#A51 INTEGERS 11 (2011) SOME REMARKS ON A PAPER OF V. A. LISKOVETS

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1 #A51 INTEGERS 11 (2011) SOME REMARKS ON A PAPER OF V. A. LISKOVETS László Tóth 1 Departent of Matheatics, University of Pécs, Ifjúság u., Pécs, Hungary and Institute of Matheatics, Departent of Integrative Biology, Universität für Bodenkultur, Wien, Austria ltoth@gaa.ttk.pte.hu Received: 1/25/11, Accepted: 7/16/11, Published: 9/19/11 Abstract We deduce new properties of the orbicyclic function E of several variables investigated in a recent paper by V. A. Liskovets. We point out that the function E and its connection to the nuber of solutions of certain linear congruences occur in the literature in a slightly different for. We investigate another siilar function considered by Deitar, Koyaa and Kurokawa by studying analytic properties of soe zeta functions of Igusa type. Siple nuber-theoretic proofs for soe known properties are also given. 1. Introduction In a recent paper Liskovets [10] investigated arithetical properties of the function E( 1,..., r ) := 1 M M c 1 (k) c r (k), (1) where 1,..., r, M N := {1, 2,...} (r N), := lc[ 1,..., r ], M and c n (k) is the Raanujan su defined as the su of k-th powers of the priitive n-th roots of unity (k, n N), i.e., c n (k) := exp(2πijk/n). (2) 1 j n gcd(j,n)=1 The function E given by (1) has been introduced by Mednykh and Nedela [13] in order to handle certain probles of enuerative cobinatorics. The function (1) also has arithetical and topological applications and it is referred to as the orbicyclic arithetic function [10]. 1 The author gratefully acknowledges support fro the Austrian Science Fund (FWF) under the project Nr. P20847-N18.

2 INTEGERS: 11 (2011) 2 For exaple, E( 1,..., r ) is the nuber of solutions (x 1,..., x r ) Z r M of the congruence x x r 0 (od M) satisfying gcd(x 1, M) = M/ 1,..., gcd(x r, M) = M/ r (see [13, Lea 4.1]). It follows fro this interpretation that all the values E( 1,..., r ) are nonnegative integers. Note that in case of one, respectively two, variables, { E( 1 ) := 1 M 1, 1 = 1, c 1 (k) = (3) M 0, otherwise, E( 1, 2 ) := 1 M M c 1 (k)c 2 (k) = { φ(), 1 = 2 =, 0, otherwise, where φ is Euler s function. Forulae (3) and (4) are well-known properties of the Raanujan sus; (4) istheir orthogonality property leading to the Raanujan expansions of arithetical functions (see, for exaple, [19]). Another function, siilar to E, is A( 1,..., r ) := 1 (4) gcd(k, 1 ) gcd(k, r ), (5) where 1,..., r N (r N) and := lc[ 1,..., r ], as above. The function (5) was entioned by Liskovets [10, section 4] and it was considered by Deitar, Koyaa and Kurokawa [9] in case j j+1 (1 j r 1) by studying analytic properties of soe zeta functions of Igusa type. The explicit forula for the values A( 1,..., r ) derived in [9, Section 3] was reproved by Minai [14] for the general case 1,..., r N, using arguents of eleentary probability theory. We reark that the corresponding forulae of both papers [9, 14] contain isprints. For r = 1 (5) reduces to the function A() := 1 gcd(k, ) = d φ(d) d (6) giving the arithetic ean of gcd(1, ),..., gcd(, ). For arithetic and analytic properties of (6) and for a survey of other gcd-su-type functions of one variable see Tóth [22]. In the present paper we deduce new properties of the functions E and A and use the to give siple nuber-theoretic proofs for soe of their known properties. We derive convolution-type identities for E and A (Propositions 3 and 12) to show that they are ultiplicative as functions of several variables. We give other identities for E and A (Propositions 9 and 14) to obtain explicit forulae for their values. We

