On distribution of subsequences of primes having prime indices with respect to the $(R)$-denseness and convergence exponent

TL;DR

This paper investigates the distribution, density, and convergence properties of recursively defined prime subsequences, analyzing their ratios, convergence exponents, and asymptotic behaviors.

Contribution

It introduces new recursive subsequences of primes and studies their distribution, density, and convergence properties, providing novel insights into their asymptotics and ratio sets.

Findings

Sets of prime subsequences have dense ratios in positive reals.

Computed convergence exponents for these prime subsequences.

Derived asymptotic formulas for their counting functions.

Abstract

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define subsequences $(p^{(k)}_n)_{n=1}^{+\infty}$ of the sequence $(p_n)_{n=1}^{+\infty}$ in the following way: let $p_n^{(1)}=p_n$ and $p_n^{(k+1)}=p_{p_n^{(k)}}$. In this paper we study and describe some interesting properties of the sets $\mathbb{P}_k=\{p_1^{(k)}<p_2^{(k)}<\dots<p_n^{(k)}<\dots\}$, $\mathbb{P}_n^{\mathrm{T}}=\{p_n^{(1)}<p_n^{(2)}<\dots<p_n^{(k)}<\dots\}$ and $\text{Diag}\mathbb{P}=\{p^{(1)}_1<p^{(2)}_2<\dots <p^{(k)}_k<\dots\}$ and their elements, for $k,n\in\mathbb{N}$. Especially, we check whether these sets have dense sets of ratios in $\mathbb{R}_+$. Moreover, we compute their exponents of convergence and asymptotics of their counting functions.