Zhaoxi Ye | Searcharxiv

arXiv2026

Involves averaging arithmetic and integral partial functions over sparse set

Let $p$ be a prime number, $k\ge 0$ and $f$ be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function $$ψ_{f,k}(x):=\sum_{n\le x}Λ(n)\frac{f\left ( \left [ \frac{x}{n} \right ] \right ) }{\left [ \frac{x}{n} \right ]^{k} } ,~~~~~~~~~~~ π_{f,k}(x):=\sum_{p\le x}\frac{f\left ( \left [ \frac{x}{p} \right ] \right ) }{\left [ \frac{x}{p} \right ]^{k} } $$ as $x\to \infty $, where $\left [ \cdot \right ] $ is the integral part function, $Λ(n)$ is the von Mangoldt function.

math.NT↗

arXiv2026

On a generalisation sum involving the Euler function

Let $j \ge1$, $k\ge 0$ be real numbers and $φ(n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum_{n\le x}\frac{φ\left ( \left [ \frac{x}{n} \right ]^{j} \right ) }{\left [ \frac{x}{n} \right ]^{k} } $$ as $x\to \infty $, where $\left [ \cdot \right ] $ is the integral part function. Our results combine and generalize the recent work of Zhai, Wu and Ma.

math.NT↗