BHUWANESH RAO PATIL
Articles written in Proceedings – Mathematical Sciences
Volume 129 Issue 5 November 2019 Article ID 0084 Research Article
Sparse subsets of the natural numbers and Euler’s totient function
MITHUN KUMAR DAS PRAMOD EYYUNNI BHUWANESH RAO PATIL
In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated to the Euler's totient function $\phi$ via the property of 'Banach density'. These sets related to the totient function are defined as follows: $V:=\phi(\mathbb{N})$ and $N_{i}:=\{N_{i}(m)\colon m\in V \}$ for $i = 1, 2, 3,$ where $N_{1}(m)=\max\{x\in \mathbb{N}\colon \phi(x)\leq m\}$, $N_{2}(m)=\max(\phi^{-1}(m))$ and $N_{3}(m)=\min(\phi^{-1}(m))$ for $m\in V$. Masser and Shiu (
Volume 130 All articles Published: 22 April 2020 Article ID 0030 Research Article
Correction to: Sparse subsets of the natural numbers and Euler’s totient function
MITHUN KUMAR DAS PRAMOD EYYUNNI BHUWANESH RAO PATIL
Correction to: Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:84https://doi.org/10.1007/s12044-019-0512-x
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