On a generalisation sum involving the Euler function

Zhaoxi Ye; Zhefeng Xu

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arXiv · 2408.01015

On a generalisation sum involving the Euler function

Zhaoxi Ye·Zhefeng Xu

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Abstract

Let $j \ge1$, $k\ge 0$ be real numbers and $\varphi(n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum_{n\le x}\frac{\varphi\left ( \left [ \frac{x}{n} \right ]^{j} \right ) }{\left [ \frac{x}{n} \right ]^{k} } $$ 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.

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Zhaoxi Ye, Zhefeng Xu. 2024-08-02. On a generalisation sum involving the Euler function. https://arxiv.org/abs/2408.01015

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