On sums of arithmetic functions involving the greatest common divisor
aa r X i v : . [ m a t h . N T ] F e b On sums of arithmetic functions involving thegreatest common divisor
Isao Kiuchi, and Sumaia Saad Eddin
Abstract
Let gcd( d , . . . , d k ) be the greatest common divisor of the positive integers d , . . . , d k , for any integer k ≥
2, and let τ and µ denote the divisor function andthe M¨obius function, respectively. For an arbitrary arithmetic function g and forany real number x > k ≥
3, we define the sum S g,k ( x ) := X n ≤ x X d ··· d k = n g (gcd( d , . . . , d k ))In this paper, we give asymptotic formulas for S τ,k ( x ) and S µ,k ( x ) for k ≥ Let gcd( d , . . . , d k ) be the greatest common divisor of the integers d , . . . , d k for anyinteger k ≥
2, and let µ and τ denote the M¨obius function and the divisor function,respectively. We recall that the Dirichlet convolution of two arithmetic functions f and g is defined by f ∗ g ( n ) = P d | n f ( d ) g ( n/d ) for all positive integers n . The arithmeticfunctions ( n ) and id( n ) are defined by ( n ) = 1 and id( n ) = n respectively. Thefunction τ k is the k -factors Piltz divisor function given by ∗ ∗ · · · . In case of k = 2,we write τ = τ . For an arbitrary arithmetic function g , we define the sum f ( g,k ) ( n ) := X d ··· d k = n g (gcd( d , . . . , d k )) . In [4], Kr¨atzel, Nowak and T´oth gave asymptotic formulas for a class of arithmeticfunctions, which describe the value distribution of the greatest common divisor function.Typically, they are generated by a Dirichlet series whose analytic behavior is determinedby the factor ζ ( s ) ζ (2 s − , where ζ ( s ) is the Riemann zeta-function. In regards to theabove formula, they proved that f ( g,k ) ( n ) = X a k b = n µ ∗ g ( a ) τ k ( b ) . Mathematics Subject Classification 2010: 11A25, 11N37, 11Y60.Keywords: The greatest common divisor function, the Piltz divisor function, Euler totient function. f ( g,k ) ( n ) for specific choicesof g such as the identity function id and the sum of divisors function σ = id ∗ . It wasshown that f (id ,k ) ( n ) = X a k b = n φ ( a ) τ k ( b ) , f ( σ,k ) ( n ) = X a k b = n aτ k ( b ) , and that, for ℜ ( s ) > ∞ X n =1 f (id ,k ) ( n ) n s = ζ k ( s ) ζ ( ks − ζ ( ks ) , ∞ X n =1 f ( σ,k ) ( n ) n s = ζ k ( s ) ζ ( ks − . In case of g = δ := µ ∗ (i.e., δ (1) = 1 or δ ( n ) = 0 else), the function f ( δ,k ) ( n ) = X d ··· d k = n gcd( d ··· d k )=1 S g,k ( x ) := X n ≤ x f ( g,k ) ( n ) = X a k b ≤ x µ ∗ g ( a ) τ k ( b )In two cases g = τ and g = µ , it is easy to see that S τ,k ( x ) = X m ≤ x k X ℓ ≤ x/m k τ k ( ℓ ) , (1)and that S µ,k ( x ) = X m ≤ x k µ ∗ µ ( m ) X n ≤ xmk τ k ( n ) . (2)In this paper, we give asymptotic formulas for S τ,k ( x ) and S µ,k ( x ) for any integer k ≥ Theorem 1.
For any real number x > , we have S τ, ( x ) = ζ (3)2 x log x + ζ (3) (cid:18) γ − ζ ′ (3) ζ (3) (cid:19) x log x (3)+ ζ (3) (cid:18) γ − γ + 3 γ + 1 + 3(3 γ − ζ ′ (3) ζ (3) + 92 ζ ′′ (3) ζ (3) (cid:19) x + O (cid:16) x + ε (cid:17) , nd S τ, ( x ) = ζ (4)6 x log x + ζ (4) (cid:18) γ −
12 + 2 ζ ′ (4) ζ (4) (cid:19) x log x (4)+ ζ (4) (cid:18) γ − γ + 4 γ + 1 + 8 (cid:18) γ − (cid:19) ζ ′ (4) ζ (4) + 8 ζ ′′ (4) ζ (4) (cid:19) x log x + ζ (4) (cid:18) γγ + 4 γ − γ + 4( γ − γ + γ ) − γ − γ + 4 γ + 1) ζ ′ (4) ζ (4) (cid:19) x + 16 ζ (4) (cid:18)(cid:18) γ − (cid:19) ζ ′′ (4) ζ (4) + 23 ζ (3) (4) ζ (4) (cid:19) x + O (cid:16) x + ε (cid:17) , where γ is the Euler constant, γ and γ are the Laurent-Stieltjes constants, (see Section 2below for details). Here the function ζ ( k ) ( s ) is the k -th derivative of the Riemann zeta-function ζ ( s ) with respect to s . Theorem 2.
