq, (13) d q ÿ| ωpqq |cdpnq| ď 2 n, (14) d q ÿ| where inequality (14) is crucial in the proof of Theorem 1, and identity (13) was pointed out by Grytczuk [5].

˚ Of course, some other properties diﬀer notably. For example, the unitary sums cq pnq do not enjoy the orthogonality property of the classical Ramanujan sums, namely

n 1 φpqq, if q1 “ q2 “ q, c 1 pkqc 2 pkq“ n q q k“1 #0, otherwise, ÿ

5 2 valid for any q1,q2,n P N with rq1,q2s | n. To see this, let p be a prime, let q1 “ p, q2 “ p and n “ p2. Then, according to (10),

p2 ˚ ˚ 2 cppkqcp2 pkq“ p pp ´ 1q‰ 0. k“1 ÿ Let Λ˚ be the unitary analog of the von Mangoldt function Λ, given by

ν log p, if k “ pν, Λ˚pkq“ #0, otherwise.

The paper of Subbarao [17] includes the formula

8 c˚pnq q “´Λ˚pqq pq ě 2q, (15) n n“1 ÿ with an incomplete proof, namely without showing that the series in (15) converges. Here (15) is the analog of Ramanujan’s formula (also see H¨older [6])

8 c pnq q “´Λpqq pq ě 2q. n n“1 ÿ For further properties and generalizations of unitary Ramanujan sums, see, e.g., Cohen [1], Johnson [7], McCarthy [10], Suryanarayana [18]. Note that a common generalization of the ˚ sums cqpnq and cq pnq, involving Narkiewicz type regular systems of divisors was investigated by McCarthy [9] and the author [19].

3.3 Arithmetic functions of several variables k For every ﬁxed k P N the set Ak of arithmetic functions f : N Ñ C of k variables is an integral domain with pointwise addition and the Dirichlet convolution deﬁned by

pf ˚ gqpn1,...,nkq“ fpd1,...,dkqgpn1{d1,...,nk{dkq, (16) d1 n1,...,d n | ÿ k| 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 (16) is µk, given by

µkpn1,...,nkq“ µpn1q¨¨¨ µpnkq pn1,...,nk P Nq, where µ is the (classical) M¨obius function.

6 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 N0. 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 Dirichlet convolution (16) preserves the multiplicativity of functions. The unitary convolution for the k variables case can be deﬁned and investigated, as well, but we will not need it in what follows. See our paper [21], which is a survey on (multiplicative) arithmetic functions of several variables.

4 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 |pµk ˚ fqpn1,...,nkq| 2ωpn1q`¨¨¨`ωpnkq ă8. (17) n1 ¨ ¨ ¨ nk n1,...,n “1 ÿk Then for every n1,...,nk P N, 8

fpn1,...,nkq“ aq1,...,qk cq1 pn1q¨¨¨ cqk pnkq, (18) q1,...,q “1 ÿk and 8 ˚ ˚ ˚ 1 1 fpn ,...,nkq“ aq1,...,qk cq1 pn q¨¨¨ cqk pnkq, (19) q1,...,q “1 ÿk where 8 pµk ˚ fqpm1q1,...,mkqkq aq1,...,qk “ , m1q1 ¨ ¨ ¨ mkqk m1,...,m “1 ÿ k 8 ˚ pµk ˚ fqpm1q1,...,mkqkq aq1,...,qk “ , (20) m1q1 ¨ ¨ ¨ mkqk m1,...,mk“1 pm1,q1q“1,...,ÿpmk,qkq“1 the series (18) and (19) being absolutely convergent. We remark that according to the generalized Wintner theorem, under conditions of Theorem 2, the mean value 1 Mpfq“ lim fpn1,...,nkq x1,...,xkÑ8 x1 ¨ ¨ ¨ x k n1ďx1,...,n ďx ÿ k k ˚ exists and a1,...,1 “ a1,...,1 “ Mpfq. See Ushiroya [22, Th. 1] and the author [21, Sect. 7.1].