3 INTEGERS: 11 (2011) 3 consider coon generalizations of these functions and point out that the function E and its connection to the nuber of solutions of certain linear congruences occur in the literature in a slightly different for. As an application of the identity of Proposition 3 we give a siple direct proof of the orthogonality property (4) of the Raanujan sus (Application 8). 2. Preliinaries We present in this section soe basic notions and properties needed throughout the paper. For the prie power factorization of an integer n N we will use the notation n = p pνp(n), where the product is over the pries p and all but a finite nuber of the exponents ν p (n) are zero. We recall that an arithetic function of r variables is a function f : N r C, notation f F r. If f, g F r, then their convolution is defined as (f g)( 1,..., r ) = f(d 1,..., d r )g( 1 /d 1,..., r /d r ). (7) d 1 1,...,d r r The set F r fors a unital ring with ordinary addition and convolution (7), the unity being the function ε (r) given by { ε (r) 1, 1 =... = r = 1, ( 1,..., r ) = (8) 0, otherwise. A function f F r is invertible iff f(1,..., 1) 0. The inverse of the constant 1 function is given by µ (r) ( 1,..., r ) = µ( 1 )... µ( r ), where µ is the Möbius function. A function f F r is said to be ultiplicative if it is nonzero and f( 1 n 1,..., r n r ) = f( 1,..., r )f(n 1,..., n r ) holds for any 1,..., r, n 1,..., n r N such that gcd( 1 r, n 1 n r ) = 1. If f is ultiplicative, then it is deterined by the values f(p a1,..., p ar ), where p is a prie and a 1,..., a r N 0 := {0, 1, 2,...}. More exactly, f(1,..., 1) = 1 and for any 1,..., r N, f( 1,..., r ) = p f(p νp(1),..., p νp(r) ). For exaple, the functions ( 1,..., r ) gcd( 1,..., r ) and ( 1,..., r ) lc[ 1,..., r ] are ultiplicative. The function µ (r) is also ultiplicative. The convolution (7) preserves the ultiplicativity of functions. Moreover, the ultiplicative functions for a subgroup of the group of invertible functions with respect to convolution (7).

4 INTEGERS: 11 (2011) 4 These properties, which are well-known in the one variable case, follow easily fro the definitions. For further properties of arithetic functions of several variables and of their ring we refer to [2], [20, Ch. VII]. In the one variable case 1, id k (k N), ε and τ will denote the functions given by 1(n) = 1, id k (n) = n k (n N), ε(1) = 1, ε(n) = 0 for n > 1 and τ = 1 1 (divisor function), respectively. The Raanujan sus c n (k) can be represented as c n (k) = dµ(n/d) (k, n N). (9) d gcd(k,n) Note that the function n c n (k) is ultiplicative for any fixed k N, and for any prie power p a (a N), p a p a 1 if p a k, c p a(k) = p a 1 if p a 1 k, p a k, (10) 0 if p a 1 k. Also, c n (k) is ultiplicative as a function of two variables, i.e., considered as the function c : N 2 Z, c(k, n) = c n (k). The inequality c n (k) gcd(k, n) holds for any k, n N. These and other general accounts of Raanujan sus can be found in the books by Apostol [6], McCarthy [12], Schwarz and Spilker [19], Sivaraakrishnan [20]. 3. The Function E The following results were proved by Liskovets [10]. Proposition 1. ([10, Leas 2, 5, Prop. 4]) (i) The function E is ultiplicative (as a function of several variables). (ii) Let p a1,..., p ar be any powers of a prie p (a 1,..., a r N). Assue that a := a 1 = a 2 =... = a s > a s+1 a s+2... a r 1 (r s 1). Then E(p a1,..., p ar ) = p v (p 1) r s+1 h s (p), (11) where the integer v is defined by v = r j=1 a j r a + 1 and h s (x) = (x 1)s 1 + ( 1) s x is a polynoial of degree s 2 (for s > 1). (12) Note that Liskovets used the ter sei-ultiplicative function, but this is reserved for another concept (see, for exaple, [20, Ch. XI]).