Under the hypotheses of Theorem 1 , we have S µ, ( x ) = 12 ζ (3) x log x + 1 ζ (3) (cid:18) γ − − ζ ′ (3) ζ (3) (cid:19) x log x (5)+ 1 ζ (3) γ − γ + 3 γ + 1 − γ − ζ ′ (3) ζ (3) − ζ ′′ (3) ζ (3) + 27 (cid:18) ζ ′ (3) ζ (3) (cid:19) ! x + O (cid:16) x + ε (cid:17) , and S µ, ( x ) = 16 ζ (4) x log x + 1 ζ (4) (cid:18) γ − − ζ ′ (4) ζ (4) (cid:19) x log x (6)+ 16 ζ (4) γ − γ + 4 γ + 116 − (cid:18) γ − (cid:19) ζ ′ (4) ζ (4) − ζ ′′ (4) ζ (4) + 3 (cid:18) ζ ′ (4) ζ (4) (cid:19) ! x log x + 1 ζ (4) (cid:18) γγ + 4 γ − γ + 4( γ − γ + γ ) − − γ − γ + 4 γ + 1) ζ ′ (4) ζ (4) (cid:19) x + 32 ζ (4) (cid:18) γ − (cid:19) (cid:18) ζ ′ (4) ζ ′ (4) (cid:19) − ζ ′′ (4) ζ (4) ! x − ζ (4) (cid:18) ζ ′ (4) ζ (4) (cid:19) − ζ ′ (4) ζ (4) · ζ ′′ (4) ζ (4) + ζ (3) (4) ζ (4) ! x + O (cid:16) x + ε (cid:17) . The following theorem states asymptotic formulas of S τ,k ( x ) and S µ,k ( x ) for anyinteger k ≥ . Theorem 3.
Let P g,k ( u ) be a polynomial in u of degree k − depending on g . For g = τ , g = µ , and k ≥ , we have S g,k ( x ) = xP g,k (log x ) + E g,k ( x ) (7)3 here E g,k ( x ) = O (cid:16) x k − k + ε (cid:17) (5 ≤ k ≤ ,E g, ( x ) = O (cid:16) x + ε (cid:17) ,E g, ( x ) = O (cid:16) x + ε (cid:17) ,E g, ( x ) = O (cid:16) x + ε (cid:17) ,E g,k ( x ) = O (cid:16) x k − k +2 + ε (cid:17) (12 ≤ k ≤ ,E g,k ( x ) = O (cid:16) x k − k +4 + ε (cid:17) (26 ≤ k ≤ ,E g,k ( x ) = O (cid:16) x k − k + ε (cid:17) (51 ≤ k ≤ ,E g,k ( x ) = O (cid:16) x k − k + ε (cid:17) ( k ≥ , for any small number ε > . Before going into the proof of Theorems, we recall that the Laurent expansion of theRiemann zeta-function at its pole s = 1 is given by ζ ( s ) = 1 s − ∞ X k =0 γ k ( s − k . Here the constants γ k are often called the Laurent-Stieltjes constants and it is knownthat γ n = ( − n n ! lim M →∞ M X m =1 (log m ) n m − (log M ) n +1 ( n + 1) ! , for all n ≥ γ = γ = 0 , · · · being the Euler–Mascheroni constant.We define the error term ∆ k ( x ) of the Piltz divisor problem by∆ k ( x ) := X n ≤ x τ k ( n ) − x P k − (log x ) , (8)where P k − ( t ) is a polynomial of degree k − t . Notice that the coefficients of P k − may be evaluated by using P k − (log x ) = Res s =1 ζ k ( s ) x s − s . (9)From Eqs. (8) and (9), one can calculate explicity the coefficients of P k − as functionsof the Laurent-Stieltjes constants. For more details, see [2, Chapter 13].In order to prove our main results, it will be necessary to give some lemmas.4 emma 1. We have X n ≤ x τ ( n ) = (cid:0) b log x + b log x + b (cid:1) x + ∆ ( x ) , (10) where ∆ ( x ) ≪ x + ε , and b = 12 , b = 3 γ − , b = 3 γ − γ + 3 γ + 1 . Furthermore, we have X n ≤ x τ ( n ) = (cid:0) c log x + c log x + c log x + c (cid:1) x + ∆ ( x ) (11) where ∆ ( x ) ≪ x + ε , and c = 16 , c = 2 γ − , c = 6 γ − γ + 4 γ + 1 ,c = 12 γγ + 4 γ − γ + 4( γ − γ + γ ) − . Proof.
