Sums Related to Euler’s Totient Function
Abstract
We obtain an upper bound for the sum $$\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$$ , where $$\varphi$$ is Euler’s totient function, $$s\in \mathbb{N}$$ , and $$a_{1},\dots, a_{N}$$ are positive integers (not necessarily distinct) with some restrictions. As applications, for any $$t>0$$ , we obtain an upper bound for the number of $$n\in [1,N]$$ such that $$a_{n}/ \varphi (a_{n})> t$$ .
Fields of Study & Subjects
Advanced Mathematical Identities
Mathematics and Applications
History and Theory of Mathematics
Publication details
- Journal / source
- Mathematical Notes
- Publisher
- Pleiades Publishing Ltd
- Volume
- 119
- Issue
- 5-6
- Pages
- 1136–1147
- Publication date
- 1 May 2026
- Language
- EN