Arithmetic functions of higher-order primes

Abstract

The sieve of Eratosthenes (SoE) is a well-known method of extracting the set of prime numbers $\mathbb{P}$ from the set positive integers $\mathbb{N}$. Applying the SoE again to the index of the prime numbers will result in the set of prime-indexed primes ${\mathbb{P}}_2=\left\{3,5,11,17,31,\dots \right\}$. More generally, the application of the SoE $k$-times will yield the set ${\mathbb{P}}_k$ of $k$-th order primes. In this paper, we give an upper bound for the $n$-th $k$-order prime as well as some results relating to number-theoretic functions over ${\mathbb{P}}_k$.