5 INTEGERS: 11 (2011) 5 Corollary 2. ([10, Th. 8, Cor. 11]) (i) For any integers 1,..., r N, E( 1,..., r ) = p p v(p) (p 1) r(p) s(p)+1 h s(p) (p), (13) where v(p) and s(p), depending on p, are the integers v and s, respectively, defined in Proposition 1. (ii) For 1 =... = r =, f r () := E(,..., ) = r 1 p (p 1)h r (p) p r 1. (14) We first give the following siple convolution representation for the function E. Proposition 3. For any 1,..., r N, E( 1,..., r ) = Proof. Using forula (9), E( 1,..., r ) = 1 M d 1 1,...,d r r M d 1 gcd(k, 1) d 1 d r lc[d 1,..., d r ] µ( 1/d 1 ) µ( r /d r ). (15) d 1 µ( 1 /d 1 ) d r gcd(k, r) d r µ( r /d r ) = 1 d 1 µ( 1 /d 1 ) d r µ( r /d r ) 1, M d 1 1,...,d r r 1 k M,d 1 k,...d r k where the inner su is 1 = M/ lc[d 1,..., d r ], ending the proof. 1 k M,lc[d 1,...,d r] k By Möbius inversion we obtain fro (15), Corollary 4. For any 1,..., r N, d 1 1,...,d r r E(d 1,..., d r ) = Forula (15) has also a nuber of other applications: 1 r lc[ 1,..., r ]. (16) Application 5. Forula (15) shows that E is integral valued and that it does not depend on M, so in (1) one can take M =, the lc of 1,..., r, rearked also by Liskovets [10, Section 1].

6 INTEGERS: 11 (2011) 6 Application 6. If 1,..., r are pairwise relatively prie, then E( 1,..., r ) = ε (r) ( 1,..., r ), defined by (8). Indeed, in this case lc[d 1,..., d r ] = d 1 d r for any d i i (1 i r) and the clai follows fro (15) using that d n µ(d) = ε(n). Application 7. Also, (15) furnishes a siple direct proof of the ultiplicativity of E. Note that Liskovets [10] used other arguents to show the ultiplicativity. Observe that, according to (15), E is the convolution of the functions F and µ (r), where F is given by 1 r F ( 1,..., r ) = lc[ 1,..., r ]. Since F and µ (r) are ultiplicative, E is ultiplicative too. Application 8. Forula (15) leads to a siple direct proof of the orthogonality property (4). Using the Gauss forula d n φ(d) = n, E( 1, 2 ) := 1 M M c 1 (k)c 2 (k) = d 1 d 2 µ( 1 /d 1 )µ( 2 /d 2 ) lc[d 1, d 2 ] d 1 1,d 2 2 = µ( 1 /d 1 )µ( 2 /d 2 ) gcd(d 1, d 2 ) d 1 1,d 2 2 = µ( 1 /d 1 )µ( 2 /d 2 ) φ(δ) = µ(k)µ(l)φ(δ) d 1 1,d 2 2 δ gcd(d 1,d 2) δak= 1,δbl= 2 = µ(l), δu= 1,δv= 2 φ(δ) ak=u µ(k) bl=v where one of the inner sus are zero, unless u = v = 1 and obtain that E( 1, 2 ) = φ() for 1 = 2 = and E( 1, 2 ) = 0 otherwise. Now we derive another identity for E which furnishes an alternative proof of forula (ii) in Proposition 1. Proposition 9. For any 1,..., r N, E( 1,..., r ) = 1 c 1 (d) c r (d)φ(/d). (17) d Proof. It follows fro (9) that c n (k) = c n (gcd(k, n)) (k, n N). Observe that for any i {1,..., r}, gcd(gcd(k, ), i ) = gcd(k, gcd(, i )) = gcd(k, i ), since i, hence c i (k) = c i (gcd(k, i )) = c i (gcd(gcd(k, ), i )) = c i (gcd(k, )). We obtain E( 1,..., r ) = 1 c 1 (gcd(k, )) c r (gcd(k, )), and by grouping the ters according to the values gcd(k, ) = d, where d, k = dj, 1 j /d, gcd(j, /d) = 1, we obtain (17).

