A Generalization of the Gcd-Sum Function

Transcription

1 Journal of Integer Sequences, Vol 6 203, Article 367 A Generalization of the Gcd-Sum Function Ulrich Abel, Waqar Awan, and Vitaliy Kushnirevych Fachbereich MND Technische Hochschule Mittelhessen University of Applied Sciences Wilhelm-Leuschner-Straße Friedberg Germany UlrichAbel@mndthmde WaqarAwan@studh-dade VitaliyKushnirevych@mndthmde Abstract In this paper we consider the generalization G d n of the Broughan gcd-sum function, ie, the sum of such gcd s that are divisors of the positive integer d Examples of Dirichlet series and asymptotic relations for G d and related functions are given Introduction In the recent article [5], Broughan studies the sum of the greatest common divisors of the first n positive integers with n, ie, the arithmetic function n Gn := gcdk,n k= This function arises in deriving asymptotic estimates for a lattice point counting problem [5, Sect 5] The function G has polynomial growth as n tends to infinity For p P throughout the paper P denotes the set of prime numbers and α N, it is not difficult to show that Gp α = α j=0 p p α j p }{{} number of gcd s equal to p j j + p α = α+p α αp α

2 cf [5, Th 22] Following [5, Cor 2] G is a multiplicative function, ie, Gmn = GmGn for coprime m,n N, that is, gcdm,n = The corresponding Dirichlet series Gs converges at all points of the complex plane, except at the zeros of the Riemann zeta function and the point s = 2, where it has a double pole Moreover, Broughan derives asymptotic expressions for the partial sums of the Dirichlet series at all real values of s The following generalization of G see [2] arises in the study of distribution of determinant values in residue class rings For d N, we introduce the function G d n := n k= gcdk,n d gcdk,n Obviously, Gn = G n n and G n = ϕn, where ϕ is Euler s totient function The purpose of this note is to study the function G d In the next section we present some elementary properties of G d Furthermore, we study the corresponding Dirichlet series G d s Some of the results will be applied in a forthcoming paper on the distribution of determinant values in residue class rings and finite fields As an example we mention that in the residue class ring Z n n N, for r Z n, H n r = {i,j Z n Z n i j = r}, the number of products equal to r having precisely two factors in Z n, is equal to { G n n = Gn, if r = 0; H n r = G d n = G gcdr,n n, if r 0 A similar problem as the calculation of the value H n r in the domain of positive integers is the so-called multiplication table problem posed by Erdős see [7]: how many integers can be written as a product i j for a given positive integer n N with positive integers i n and j n? Erdős [7, 8] gave the first estimates of this quantity Tenenbaum [3] had made the results of Erdős more precise Ford [9, 0] derived the exact order of magnitude of the n n multiplication table size completely Koukoulopolous [, 2] presents a perfect overview of the actual situation and the further development of Ford-Erdős results 2 Properties of G d The following lemma gathers some elementary properties of G d n Lemma i For m,n N, we have G m n = G gcdm,n n In particular, for m,α N, p P, we have G gcdm,p α p α = G m p α ; 2

3 ii for coprime d,n N, we have G d n = ϕn; iii for d = d d 2 with gcdd,n =, we have G d n = G d2 n Proof i Since gcdk,n gcdm,n gcdk,n m for all m,n,k N, the first formula follows from the definition One obtains the second one by substituting n = p α ii If gcdd,n =, using i we get G d n = G n = ϕn iii Since gcdd,n = gcdd d 2,n = gcdd 2,n, it follows that G d n = G d d 2 n = G gcdd d 2,nn = G gcdd2,nn = G d2 n, where we used i twice The proof of the lemma is completed Let ρ d denote the multiplicative function { w, if w d; ρ d w = 0, if w d Then we have the representation where denotes Dirichlet product Indeed, G d n = n k= gcdk,n d gcdk,n = w d w n G d = ρ d ϕ, n wϕ = w n ρ d wϕ = ρ d ϕn w w n Therefore, G d is multiplicative as it is the Dirichlet product of multiplicative functions [, Th 25c and Th 24] Theorem 2 G d is a multiplicative function, ie, for coprime m,n N, we have G d mn = G d mg d n We also give a direct proof of the preceding theorem Proof Let d n n 2 with coprime n,n 2 N This implies d = d d 2 with d n and d 2 n 2, so that d and d 2 are coprime One has G d n n 2 = G d d 2 n n 2 = n n 2 wϕ = w d n n 2 w w 2 ϕ w w w 2 w d d 2 w 2 d 2 3

