arXiv subjects

Isao Kiuchi

Publications and source records attributed to Isao Kiuchi.

8 recordsLinked to original sources

arXiv2024

The summatory function of the M\"obius function involving the greatest common divisor

Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $\mu$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the greatest common divisor as $ S_{\mu}(x) := \sum_{mn\leq x} \mu(\gcd(m,n)). $ In this paper, we present an asymptotic formula for $S_{\mu}(x)$. Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of $S_{\mu}(x)$ and derive a mean square estimate for its error term. Our proof employs the Perron formula, Parseval's theorem, complex integration techniques, and the properties of the Riemann zeta-function.

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arXiv2021

On sums of arithmetic functions involving the greatest common divisor

Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, 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 $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{τ,k}(x)$ and $S_{μ,k}(x)$ for $k\geq 3$.

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arXiv2021

On the multivariable generalization of Anderson-Apostol sums

In this paper, we give various identities for the weighted average of the product of generalized Anderson-Apostol sums with weights concerning completely multiplicative function, completely additive function, logarithms, the Gamma function, Bernoulli polynomials and binomial coefficients respectively. Our results generalize many known results.

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arXiv2020

On the weighted average number of subgroups of ${\mathbb {Z}}_{m}\times {\mathbb {Z}}_{n}$ with $mn\leq x$

Let $\mathbb{Z}_{m}$ be the additive group of residue classes modulo $m$. For any positive integers $m$ and $n$, let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and cyclic subgroups of the group ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{n}$, respectively. Define $$ \widetilde{D}_{s}(x) = \sum_{mn\leq x}s(m,n)\log\frac{x}{mn} \quad \quad \widetilde{D}_{c}(x) = \sum_{mn\leq x}c(m,n)\log\frac{x}{mn}. $$ In this paper, we study the asymptotic behaviour of functions $\widetilde{D}_{s}(x)$ and $\widetilde{D}_{c}(x)$.

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arXiv2020

Sums of averages of gcd-sum functions II

Let $\gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) $$ for any large real number $x\geq 5$, where $f$ is any arithmetical function. Let $ϕ$, and $ψ$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {\rm id})$, $M_r(x;ϕ)$ and $M_r(x;ψ)$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{\rm id})$ for any large positive number $x>5$ satisfying $x=[x]+\frac{1}{2}$.

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arXiv2019

Sums of weighted averages of gcd-sum functions II

In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\logΓ\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\ d^{s}|j}}f*μ(d) \quad {\rm and } \quad \sum_{k\leq x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1} B_{m}\sum_{\substack{d|k \\ d^{s}|j}} f*μ(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of the Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.

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arXiv2018

On sums of logarithmic averages of gcd-sum functions

Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $ϕ$, $id_{1+a}$ and $ϕ_{1+a}$ with $-1<a<0$. We also establish some formulas of Dirichlet series having coefficients of the sum function $\sum_{j=1}^{k}s_{k}(j)\log j$ where $s_{k}(j)$ is Anderson--Apostol sums.

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arXiv2018

On sums of weighted averages of $\gcd$-sum functions

Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k, d^{s}|j}(f*μ)(d) $ for any positive integers $j$ and $k$, namely $$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad \text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r} \sum_{\substack{d|k d^{s}|j}}(f*μ)(d), $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of $\gcd$-sum functions $f(\gcd(j,k)). $

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