7 INTEGERS: 11 (2011) 7 Application 10. By (17), with the notation of Proposition 1, E(p a1,..., p ar ) = 1 p a d p a c p a1 (d) c p ar (d)φ(p a /d). (18) Using (10) we see that only two ters of (18) are nonzero, naely those for d = p a and d = p a 1. Hence, E(p a1,..., p ar ) = 1 p a ( cp a 1 (p a ) c p ar (p a )φ(1) + c p a 1 (p a 1 ) c p ar (p a 1 )φ(p) ) = 1 p a ( (p 1)p a 1 1 (p 1)p ar 1 + ( p a 1 ) s (p 1)p as+1 1 (p 1)p ar 1 (p 1) ), and a short coputation gives forula (ii) in Proposition 1. Consider the function f r () defined in Corollary 2 (case 1 =... = r = ). Here f 1 = ε, f 2 = φ. Proposition 11. Let r 3. The average order of the function f r () is C r r 1, where C r := (1 + (p 1)h r(p) p r 1 ) p r. (19) p More exactly, for any 0 < ε < 1, x f r () = C r r xr + O(x r 1+ε ). (20) Proof. The function f r is ultiplicative and by (14), f r () s = ζ(s r + 1) (1 + (p 1)h r(p) p r 1 ) p s p =1 for s C, Re s > r, where the infinite product is absolutely convergent for Re s > r 1. Hence f r = g r id r 1 in ters of the Dirichlet convolution, where g r is ultiplicative and for any prie power p a (a N), { g r (p a (p 1)h r (p) p r 1, a = 1, ) = 0, a 2. We obtain x f r () = d x g r (d) e x/d and (20) follows by usual estiates. e r 1 = xr r d x g r (d) d r + O x r 1 d x g r (d) d r 1

8 INTEGERS: 11 (2011) 8 4. The Function A Consider now the function A given by (5). The next forulae are siilar to (15) and (17). The following one was already given in [22, Section 3]. Proposition 12. For any 1,..., r N, φ(d 1 ) φ(d r ) A( 1,..., r ) = lc[d 1,..., d r ]. (21) d 1 1,...,d r r Proof. The proof is siilar to the proof of Proposition 3, obtained by inserting gcd(k, i ) = d i gcd(k, i) φ(d i) (1 i r). Corollary 13. The function A is ultiplicative (as a function of several variables). Proof. According to (21), A is the convolution of the functions G and the constant 1 function, where G is given by G( 1,..., r ) = φ( 1) φ( r ) lc[ 1,..., r ], both being ultiplicative. Therefore A is also ultiplicative. Proposition 14. For any 1,..., r N, A( 1,..., r ) = 1 gcd(d, 1 ) gcd(d, r )φ(/d). (22) d Proof. Siilar to the proof of Proposition 9. Using that gcd(k, i ) = gcd(gcd(k, ), i ) (1 i r) we have A( 1,..., r ) = 1 gcd(gcd(k, ), 1 ) gcd(gcd(k, ), r ), and by grouping the ters according to the values gcd(k, ) = d we obtain the forula. Corollary 15. Let p be a prie and let a 1,..., a r N. Assue that a 0 := 0 < a 1 a 2... a r. Then ( A(p a1,..., p ar ) = p a0+a1+...+ar ) r p a0+a1+...+a l 1 p l=1 a l 1 j=a l 1 p (r l)j. (23)