4 n Because ϕ is multiplicative and gcd w, n 2 w 2 =, one obtains G d n n 2 = n n2 w ϕ w 2 ϕ = G d n G d2 n 2 = G d n G d n 2 w w 2 w d w 2 d 2 This completes the proof Theorem 3 For n N and for coprime d,d 2 N, we have In particular, G d d 2 n G d ng d2 n G d n G d2 n = ϕn G d d 2 n Proof Let d = d d 2 with gcdd,d 2 = By Equation we have G d n = ρ d ϕn = n ρ d d 2 wϕ = w n ρ d w ρ d2 w 2 ϕ w w 2 w n w n w 2 n Now, decompose n = kn n 2 in a product of three pairwise coprime factors k, n, n 2 such that d i n i i =,2 If w i d i i =,2 we conclude that n n n n n n n2 ϕ w ϕ w 2 ϕ ϕ = ϕkϕ ϕ = ϕk w w 2 w w 2 ϕkn 2 ϕkn = w ϕ w 2 ϕn Hence, we obtain ϕng d n = n d w ϕ w nρ which is the desired formula w w 2 n We close this section with the following nice formula Theorem 4 For all n N, we have n G i n = n 2 i= n ρ d2 w 2 ϕ = G d n G d2 n w 2 Proof Analogously to the proof of Theorem 2 one has n n n G i n = ρ i wϕ = w n n ϕ ρ i w w w n i= i= w n = n ϕ ww w n where we used that w n ϕw = n i n w i i= = n ϕ w w n w = n ϕw = n 2, w n w n 4

5 3 Evaluation of G d at positive integers In this section we consider the problem how to calculate the values of G d n for positive integers We start with the special { case of prime powers In the following δ αβ denotes the, if α = β; Kronecker symbol defined by δ αβ = 0, otherwise Proposition 5 For prime powers n = p α α N and d = p β, β α β N {0}, we have G p βp α = ϕp α +β + δ αβ p For prime powers n = p α α N and d = p β, β > α β N, we have G p βp α = G p αp α = ϕp α +α+ p Proof For 0 < β < α, we have and, for β = α, G p βp α = β p α j p α j p j = p α p α +β = ϕp α +β, j=0 G p βp α = G p αp α = Gp α = α+p α αp α = α+p α p α +p α = ϕp α +β+p α In the case β = 0 application of Lemma ii leads to G p α = ϕp α = ϕp α +β Thus, for all 0 β α, one has G p βp α = ϕp α +β+p α δ αβ = ϕp α +β + pα δ αβ ϕp α Taking into account that ϕp α = p α p α one obtains the first result For β > α, we have gcdk,p α p β gcdk,p α p α Hence, G p βp α = p α gcdk,p k= gcdk,p α p β α = p α gcdk,p α = G p αp α k= gcdk,p α p α and the second result follows by application of the first formula Remark 6 The result of Proposition 5 can be written in one single formula: for p P, α N and β N {0}, we have G p βp α = ϕp α +minα,β+ δ α,minα,β p 5