The proof of Eqs. (10) and (11) can be found in [3] and [2, Theorems 13.2],respectively.
Lemma 2.
Let α k be the infimum of numbers a k such that ∆ k ( x ) ≪ ( x a k + ε ) for anysmall ε > . Then α k ≤ k − k (5 ≤ k ≤ ,α ≤ ,α ≤ ,α ≤ ,α k ≤ k − k + 2 (12 ≤ k ≤ ,α k ≤ k − k + 4 (26 ≤ k ≤ ,α k ≤ k − k (51 ≤ k ≤ ,α k ≤ k − k ( k ≥ . Proof.
The proof of this lemma can be found in [2, Theorem 13.2].5 emma 3.
For any real number x > and any integer k ≥ , we have X n ≤ x k n k = ζ ( k ) + 11 − k x − kk + O (cid:0) x − (cid:1) . (12) Furthermore, we have X n ≤ x k log nn k = − ζ ′ ( k ) + 1 k (1 − k ) x − kk log x − − k ) x − kk + O (cid:0) x − log x (cid:1) , (13) X n ≤ x k log nn k = ζ ′′ ( k ) + 1(1 − k ) k x − kk log x − k (1 − k ) x − kk log x + 2(1 − k ) x − kk + O (cid:0) x − log x (cid:1) , (14) X n ≤ x k log nn k = − ζ (3) ( k ) + 1(1 − k ) k x − kk log x − k (1 − k ) x − kk log x + 6 k (1 − k ) x − kk log x + 6(1 − k ) x − kk + O (cid:0) x − log x (cid:1) . (15) Proof.
Eq. (12) is given in [2, Eq. (14.40)]. Let z be any large real number, and let r beany positive integer. Then we have X n ≤ z log r nn s = ( − r ζ ( r ) ( s ) − lim y →∞ X z
1. Using the fact that X n ≤ z log n = z log z − z + O (log z )and the partial summation, we getlim y →∞ X z
5, we define M ( x ) := X n ≤ x µ ∗ µ ( n )It is known that M ( x ) can be estimate by, see [1, Eq. (4.11)], M ( x ) = O ( xε ( x )) , (20)where ε ( x ) = exp (cid:18) − C (log x ) / (log log x ) / (cid:19) (21)with C being a positive constant. Under the above hypotheses, we are ready to statethe following result. Lemma 4.
For any real number x > and any integers k ≥ , we have X n ≤ x k µ ∗ µ ( n ) n k = 1 ζ ( k ) + O (cid:16) x − kk ε ( x ) (cid:17) , (22)7 nd, for r any positive integer, X n ≤ x k µ ∗ µ ( n ) log r nn k = M k,r ( k ) + O (cid:16) x − kk ε ( x ) (cid:17) (23) where ε ( x ) is given by (21) . Here M k,r ( k ) are certain constants depending on the Rie-mann zeta-function. Moreover, we have M k, ( k ) = 2 ζ ′ ( k ) ζ ( k ) , M k, ( k ) = 2 3( ζ ′ ( k )) − ζ ′′ ( k ) ζ ( k ) ζ ( k ) ,M k, ( k ) = 2 ζ ( k ) (cid:18) ζ ′ ( k ) ζ ( k ) (cid:19) + 3 ζ ′ ( k ) ζ ( k ) · ζ ′′ ( k ) ζ ( k ) − ζ (3) ( k ) ζ ( k ) ! . Proof.