9 INTEGERS: 11 (2011) 9 Proof. Let a := a r. Then lc[p a1,..., p ar ] = p a. Fro (22) we obtain where the last su is a 1 1 j=0 A(p a1,..., p ar ) = 1 p a a gcd(p j, p a1 ) gcd(p j, p ar )φ(p a j ) j=0 ( = p a0+a1+...+ar ) a 1 p in(j,a1)+...+in(j,ar) j, p j=0 a 2 1 p rj j + p a1+(r 1)j j j=a 1 a r 1 j=a r 1 p a1+a2+...+ar 1+1j j finishing the proof. = r l=1 a l 1 j=a l 1 p a0+a1+...+al 1+(r l)j, Application 16. Fro (21) we have for any 1,..., r N, φ(d 1 ) φ(d r ) A( 1,..., r ) = φ(d 1 ) φ(d r ) (24) d 1 d r d 1 d r d 1 1,...,d r r d 1 1 d r r = A( 1 ) A( r ), cf. (6), with equality if 1,..., r are pairwise relatively prie. Note that if 1 =... = r =, then fro its definition, A r () := A(,..., ) = 1 (gcd(k, )) r = 1 d r φ(/d), (25) which is a ultiplicative function. An asyptotic forula for x A r() was given by Alladi [3]. See also [22, Section 2]. A siple inequality for the functions E and A is given by Proposition 17. For any 1,..., r N, d E( 1,..., r ) A( 1,..., r ). (26) Proof. Using the inequality c n (k) gcd(k, n), entioned in the Introduction, we obtain E( 1,..., r ) 1 c 1 (k) c r (k) 1 n gcd(k, 1 ) gcd(k, r ) = A( 1,..., r ).

10 INTEGERS: 11 (2011) Generalizations Let f F 2 be a function of two variables and consider the function F f ( 1,..., r ) := 1 f(k, 1 ) f(k, r ). (27) Proposition 18. ([10, Lea 5]) If n f(k, n) is ultiplicative for any k N and k f(k, n) is periodic (od n) for any n N, then F f is ultiplicative. Now suppose that f has the following representation: f(k, n) = g(d)h(n/d) (k, n N) (28) d gcd(k,n) where g, h F 1 are arbitrary functions. Functions f defined in this way, as generalizations of the Raanujan sus, were investigated in [4, 5]. See also Apostol [6, Section 8.3]. Proposition 19. Assue that f is given by (28). Then (i) F f has the representations g(d 1 ) g(d r ) F f ( 1,..., r ) = lc[d 1,..., d r ] h( 1/d 1 ) h( r /d r ), (29) d 1 1,...,d r r F f ( 1,..., r ) = 1 f(gcd(d, 1 )) f(gcd(d, r ))φ(/d). (30) d (ii) If g and h are ultiplicative functions, then F f is ultiplicative (as a function of several variables). Proof. (i) follows by the sae arguents as in the proofs of Propositions 3, 9, 12 and 14. (ii) is a direct consequence of (29). The functions E and A are recovered choosing g = id, h = µ and g = φ, h = 1, respectively, where note that gcd(k, n) = d gcd(k,n) φ(d). Now let f(k, n) = f(gcd(k, n)) (k, n N), where f F 1 is an arbitrary function. Then f is of type (28) with g = f µ, h = 1, since f(gcd(k, n)) = d gcd(k,n) (f µ)(d). Special choices of f can be considered. For f = id we reobtain the function A. As another exaple, let f = τ, with g = h = 1. Then we obtain Corollary 20. The function F τ F r is ultiplicative and 1 F τ ( 1,..., r ) = lc[d 1,..., d r ], (31) d 1 1,...,d r r