6 Theorem 7 For n N with prime powers decomposition n = p λ p λt t and positive integer d = c p κ p κt t with p j c for all j =,,t, and 0 κ j we have the representation G d n = ϕn t +minκ j,λ j +δ λj,minκ j,λ j p j j= Proof Because G d is multiplicative, by Theorem 2, and applying Lemma iii, we obtain G d n = G d t j= = = = t j= t j= t j= p λ j j G κ c p pκ t t G p κ j j ϕ p λ j j p λ j j p λ j j where the last equation is a consequence of Rem 6 +minκ j,λ j + δ λ j minκ j,λ j, p j We note that under the notation of Theorem 7 the equation gcdd,n = p κ p κt t defines unique numbers κ j j =,,t with 0 κ j λ j, such that the result can be written in the form t G d n = G gcdd,n n = ϕn +κ j +δ λj,κ j p j 4 Dirichlet series, averages and asymptotic properties Some asymptotic formulas of the Broughan s gcd-sum function were derived by Broughan [5] and Bordellès [4] The average order of the Dirichlet series of the Broughan s gcd-sum function was studied by Broughan[6] and Bordellès[3] In this section we give some examples of Dirichlet series of arithmetic functions connected with G d n We calculate the average functions and derive some asymptotic formulas for these examples κ j = 0 means that p j is not present in the decomposition of d, ie, p j d j= 6

7 The Dirichlet series for an arithmetic function fn is defined see, eg, [,, p 224] by fn Fs := n s The most prominent example is the Riemann ζ function ζs = ns It is clear, that ζs is the Dirichlet series associated to fn =, for all n N For any prime number p, the Bell series [, Sect 25, p 42ff] of an arithmetic function f is the formal power series f p x = fp n x n If f is multiplicative the corresponding Dirichlet series is given by Fs = fnn s = p f p p s provided that the Dirichlet series converges absolutely for Res > a see, eg, [, Th 7, p 23] The number e N {0} is called the m-adic order of n N m N, if m e n and m e+ n It is denoted by e = ν m n 4 The arithmetic function G d 4 Dirichlet series Since G d = ρ d ϕ see and P d s := Φs := ρ d n n s = n d n s ; ϕn = ζs n s ζs = p p s p s, [, Ex 4, p 229 and p 23], we have according to [, Th 5]: for Res > 2, so G d s := G d s = ζs ζs G d n n s = n d ρ d n ϕn n s n s = p = P d sφs, p s p s n s n d 7

8 If d = one obviously has G s = ζs ζs 42 Average functions G d s = ζs ζs cf [, Ex 3, p 23] For d P, one has + d s We study the asymptotic behaviour of the average function G [α] d x := n xn α G d n as n tends to infinity Taking advantage of the representation G d = ρ d ϕ we obtain G [α] d x = ρ d w ϕ n w w α n α n x w n w By application of [, Th 30, p 65], we conclude that G [α] d x = x n α ρ d nφ [α] = n n x w d w α Φ [α] x w, where Φ [α] denotes the average Φ [α] x := n xn α ϕn of Euler s totient function ϕ We distinguish 3 cases Because, for α, [, Ex 8, p 7], we have Φ [α] x x2 α 2 α ζ 2+O x α logx G [α] d x x2 α 2 α ζ 2 w d Because, for α >,α 2, w +O x α logx x, x Φ [α] x x2 α 2 α ζ 2+ ζα +O x α logx x, ζα [, Ex 7, p 7], we have G [α] d x x2 α 2 α ζ 2 w d w + ζα ζα w α +O x α logx w d x 8

9 Finally, for α = 2, we have Φ [2] x logx ζ2 + γ ζ2 A+O logx x x, where γ is Euler s constant and A = µnn 2 logn [, Ex 6, p 7], and we conclude that G [2] d x logx/w γ logx + ζ2 w ζα A w +O x x w d w d 42 The arithmetic function G n/gcdr,n n Let r N be given Consider the arithmetic function b r n := G n/gcdr,n n which is easily seen to be multiplicative Let p be a prime number and put β = ν p r According to Prop 5 one has b r p n = G p n /gcdr,p n p n = G p n βp n = ϕp n +n β + δ n,n β p So, if β = 0 one has δ n,n β = and b r p n = ϕp n +n+ = n+p n np n p Therefore, for β = 0, the Bell series is given by b r p x = b r p n x n = If β > 0 one has δ n,n β = 0 and b r p x = = b r p n x n = n+p n np n x n = x px 2 ϕp n +n βx n p n p n +n βx n = Hence, the Dirichlet series is given by where βp = ν p r B r s := ζ2 s ζs p r p pxβ β + ppx 2 p p s βp p p s Res > 2, 9