7 ˚ N Proof. We consider the case of the unitary Ramanujan sums cq pnq. For any n1,...,nk P we have, by using property (9),

fpn1,...,nkq“ pµk ˚ fqpd1,...,dkq d1 n1,...,d n | ÿ k| k 8 pµk ˚ fqpd1,...,dkq ˚ ˚ c n1 c n “ q1 p q¨¨¨ qk p kq d1 ¨ ¨ ¨ dk d1,...,d “1 q1 d1 q d ÿk ÿ|| kÿ|| k 8 8 ˚ ˚ pµk ˚ fqpd1,...,dkq c n1 c n , “ q1 p q¨¨¨ qk p kq d1 ¨ ¨ ¨ dk q1,...,qk“1 d1,...,dk“1 ÿ q1||d1,...,qÿ k||dk giving expansion (19) with the coeﬃcients (20), by denoting d1 “ m1q1,...,dk “ mkqk. The rearranging of the terms is justiﬁed by the absolute convergence of the multiple series, shown hereinafter: 8 ˚ ˚ ˚ 1 |aq1,...,qk ||cq1 pn q| ¨ ¨ ¨ |cqk pnkq| q1,...,q “1 ÿk 8 |pµk ˚ fqpm1q1,...,mkqkq| ˚ ˚ 1 ď |cq1 pn q| ¨ ¨ ¨ |cqk pnkq| m1q1 ¨ ¨ ¨ mkqk q1,...,qk“1 m1,...,mÿ k“1 pm1,q1q“1,...,pmk,qkq“1 8 |pµk ˚ fqpt1,...,tkq| ˚ ˚ c n1 c n “ | q1 p q| ¨ ¨ ¨ | qk p kq| t1 ¨ ¨ ¨ tk t1,...,tk“1 m1q1“t1 mkqk“tk ÿ pm1ÿ,q1q“1 pmkÿ,qkq“1 8 µ f t1,...,t ωpt1q`¨¨¨`ωptkq |p k ˚ qp kq| ď n1 ¨ ¨ ¨ nk 2 ă8, t1 ¨ ¨ ¨ tk t1,...,t “1 ÿk by using inequality (12) and condition (17). For the Ramanujan sums cqpnq the proof is along the same lines, by using inequality (14). In the case k “ 2 the proof of (18) was given by Ushiroya [23].

For multiplicative functions f, condition (17) is equivalent to

8 |pµ ˚ fqpn1,...,n q| k k ă8 (21) n1 ¨ ¨ ¨ nk n1,...,n “1 ÿk and to 8 |pµ ˚ fqppν1 ,...,pνk q| k ă8. (22) pν1`¨¨¨`νk pPP ν1,...,νk“0 ÿ ν1`...ÿ`νkě1 We deduce the following result:

8 Corollary 1. Let f : Nk Ñ C be a multiplicative function (k P N). Assume that condition (21) or (22) holds. Then for every n1,...,nk P N one has the absolutely convergent expansions (18), (19), and the coeﬃcients can be written as

ν1 ν pµk ˚ fqpp ,...,p k q aq1,...,q “ , k pν1`¨¨¨`νk pPP ν1 ν q1 ,...,ν ν q ź ě pp qÿ kě pp kq

ν1 νk ˚ 1 pµk ˚ fqpp ,...,p q aq1,...,q “ , k pν1`¨¨¨`νk pPP ν1,...,νk ź ÿ 1 where means that for ﬁxed p and j, νj takes all values νj ě 0 if νppqjq “ 0, and takes only the value νj “ 0 if νppqjqě 1. ř Proof. This is a direct consequence of Theorem 2 and the deﬁnition of multiplicative functions. In the cases k “ 1 and k “ 2, with the sums cqpnq see [3, Eq. (7)] and [23, Th. 2.4], respectively.

Next we consider the case fpn1,...,nkq“ gppn1,...,nkqq. Theorem 3. Let g : N Ñ C be an arithmetic function and let k P N. Assume that

8 |pµ ˚ gqpnq| 2kωpnq ă8. (23) nk n“1 ÿ Then for every n1,...,nk P N,

gppn1,...,nkqq “ aq1,...,qk cq1 pn1q¨¨¨ cqk pnkq, (24) q1,...,q “1 ÿk 8 ˚ ˚ ˚ 1 1 gppn ,...,nkqq “ aq1,...,qk cq1 pn q¨¨¨ cqk pnkq (25) 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. Proof. We apply Theorem 2. The identity

gppn1,...,nkqq “ pµ ˚ gqpdq d n1,...,d n | ÿ | k 9 shows that now

pµ ˚ gqpnq, if n1 “¨¨¨“ nk “ n, pµk ˚ fqpn1,...,nkq“ #0, otherwise. In the unitary case the coeﬃcients are

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

8 pµ ˚ gqpnq “ , nk n“1 q1||n,...,qÿ k||n and take into account that q1 || n,...,qk || n holds if and only if rq1,...,qks “ Q || n, that is, n “ mQ with pm,Qq“ 1.