11 INTEGERS: 11 (2011) 11 F f ( 1,..., r ) = 1 τ(gcd(d, 1 )) τ(gcd(d, r ))φ(/d). (32) d Further coon generalizations of the functions E and A can be given using r-even functions. See [12, 19, 23] for their definitions and properties. 6. Linear Congruences With Constraints A direct generalization of the interpretation of E given in the Introduction is the following. Let M N and let D k (M) (1 k r) be arbitrary nonepty subsets of the set of (positive) divisors of M. For an integer n let N n (M, D 1,..., D r ) stand for the nuber of solutions (x 1,..., x r ) Z r M of the congruence x x r n (od M) (33) satisfying gcd(x 1, M) D 1,..., gcd(x r, M) D r. If D k = {M/ k } (1 k r) and n = 0, then we re-obtain the function E. Special cases of the function N n (M, D 1,..., D r ) were investigated earlier by several authors. The case D k = {1}, i.e., gcd(x k, M) = 1 (1 k r) was considered for the first tie by Radeacher [17] in 1925 and Brauer [7] in It was recovered by Nicol and Vandiver [16] in 1954, Cohen [8] in 1955, Rearick [18] in 1963, and others. The case r = 2 was treated by Alder [1] in The general case of arbitrary subsets D k was investigated, aong others, by McCarthy [11] in 1975 and by Spilker [21] in One has N n (M, D 1,..., D r ) = 1 r c M/d (n) M d M i=1 e D i(m) c M/e (d). (34) For M =, D k = {/ k } (1 k r) and n = 0 (34) reduces to our Proposition 9. The proof of forula (34) given in [21, Section 4], see also [12, Ch. 3], uses properties of r-even functions, Cauchy products and Raanujan Fourier expansions of functions. See Chapter 3 of the book of McCarthy [12] for a survey of this topic. It is well known that the nuber of solutions of polynoial congruences can be expressed using exponential sus, see for ex. [15, Th. 1.31]. Although the obtained expression can be easily transfored by eans of Raanujan sus in case of linear congruences with side conditions, this is not used in the literature cited in this section.

12 INTEGERS: 11 (2011) 12 In what follows we give a siple direct proof of (34) in the case D k (M) = {d k } (1 k r), applying the ethod of above. Proposition 21. Let M N, n Z and let d 1,..., d r M. Then N n (M, {d 1 },..., {d r }) = 1 M Proof. Only the siple fact M c M/d1 (k) c M/dr (k) exp( 2πikn/M) (35) = 1 c M/d1 (δ) c M/dr (δ)c M/δ (n). (36) M δ M M exp(2πikn/m) = { M if M n 0 if M n and the definition of Raanujan sus are required. By the definition of N n (M, {d 1 },..., {d r }), N := N n (M, {d 1 },..., {d r }) = 1 M exp(2πik(x x r n)/m) M 1 x 1 M 1 x r M gcd(x 1,M)=d 1 = 1 M M exp( 2πikn/M) gcd(x 1,M)=d r exp(2πikx 1 /M) 1 x 1 M gcd(x 1,M)=d 1 and denoting x k = d k y k, gcd(y k, M/d k ) = 1 (1 k M), N = 1 M M exp( 2πikn/M) = 1 M 1 y r M/d r gcd(y r,m/d r)=1 1 y 1 M/d 1 gcd(y 1,M/d 1)=1 exp(2πiky r /(M/d r )) (37) exp(2πikx r /M), 1 x r M gcd(x r,m)=d r exp(2πiky 1 /(M/d 1 )) M exp( 2πikn/M)c M/d1 (k) c M/dr (k). The second forula is obtained by using the fact that c M/di (k) = c M/di (gcd(k, M)) and grouping the ters according to the values of gcd(k, M) = δ (see the proof of Proposition 9). For n = 0 and d k = M/ k (1 k r) we re-obtain the interpretation of the function E and forula (17). We reark that the proof of Proposition 21 is a slight siplification of the proof of [13, Lea 4.1].

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14 INTEGERS: 11 (2011) 14 [22] L. Tóth, A survey of gcd su functions, J. Integer Sequences 13 (2010), Article , 23 pp. [23] L. Tóth and P. Haukkanen, The discrete Fourier transfor of r-even functions, 2010, subitted,