在由\Big ｛{[\sqrt 2n{] }\over n }\mid n \in\mathbb{N}\Big｝生成的乘法组上。v

IF 0.4 Q4 MATHEMATICS Notes on Number Theory and Discrete Mathematics Pub Date : 2023-05-12 DOI:10.7546/nntdm.2023.29.2.348-353

I. Kátai, B. M. Phong

引用次数: 0

摘要

设$f,g$是完全乘法函数$\vert f(n)\vert=\vert g(n)\vert =1 (n\in\mathbb{N})$。假设$${1\over {\log x}}\sum_{n\le x}{\vert g([\sqrt{2}n])-Cf(n)\vert\over n}\to 0 \quad (x\to\infty).$$那么 $$f(n)=g(n)=n^{i\tau},\quad C=(\sqrt{2})^{i\tau}, \tau\in \mathbb{R}.$$

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On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\}. V

Let $f,g$ be completely multiplicative functions, $\vert f(n)\vert=\vert g(n)\vert =1 (n\in\mathbb{N})$. Assume that $${1\over {\log x}}\sum_{n\le x}{\vert g([\sqrt{2}n])-Cf(n)\vert\over n}\to 0 \quad (x\to\infty).$$ Then $$f(n)=g(n)=n^{i\tau},\quad C=(\sqrt{2})^{i\tau}, \tau\in \mathbb{R}.$$

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Notes on Number Theory and Discrete Mathematics MATHEMATICS-

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