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On a generalisation sum involving the Euler function

Autor: Ye, Zhaoxi, Xu, Zhefeng

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 ) }{\l

Externí odkaz: http://arxiv.org/abs/2408.01015

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Involves averaging arithmetic and integral partial functions over sparse set

Autor: Ye, Zhaoxi, Xu, Zhefeng

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 $$\psi_{f,k}(x):=\sum_{n\le x}\Lambda (n)\frac{f\left ( \

Externí odkaz: http://arxiv.org/abs/2406.19788

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Maz'ya-Shaposhnikova meet Bishop-Gromov

Autor: Han, Bang-Xian, Pinamonti, Andrea, Xu, Zhefeng, Zambanini, Kilian

We find a surprising link between Maz'ya-Shaposhnikova's well-known asymptotic formula concerning fractional Sobolev seminorms and the generalized Bishop-Gromov inequality. In the setting of abstract metric measure spaces we prove the validity of a l

Externí odkaz: http://arxiv.org/abs/2402.11174

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Sharp uncertainty principles on metric measure spaces

Autor: Han, Bang-Xian, Xu, Zhefeng

We study the Heisenberg-Pauli-Weyl uncertainty principle and the Caffarelli-Kohn-Nirenberg interpolation inequalities, on metric measure spaces satisfying measure contraction property. Using localization techniques, we show that these inequalities ar

Externí odkaz: http://arxiv.org/abs/2309.00847

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The 4-Adic Complexity of Interleaved Quaternary Sequences of Even Length with Optimal Autocorrelation

Autor: Jing, Xiaoyan, Xu, Zhefeng, Yang, Minghui, Feng, Keqin

Su et al. proposed several new classes of quaternary sequences of even length with optimal autocorrelation interleaved by twin-prime sequences pairs, GMW sequences pairs or binary cyclotomic sequences of order four in \cite{S1}. In this paper, we det

Externí odkaz: http://arxiv.org/abs/2209.10279

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