On the unitary and bi-unitary analogues of generalized Ramanujan sums Isao Kiuchi and Sumaia Saad Eddin May 03, 2018 Abstract A positive integer d is called an unitary divisor of any positive integer n if djn and (n; n=d) = 1; notation djjn: Let (k; n)∗ be the greatest divisor of the integer k which is an unitary divisor of n. Let (k; n)∗∗ be the greatest common unitary divisor of k and n. We introduce the following two functions X k X k t (j) = f(d)g ; v (j) = f(d)g ; k d k d djj(j;k)∗ dk(j;k)∗∗ for any arithmetical functions f and g. Here dk(j; k)∗ holds if and only if djj and dkk. In this paper, we give some asymptotic formulas for the weighted averages of tk(j) and vk(j) with weights concerning the monomial factor, the Gamma function, and the Bernoulli polynomials. We also derive useful formulas for the unitary and the bi-unitary analogues of the gcd-sum function. 1 Introduction and statements of the results Let (k; n) be the greatest common divisor of the integers k and n. In 1885, Ces`aropub- lished an important result on the arithmetic function, showing that for any arithmetic function f, we have n X X n P (n) := f ((k; n)) = f(d)φ : (1) d k=1 djn In a special case when f = id, one can write n X X n (k; n) = dφ = (id ∗φ)(n): (2) d k=1 djn Mathematics Subject Classification 2010: 11A25, 11N37, 11Y60. Keywords: Ramanujan's sums; the gcd-sum function; unitary divisor; Euler totient function. 1 This latter sum is called by the gcd-sum function (sometime by Pallai's arithmetic function) and due to Pallai (1937). Of course, the symbol ∗ denotes the Dirichlet con- P volution of two arithmetical functions f and g defined by (f ∗ g)(n) = djn f(d)g(n=d), for every positive integer n. Various generalizations and analogues of Eqs. (1) and (2) have been widely studied by many authors. For a nice survey see [5]. It is well known that the Ramanujan sum is defined by X k c (j) = dµ ; k d dj(k;j) for any positive integers k and j. One of the most common generalization of the Ramanujan sum is due to Anderson-Apostol [1] and defined by X k s (j) := f(d)g ; (3) k d dj(k;j) for any arithmetical functions f and g. In this paper, we introduce the unitary and bi-unitary analogues of the Anderson-Apostol sums. The next two subsections describe precisely our functions and results. 1.1 Unitary analogues A positive integer d is called an unitary divisor of any positive integer n if djn and (n; n=d) = 1; notation djjn: Let (k; n)∗ be the greatest divisor of k which is an unitary divisor of n, namely (k; n)∗ = max fd 2 N : djk; djjng : In 1989, T´oth[4] introduced the unitary analogue of Eq. (2) as follows n ∗ X P (n) = (k; n)∗: (4) k=1 In that paper, he showed that the function P ∗(n) is multiplicative and that X n P ∗(n) = dφ∗ ; (5) d djjn where φ∗ denotes the unitary analogue of Euler's function and defined by ∗ φ (n) = # fk 2 N : 1 ≤ k ≤ n; (k; n)∗ = 1g : (6) In other words, φ∗(n) is rewritten in the form X n φ∗(n) = dµ∗ = (id ∗µ∗)(n): d djn 2 Here µ∗(n) the unitary analogue of the M¨obiusfunction given by µ∗(n) = (−1)!(n) where !(n) is the he number of distinct prime factors of n. We introduce the function tk(j) defined by X k t (j) = f(d)g ; (7) k d djj(j;k)∗ such that dk(j; k)∗ holds if and only if djj and dkk. The function tk(j) is an unitary analogue of Eq.(3). From the above, we immediately deduce the identity k k=d X X k X t (j) = f(d)g 1 = (f ? g · id)(k): (8) k d j=1 dkk `=1 Here the symbol ? denotes the unitary convolution of two arithmetical functions f and P g defined by (f ? g)(n) = djjn f(d)g(n=d), for every positive integer n. Eq.