• MITHUN KUMAR DAS

Articles written in Proceedings – Mathematical Sciences

• Sparse subsets of the natural numbers and Euler’s totient function

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 (Pacific J. Math. 121(2) (1986) 407-426) called the elements of $N_{1}$ as sparsely totient numbers' and constructed an infinite family of these numbers. Here we construct several infinite families of numbers in $N_{2}\setminus N_{1}$ and an infinite family of composite numbers in $N_{3}$. We also study (i) the ratio $\frac{N_{2}(m)}{N_{3}(m)}$ which is linked to the Carmichael's conjecture, namely, $|\phi^{-1}(m)|\geq 2 ~\forall ~ m\in V$, and (ii) arithmetic and geometric progressions in $N_{2}$ and $N_{3}$. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of $\mathbb{N}$, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.

• Correction to: Sparse subsets of the natural numbers and Euler’s totient function

Correction to: Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:84https://doi.org/10.1007/s12044-019-0512-x

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Volume 131, 2021
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