Ramanujan Expansions of Arithmetic Functions of Several Variables

Ramanujan expansions of arithmetic functions of several variables LászlóTóth Department of Mathematics, University of Pécs Ifjúság útja 6, 7624 Pécs, Hungary E-mail: [email protected] The Ramanujan Journal 47 (2018), Issue 3, 589–603 Abstract We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel. 2010 Mathematics Subject Classification: 11A25, 11N37 Key Words and Phrases: Ramanujan expansion of arithmetic functions, arithmetic function of several variables, multiplicative function, unitary divisor, unitary Ramanujan sum 1 Introduction Basic notations, used throughout the paper are included in Section 2. Further notations are explained by their first appearance. Refining results of Wintner [25], Delange [3] proved the following general theorem concerning Ramanujan expansions of arithmetic functions. Theorem 1. Let f : N Ñ C be an arithmetic function. Assume that 8 ωpnq |pµ ˚ fqpnq| arXiv:1704.02881v3 [math.NT] 12 Nov 2018 2 ă8. (1) n n“1 ÿ Then for every n P N we have the absolutely convergent Ramanujan expansion 8 fpnq“ aqcqpnq, (2) q“1 ÿ where the coefficients aq are given by 8 pµ ˚ fqpmqq a “ pq P Nq. q mq m“1 ÿ 1 Delange also pointed out how this result can be formulated for multiplicative functions f. By Wintner’s theorem ([25, Part I], also see, e.g., Postnikov [11, Ch. 3], Schwarz and Spilker [12, Ch. II]), condition (1) ensures that the mean value 1 Mpfq“ lim fpnq xÑ8 x nďx ÿ exists and a1 “ Mpfq. Recently, Ushiroya [23] proved the analog of Theorem 1 for arithmetic functions of two variables and as applications, derived Ramanujan expansions of certain special functions. For example, he deduced that for any fixed n1,n2 P N, 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 Möbius 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.

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