On the asymptotic behavior of the integral T(x)
In this paper, we estimate the integral T(x) mentioned in the title, where {t} denotes the fractional part of the real number t, and x is any positive real number.
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arXiv subjects
Meselem Karras
Publications and source records attributed to Meselem Karras.
On the Mean Value of $D_k(n)$ in Arithmetic Progressions
Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{\omega(n)}$ is its unitary analogue, where $\omega(n)$ is the number of distinct prime divisors of $n$. We establish an asymptotic formula for the sum \[ \sum_{\substack{n \le x \\ n \equiv a \pmod q}} D_k(n), \] where $\gcd(a,q)=1$. This result is a generalization of the study presented in \cite{Derbal 2023}. \noindent
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Note on Fractional Sums with Fixed GCD
We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for some $0 \le \alpha < 1$. For $r \ge 2$, let $\tau_r(n)$ denote the number of representations of $n$ as a product of $r$ positive integers, and more generally, $\tau_r^{(d)}(n)$ the number of representations with $\gcd$ factors equal to $d$. We establish asymptotic formulas for the fractional sums \[ S_{f,r}^{(d)}(x) = \sum_{n \le x} \tau_r^{(d)}(n) f\!\left(\left\lfloor \frac{x}{n}\right\rfloor \right), \] in the cases $r=2$ and $r=3$.
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Hyperbolic summation involving certain arithmetic functions and the integer part function
Let f be an arithmetic function satisfying certain conditions. In this paper, we give an asymptotic formula for the sum \[\sum_{n_1 n_2 \cdots n_r \leq x} f\left(\left\lfloor \frac{x}{n_1 n_2 \cdots n_r} \right\rfloor\right), \quad r \geq 2.\], where $\lfloor . \rfloor$ denotes the integer part function.
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Hyperbolic Summation for Fractional Sums
Let $f(n)$ be an arithmetic function with $f(n) \ll n^α$ for some $α\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left( \left\lfloor \frac{x}{n_{1}n_{2}} \right\rfloor \right),$$ we use the estimation of three-dimensional exponential sums due to Robert and Sargos.
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Hyperbolic summation involving the function $\Omega(n)$ and lcm
We study the sum $\sum_{abc \leq x} \Omega([a,b,c])$, where $\Omega(n)$ denotes the number of distinct prime divisors of $n \in \mathbb{Z}_{\geq 1}$, counted with multiplicity, and where $(a,b,c) = \gcd(a,b,c)$ and $[a,b,c] = \operatorname{lcm}(a,b,c)$. An asymptotic formula is derived for this sum over the hyperbolic region $\{(a,b,c) \in \mathbb{Z}_{\geq 1}^3 : abc \leq x\}$.
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Asymptotic formula for the multiplicative function $\frac{d(n)}{k^{ω(n)}}$
For a fixed integer $k$, we define the multiplicative function \[D_{k,ω}(n) := \frac{d(n)}{k^{ω(n)}}, \]where $d(n)$ is the divisor function and $ω(n)$ is the number of distinct prime divisors of $n$. The main purpose of this paper is the study of the mean value of the function $D_{k,ω}(n)$ by using elementary methods.
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