On certain arithmetic functions involving the greatest common divisor

Krätzel, Ekkehard; Nowak, Werner Georg; Tóth, László

doi:10.2478/s11533-011-0144-6

Cited by 11 publications

(29 citation statements)

References 7 publications

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“…In [4], Krätzel, Nowak and Tóth gave asymptotic formulas for a class of arithmetic functions, which describe the value distribution of the greatest common divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ 2 (s)ζ(2s − 1), where ζ(s) is the Riemann zeta-function.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On sums of arithmetic functions involving the greatest common divisor

Kiuchi¹,

Eddin²

2021

Preprint

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Let gcd(d 1 , . . . , d k ) be the greatest common divisor of the positive integers d 1 , . . . , d k , for any integer k ≥ 2, and let τ and µ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function g and for any real number x > 5 and any integer k ≥ 3, we define the sumIn this paper, we give asymptotic formulas for S τ,k (x) and S µ,k (x) for k ≥ 3.a k b=n µ * g(a)τ k (b).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On sums of arithmetic functions involving the greatest common divisor

Kiuchi¹,

Eddin²

2021

Preprint

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“…Also see [8]. We also have For other related asymptotic results for functions involving the gcd and lcm of two (and several) integers see the papers [2,8,11,17,19] and their references. For summations of some other functions of two variables see [3,14,20].…”

Section: Introductionmentioning

confidence: 98%

“…3.1] using elementary arguments. Formula (1.2) was deduced applying analytic methods by Krätzel et al [11,Th. 3.5].…”

Section: Introductionmentioning

confidence: 99%

“…Formula (1.2) represents its special case when f (k) = k (k ∈ N). The sum of divisors function f (k) = σ(k), giving an estimate similar to (1.2), with the same error term, was considered in paper [11]. The case of the divisor function f (k) = τ (k) was discussed by Heyman [7], obtaining an asymptotic formula with error term O(x 1/2 ) by using elementary estimates.…”

Section: Introductionmentioning

confidence: 99%

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On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers

Heyman

Tóth

2021

Results Math

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We obtain asymptotic formulas with remainder terms for the hyperbolic summations $$\sum _{mn\le x} f((m,n))$$ ∑ m n ≤ x f ( ( m , n ) ) and $$\sum _{mn\le x} f([m,n])$$ ∑ m n ≤ x f ( [ m , n ] ) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions $$f(n)=\tau (n), \log n, \omega (n)$$ f ( n ) = τ ( n ) , log n , ω ( n ) and $$\Omega (n)$$ Ω ( n ) . We also define a common generalization of the latter three functions, and prove a corresponding result.

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“…3.1] using elementary arguments. Formula (1.2) was deduced applying analytic methods by Krätzel, Nowak and Tóth [11,Th. 3.5].…”

Section: Introductionmentioning

confidence: 99%