10 43 The arithmetic function G n gcdr,nn Let r N be given Consider the arithmetic function a r n := G n gcdr,n n which is easily seen to be multiplicative Let p be a prime number and put β = ν p r According to Remark 6 one has a r p n = G p ngcdr,p n p n = G p n p n+minβ,n = ϕ p n+minβ,n +n+δ 0,minβ,n p If β = 0, one has δ 0,minβ,n = and a r p n = ϕp n +n+ = n+p n np n p Therefore, for β = 0, the Bell series is given by a p r x = a r p n x n = n+p n np n x n = x px 2 If β > 0 one has δ 0,minβ,n = 0 and a r p x = + = + = + +nϕp n+minβ,n x n β +nϕp 2n x n + n=β+ β +np 2n p 2n x n + = + p p +nϕp n+β x n n=β+ β n+p 2 x n +p β p +np n+β p n+β x n n=β+ n+px n = + p pxβ +p2 x β+ β +2p 2 x β p 2 x+2 p 2 x 2 p p2β x β+ β +px β +2 px 2 Hence, the Dirichlet series is given by A r s := ζ2 s p s 2 a r ζs p s p p s p r 0 Res > 2

11 where a r p p s = p p2β p sβ+ β +p s β +2 p s 2 and β = βp = ν p r + p p s β +p 2 sβ+ β +2p 2 sβ p 2 x+2 p 2 x 2 5 Acknowledgment The authors are grateful to the referee for valuable remarks, in particular for pointing out the similarity of our function H n r to the famous Erdős multiplication table problem References [] T M Apostol, Introduction to Analytic Number Theory, Springer, 976 [2] W Awan, Werteverteilung der Determinanten von Matrizen über Restklassenringen und endlichen Körpern, Bachelor thesis, Friedberg, Germany, 202 [3] O Bordellès, A note on the average order of the gcd-sum function, J Integer Seq , Article 0733 [4] O Bordellès, An asymptotic formula for short sums of gcd-sum functions, J Integer Seq 5 202, Article 268 [5] K A Broughan, The gcd-sum function, J Integer Seq 4 200, Article 022 [6] K A Broughan, The average order of the Dirichlet series of the gcd-sum function, J Integer Seq , Article 0742 [7] P Erdős, Some remarks on number theory, Riveon Lematematika 9 955, [8] P Erdős, An asymptotic inequality in the theory of numbers, Vestnik Leningrad Univ 5 960, 4 49 [9] K Ford, Integers with a divisor in y,2y] in J-M Koninck, A Granville, F Luca, eds, Anatomy of Integers, CRM Proc and Lect Notes 46, Amer Math Soc, Providence, RI, 2008, pp 65 8 [0] K Ford, The distribution of integers with a divisor in a given interval, Annals of Math ,

12 [] D Koukoulopolous, Generalized and restricted multiplication tables of integers, PhD thesis, Univ Illinois, 200, [2] D Koukoulopolous, On the number of integers in a generalized multiplication table, preprint, [3] G Tenenbaum, Sur la probabilité qu un entier possède un diviseur dans un intervalle donné, Compositio Math 5 984, Mathematics Subject Classification: Primary A05; Secondary A25, F66, N37, N56 Keywords: multiplicative structure, arithmetic function, one-variable Dirichlet series, asymptotic results on arithmetic functions Received February 6 203; revised version received June Published in Journal of Integer Sequences, July Return to Journal of Integer Sequences home page 2