(8) is a generalization of Eq. (5). Moreover, we have k X 1 X X f(d) t (j) = g(`): (9) k k d k≤x j=1 d`≤x (d;`)=1 Since φ is a multiplicative function, one can also see that k X 1 X X f(d) g(`)` t (j) = : (10) φ(k) k φ(d) φ(`) k≤x j=1 d`≤x (d;`)=1 For any complex z we define the functions Bn(x) by the generating function xz 1 ze X Bn(x) = zn ez − 1 n! n=0 for jzj < 2π. The functions Bn(x) are known as Bernoulli polynomials, and the numbers Bn(0) are called Bernoulli numbers and are denoted by Bn. Our first goal of this paper is to derive some formulas of the weighted averages of tk(j) with weight function w concerning the monomial factor, the Gamma function Γ(:), and the Bernoulli polynomials Bn(:). This is k X 1 X w(j)t (j) (11) W (k) k k≤x j=1 3 with certain weight function W . For any fixed positive integer r, we define the following function: k X 1 X T (x; f; g) := jrt (j); (12) r kr+1 k k≤x j=1 Then we prove that Theorem 1. For any positive real number x ≥ 2 and any fixed positive integer r, we have 1 X f(d) g(`) 1 X f(d) T (x; f; g) = + g(`) r 2 d ` r + 1 d d`≤x d`≤x (d;`)=1 (d;`)=1 [r=2] 1 X r + 1 X f(d) g(`) + B (13) r + 1 2m 2m d `2m m=1 d`≤x (d;`)=1 In a special case of Theorem 1 when f = f ∗ µ and g = 1, we get an useful formula for the unitary analogue of gcd-sum function, that is Corollary 1. Under the hypotheses of Theorem 1, we have 1 X (f ∗ µ)(d) 1 1 X (f ∗ µ)(d) T (x; f ∗ µ, 1) = + r 2 d ` r + 1 d d`≤x d`≤x (d;`)=1 (d;`)=1 [r=2] 1 X r + 1 X (f ∗ µ)(d) 1 + B : (14) r + 1 2m 2m d `2m m=1 d`≤x (d;`)=1 Now, we define the following two functions k X 1 X j G (x) = log Γ t (j); f;g k k k k≤x j=1 and k−1 X 1 X j Y (x) = B t (j); f;g k m k k k≤x j=0 for any fixed positive integer m. We also prove that Theorem 2. for any positive real number x ≥ 2 and any fixed positive integer m, we have p X f(d) X f(d) g(`) p G (x) = log 2π g(`) − log 2π`; (15) f;g d d ` d`≤x d`≤x (d;`)=1 (d;`)=1 and X f(d) g(`) Y (x) = B (16) f;g m d `m d`≤x (d;`)=1 4 By taking f ∗µ in place of f and g = 1 in the above, we get the following interesting formulas. Corollary 2. Under the hypotheses of Theorem 2, we have p p X (f ∗ µ)(d) X (f ∗ µ)(d) log 2π` G (x) = log 2π − ; f∗µ,1 d d ` d`≤x d`≤x (d;`)=1 (d;`)=1 and X (f ∗ µ)(d) 1 Y (x) = B : f∗µ,1 m d `m d`≤x (d;`)=1 1.2 Bi-unitary analogues Let (k; n)∗∗ be the greatest common unitary divisor of k and n, namely (k; n)∗∗ = max fd 2 N djjk; djjng : In 2008, Haukkanen [3] defined a generalization of the gcd-sum function as follows: n ∗∗ X P (n) = (k; n)∗∗: (17) k=1 It is called by the bi-unitary gcd-sum function. Not surprisingly the study of P ∗∗(n) has a lot of similarities with that of P ∗(n) or even P (n). Haukkanen also showed that X n P ∗∗(n) = φ∗(d)φ ; d ; (18) d djjn where φ(x; d) is the Legendre function. More recently, T´oth [6] gave an asymptotic formula for the partial sum of P ∗∗(n). He proved that X 1 P ∗∗(n) = Bx2 log x + O x2 ; 2 n≤x where Y 3p − 1 Y (2p − 1)2 B = 1 − = ζ(2) 1 − : p2(p + 1) p4 p p Using the above and the partial summation, he deduced that X P ∗∗(n) = Bx log x + O (x) : (19) n n≤x 5 By similar considerations, we introduce the bi-unitary analogue of Anderson-Apostol sums vk(j) defined by X k v (j) = f(d)g : (20) k d dk(j;k)∗∗ With some careful calculations, one can check that k k=(dδ) X X k X X v (j) = f(d)g µ(δ) 1 (21) k d j=1 dkk δjd `=1 = (f · φ · id−1 ?g · id)(k): (22) Then, we get k X 1 X X f(d) φ(d) v (j) = g(`); (23) k k d d k≤x j=1 d`≤x (d;`)=1 k X 1 X X f(d) g(`)` v (j) = : (24) φ(k) k d φ(`) k≤x j=1 d`≤x (d;`)=1 For any positive integer r, we consider the function Vr(x; f; g) k X 1 X V (x; f; g) := jrv (j) (25) r kr+1 k k≤x j=1 to get the following result.