We recall that, for z any large real number and any s > X n ≤ z µ ∗ µ ( n ) n s = 1 ζ ( s ) − lim y →∞ X z In case of k = 3, we substitute Eq. (10) into Eq. (1) and then we use Eqs. (12), (13),and (14) of Lemma 3 to obtain S τ, ( x ) = (cid:0) b log x + b log x + b (cid:1) x X n ≤ x n − (6 b log x + 3 b ) x X n ≤ x log nn + 9 b x X n ≤ x log nn + X n ≤ x ∆ (cid:16) xn (cid:17) = ζ (3)2 x log x + ζ (2) (cid:18) γ − ζ ′ (3) ζ (3) (cid:19) x log x + ζ (3) (cid:18) γ − γ + 3 γ + 1 + 3(3 γ − ζ ′ (3) ζ (3) + 92 ζ ′′ (3) ζ (3) (cid:19) x − (cid:0) γ − γ + 12 γ + 19 (cid:1) x / + X m ≤ x ∆ (cid:16) xn (cid:17) + O (cid:0) log x (cid:1) . Now, we use the estimate, see [3], ∆ ( x ) ≪ x + ε to get X n ≤ x ∆ (cid:16) xn (cid:17) ≪ x + ε X n ≤ x n ≪ x + ε , for any small number ε > 0. This completes the proof of Eq. (3).In the case of k = 4, we substitute Eq. (11) into Eq. (1) S τ, ( x ) = (cid:0) c log x + c log x + c log x + c (cid:1) x X n ≤ x n − (cid:0) c log x + 2 c log x + c (cid:1) x X n ≤ x log nn + 4 (3 c log x + c ) x X n ≤ x log nn − c x X n ≤ x log nn + X n ≤ x ∆ (cid:16) xn (cid:17) , S τ, ( x ) = ζ (4)6 x log x + ζ (4) (cid:18) γ − 12 + 2 ζ ′ (4) ζ (4) (cid:19) x log x + ζ (4) (cid:18) γ − γ + 4 γ + 1 + 8 (cid:18) γ − (cid:19) ζ ′ (4) ζ (4) + 8 ζ ′′ (4) ζ (4) (cid:19) x log x + ζ (4) (cid:18) γγ + 4 γ − γ + 4( γ − γ + γ ) − γ − γ + 4 γ + 1) ζ ′ (4) ζ (4) (cid:19) x + 16 ζ (4) (cid:18)(cid:18) γ − (cid:19) ζ ′′ (4) ζ (4) + 23 ζ (3) (4) ζ (4) (cid:19) x + ρx + X m ≤ x ∆ (cid:16) xm (cid:17) + O (cid:0) log x (cid:1) , with ρ being a computable constant. By Lemma 2, we have X m ≤ x ∆ (cid:16) xm (cid:17) ≪ x + ε X m ≤ x m ≪ x + ε . Therefore Eq. (4) is proved. In much the same way as in Subsection 3.1, we prove Eqs. (5) and (6). First, in the caseof k = 3, we substitute Eqs. (10), (22), and (23) into Eq. (2) with r = 1 , S µ, ( x ) = 12 ζ (3) x log x + 1 ζ (3) (cid:18) γ − − ζ ′ (3) ζ (3) (cid:19) x log x + 1 ζ (3) γ − γ + 3 γ + 1 − γ − ζ ′ (3) ζ (3) − ζ ′′ (3) ζ (3) + 27 (cid:18) ζ ′ (3) ζ (3) (cid:19) ! x + X n ≤ x µ ∗ µ ( n )∆ (cid:16) xn (cid:17) + O (cid:0) x / ε ( x ) (cid:1) . Again we use ∆ ( x ) ≪ x + ε to get X n ≤ x µ ∗ µ ( n )∆ (cid:16) xn (cid:17) ≪ x + ε X n ≤ x τ ( n ) n ≪ x + ε . This completes the proof of Eq. (5).Second, for the case of k = 4 and r = 1 , , 3, we substitute Eqs. (11), (22) and (23)10nto Eq. (2) to obtain S µ, ( x ) = 16 ζ (4) x log x + 1 ζ (4) (cid:18) γ − − ζ ′ (4) ζ (4) (cid:19) x log x (28)+ 16 ζ (4) γ − γ + 4 γ + 116 − (cid:18) γ − (cid:19) ζ ′ (4) ζ (4) − ζ ′′ (4) ζ (4) + 3 (cid:18) ζ ′ (4) ζ (4) (cid:19) ! x log x + 1 ζ (4) (cid:18) γγ + 4 γ − γ + 4( γ − γ + γ ) − − γ − γ + 4 γ + 1) ζ ′ (4) ζ (4) (cid:19) x + 32 ζ (4) (cid:18) γ − (cid:19) (cid:18) ζ ′ (4) ζ ′ (4) (cid:19) − ζ ′′ (4) ζ (4) ! x − ζ (4) (cid:18) ζ ′ (4) ζ (4) (cid:19) − ζ ′ (4) ζ (4) · ζ ′′ (4) ζ (4) + ζ (3) (4) ζ (4) ! x + X m ≤ x µ ∗ µ ( m )∆ (cid:16) xm (cid:17) + O (cid:0) x / ε ( x ) (cid:1) . Using ∆ ( x ) ≪ x + ε , we get X m ≤ x µ ∗ µ ( m )∆ (cid:16) xm (cid:17) ≪ x + ε X m ≤ x τ ( m ) m ≪ x + ε . This completes the proof of Eq. (6). By the following formula and the partial summation X n ≤ z log r z = z log r z + d z log r − z + d z log r − z + · · · + d r z + O (log r z ) , (29)we find that lim y →∞ X z 5, we have X n ≤ x k ∆ k (cid:16) xn k (cid:17) ≪ x α k + ε X n ≤ x /k n kα k + ε ≪ x α k + ε , where we used the fact that kα k > 2. This completes the proof of Theorem 3 in the case g = τ .Similar arguments apply to the case g = µ. From Eqs. (2), (22) and (23), we get S µ,k ( x ) = x k − X r =0 q r r X ℓ =0 (cid:18) rℓ (cid:19) (log x ) r − ℓ k ℓ X n ≤ x /k µ ∗ µ ( n ) log ℓ nn k + X n ≤ x /k µ ∗ µ ( n )∆ k (cid:16) xn k (cid:17) = x k − X r =0 q r r X ℓ =0 (cid:18) rℓ (cid:19) (log x ) r − ℓ k ℓ M k,ℓ ( k ) + X m ≤ x /k µ ∗ µ ( m )∆ k (cid:16) xm k (cid:17) + O (cid:16) x k ε ( x ) (cid:17) = xP µ,k (log x ) + E µ,k ( x ) , where E µ,k ( x ) = X n ≤ x k µ ∗ µ ( n )∆ k (cid:16) xn k (cid:17) + O (cid:0) x /k ε ( x ) (cid:1) , and M k, ( x ) = 1 /ζ (2). From Lemma 2, the above sums can be estimated by x α k + ε X n ≤ x /k τ ( n ) n kα k + ε ≪ x α k + ε . This completes the proof of our theorem. 12 cknowledgement The authors express their gratitude to Professor W ladys law Narkiewicz for carefullyreading the paper and useful comments. The second author is supported by the AustrianScience Fund (FWF): Projects F5507-N26 and F5505-N26, which are part of the SpecialResearch Program “Quasi Monte Carlo Methods: Theory and Applications”. References [1] A. Ivi´c, On the asymptotic formulae for some functions connected with powers ofthe zeta function, Matematiˇcki Vesnik (Belgrade) (29). 63 (1977), 79–90.[2] A. Ivi´c, The Riemann Zeta-Function: Theory and Applications , Dover publicationsInc., Mineola, New York, 1985.[3] G. Kolesnik, On the estimation of multiple exponential sums, Recent Progress inAnalytic Number Theory, Vol. 1 (eds. H. Halberstam and C. Hooley, AcademicPress, New York, 1981) 247–256.[4] E. Kr¨atzel, W. G. Nowak and L. T´oth, On certain arithmetic functions involvingthe greatest common divisor, Cent. Eur. J. Math. (2012), 761–774.[5] M. B. Nathanson, Elementary Methods in Number Theory , Graduate Texts in Math. , Springer, 2000. Isao Kiuchi: Department of Mathematical Sciences, Faculty of Science, Yamaguchi University, Yoshida1677-1, Yamaguchi 753-8512, Japan. e-mail: [email protected] Sumaia Saad Eddin: Institute of Financial Mathematics and Applied Number Theory, Johannes KeplerUniversity, Altenbergerstrasse 69, 4040 Linz, Austria. e-mail: sumaia:saad [email protected]:saad [email protected]