In the classical case, namely for the coeﬃcients aq1,...,qk , the proof is analog. If the function g is multiplicative, then condition (23) is equivalent to

8 |pµ ˚ gqpnq| ă8 (26) nk n“1 ÿ and to 8 |pµ ˚ gqppν q| ă8, (27) pkν pPP ν“1 ÿ ÿ and Theorem 3 can be rewritten according to Corollary 1. We continue, instead, with the following result, having direct applications to special functions.

Corollary 2. Let g : N Ñ C be a multiplicative function and let k P N. Assume that condition (26) or (27) holds. If µ ˚ g is completely multiplicative, then (24) holds for every n1,...,nk P N with

pµ ˚ gqpQq 8 pµ ˚ gqpmq a 1 “ . q ,...,qk Qk mk m“1 ÿ Identity (25) holds for every n1,...,nk P N (and any multiplicative g) with

pµ ˚ gqpQq 8 pµ ˚ gqpmq a˚ “ . q1,...,qk Qk mk m“1 pm,Qÿq“1

Proof. This is immediate by Theorem 3. By making use of condition pm,Qq“ 1, it follows for a˚ µ g mQ µ g m µ g Q the coeﬃcients q1,...,qk that p ˚ qp q “ p ˚ qp qp ˚ qp q for any multiplicative function g.

10 Corollary 3. For every n1,...,nk P N the following series are absolutely convergent:

8 σ ppn1,...,n qq c 1 pn1q¨¨¨ c pn q s k q qk k R s “ ζps ` kq s`k ps P ,s ` k ą 1q, pn1,...,nkq Q q1,...,q “1 ÿk 8 σppn1,...,nkqq cq1 pn1q¨¨¨ cqk pnkq “ ζpk ` 1q k`1 pk ě 1q, (28) pn1,...,nkq Q q1,...,q “1 ÿk 8 cq1 pn1q¨¨¨ cq pnkq τppn1,...,n qq “ ζpkq k pk ě 2q. (29) k Qk q1,...,q “1 ÿk s s Proof. Apply Corollary 2 for gpnq“ σspnq{n , where the function pµ˚gqpnq“ 1{n is completely multiplicative.

In the case k “ 2 the identities of Corollary 3 were given in [23, Ex. 3.8, 3.9]. For k “ 2, (28) and (29) reduce to (3) and (4), respectively.

Corollary 4. For every n1,...,nk P N the following series are absolutely convergent:

8 ˚ ˚ σ ppn1,...,n qq φs`kpQqc 1 pn1q¨¨¨ c pnkq s k ζ s k q qk s “ p ` q 2ps`kq pn1,...,nkq Q q1,...,q “1 ÿk with s P R,s ` k ą 1,

8 ˚ ˚ σppn1,...,n qq φk`1pQqc 1 pn1q¨¨¨ c pnkq k “ ζpk ` 1q q qk pk ě 1q, (30) 2pk`1q pn1,...,nkq Q q1,...,q “1 ÿk 8 ˚ ˚ φkpQqcq1 pn1q¨¨¨ cq pnkq τppn1,...,n qq “ ζpkq k pk ě 2q. (31) k Q2k q1,...,q “1 ÿk s s Proof. Apply the second part of Corollary 2 for gpnq “ σspnq{n . Since pµ ˚ gqpnq “ 1{n , we deduce that 1 8 1 φ pQq a˚ “ “ ζps ` kq s`k . q1,...,qk Qs`k ms`k Q2ps`kq m“1 pm,Qÿq“1

If k “ 2, then identities (30) and (31) recover (6) and (7), respectively. Corollary 2 can be applied for several other special functions g. Further examples are gpnq“ s R s βspnq{n (s P ) and gpnq “ rpnq, where βspnq “ d|n d λpn{dq and rpnq denotes the number of solutions px,yq P Z2 of the equation x2 ` y2 “ n. ř

11 Corollary 5. For every n1,...,nk P N and s P R,s ` k ą 1 the following series are absolutely convergent: 8 βsppn1,...,nkqq ζp2ps ` kqq λpQqcq1 pn1q¨¨¨ cqk pnkq s “ s`k , (32) pn1,...,nkq ζps ` kq Q q1,...,q “1 ÿk 8 ˚ ˚ β ppn1,...,n qq ζp2ps ` kqq λpQqψs`kpQqc 1 pn1q¨¨¨ c pnkq s k “ q qk . s 2ps`kq pn1,...,nkq ζps ` kq Q q1,...,q “1 ÿk In the case k “ s “ 1, identity (32) was derived by Cohen [2, Eq. (12)]. See also the author [20, Eq. (16)].

Corollary 6. For every n1,...,nk P N (k ě 1) the following series are absolutely convergent:

8 pQ´1q{2 p´1q cq1 pn1q¨¨¨ cq pnkq rppn1,...,n qq “ 4Lpχ, kq k , (33) k Qk q1,...,qk“1 Qÿodd

8 pQ´1q{2 p´1q F pQq ˚ ˚ rppn1,...,n qq “ 4Lpχ, kq c pn1q¨¨¨ c pn q, k Qk q1 qk k q1,...,qk“1 Qÿodd where ´1 8 p´1qn p´1qpp´1q{2 Lpχ, kq“ “ 1 ´ , p2n ` 1qk pk n“0 pPP ˜ ¸ ÿ pźą2

χ “ χ4 being the nonprincipal character (mod 4) and

1 2 F pQq“ 1 ´ p´1qpp´ q{ {pk . p|Q pźą2 ´ ¯

Proof. Use that rpnq “ 4 d|n χpdq, thus µ ˚ r{4 “ χ, completely multiplicative. Here (33) is well known for k “ 1, and was obtained in [23, Ex. 3.13] in the case k “ 2. ř If g is multiplicative such that µ ˚ g is not completely multiplicative, then Corollary 2 can ˚ be applied for the sums cq pnq, but not for cqpnq, in general.

Corollary 7. For every n1,...,nk P N the following series are absolutely convergent:

8 µ Q c˚ n1 c˚ n φsppn1,...,nkqq 1 p q q1 p q¨¨¨ qk p kq s “ (34) pn1,...,nkq ζps ` kq φs kpQq q1,...,q “1 ` ÿk with s P R,s ` k ą 1,

8 ˚ ˚ φppn1,...,n qq 1 µpQqc 1 pn1q¨¨¨ c pnkq k “ q qk pk ě 1q. (35) pn1,...,nkq ζpk ` 1q φk 1pQq q1,...,q “1 ` ÿk

12 s s Proof. Similar to the proof of Corollary 4, selecting gpnq“ φspnq{n , where pµ˚gqpnq“ µpnq{n .

However, observe that if Q is not squarefree, then µpQq “ 0 and all terms of the sums in ˚ (34) and (35) are zero. Now if Q is squarefree, then qj is squarefree and cqj pnjq “ cqj pnjq for any j and any nj. Thus we deduce the next identities:

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

8 φsppn1,...,nkqq 1 µpQqcq1 pn1q¨¨¨ cqk pnkq s “ (36) pn1,...,nkq ζps ` kq φs kpQq q1,...,q “1 ` ÿk with s P R,s ` k ą 1,

8 φppn1,...,n qq 1 µpQqc 1 pn1q¨¨¨ c pn q k “ q qk k pk ě 1q, (37) pn1,...,nkq ζpk ` 1q φk 1pQq q1,...,q “1 ` ÿk which are formally the same identities as (34) and (35).

Note that the special identities (36) and (37) can also be derived by the ﬁrst part of Corollary ˚ 2, without considering the unitary sums cq pnq. See Ushiroya [23, Ex. 3.10, 3.11] for the case k “ 2, using diﬀerent arguments. s Similar formulas can be deduced for the generalized Dedekind function ψspnq“ n p|np1 ` 1{psq (s P R). See Cohen [2, Eq. (13)] in the case k “ s “ 1. ś Finally, consider the Piltz divisor function τm. The following formula, concerning the sums ˚ cq pnq, deduced along the same lines with the previous ones, has no simple counterpart in the classical case.

Corollary 9. For any n1,...,nk,m P N with m ě 2, k ě 2,

8 m´1 m´1 τm´1pQqpφkpQqq ˚ ˚ τmppn1,...,n qq “ pζpkqq c pn1q¨¨¨ c pn q. (38) k Qkm q1 qk k q1,...,q “1 ÿk If m “ 2, then (38) reduces to formula (31), while for m “ 3 and k “ 2 it gives (8).

5 Acknowledgement

The author thanks the anonymous referee for careful reading of the manuscript and helpful comments.

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