On an inequality about Euler’s totient function

In this paper, we show that when $N_k$ is a primorial and $\varphi(N_k)$ is Euler's totient function, the inequality $\varphi(N_k) < \frac{N_k}{e^{\gamma}\log{\log{N_k}}}$ holds for all positive integer $k > 44.$

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Publication Details

Journal
Notes on Number Theory and Discrete Mathematics
Published
2025-01-01
DOI
https://doi.org/10.7546/nntdm.2025.31.2.299-304
Primary Topic
Mathematical Inequalities and Applications
Type
article
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article

On an inequality about Euler’s totient function

Cheng-Ting Wang
Notes on Number Theory and Discrete Mathematics
Mathematical Inequalities and Applications
article

On an inequality about Euler’s totient function

Cheng-Ting Wang
article en

Abstract

In this paper, we show that when $N_k$ is a primorial and $\varphi(N_k)$ is Euler's totient function, the inequality $\varphi(N_k) < \frac{N_k}{e^{\gamma}\log{\log{N_k}}}$ holds for all positive integer $k > 44.$

Notes on Number Theory and Discrete MathematicsVol. 31(2)
Taipei City Government (TW)
Openalex Percentile: Top 83%
Mathematical Inequalities and Applications
DOIOpen Access PDFOpenAlex
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On an inequality about Euler’s totient function — Cheng-Ting Wang · Notes on Number Theory and Discrete Mathematics (2025) | TGRS Research Map | TGRS