Mean value of an arithmetic function associated with the Piltz divisor function

Karras, Meselem; Derbal, Abdallah

doi:10.1142/s179355712050062x

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2022

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“…For all integers m, n ≥ 1, we denote by gcd(m, n) = (m, n) the greatest common divisor of the integers m and n. The details of the function D k (n) is given in [4] and for many properties of the classical functions τ k (n), τ * k (n) see, e.g. [5,6].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic formula of a “hyperbolic” summation related to the Piltz divisor function

Bouderbala¹,

Karras²

2022

NNTDM

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In this paper, we obtain asymptotic formula on the "hyperbolic" summation \begin{equation*} \underset{mn\leq x}{\sum }D_{k}\left( \gcd \left( m,n\right) \right) \text{ \ \ }\left( k\in \mathbb{Z}_{\geq 2}\right), \end{equation*} such that $D_{k}\left( n\right) = \dfrac{\tau _{k}\left( n\right) }{\tau_{k}^{\ast }\left( n\right) }$, where $\tau _{k}\left( n\right) =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1$ denotes the Piltz divisor function, and $\tau _{k}^{\ast }\left( n\right) $ is the unitary analogue function of $\tau _{k}\left( n\right) $.

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Section: Introductionmentioning

confidence: 99%