Arithmetic functions of higher-order primes

Czarnecki, Kyle; Giddings, Andrew

doi:10.2140/involve.2020.13.181

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Arithmetic functions of higher-order primes

Kyle Czarnecki and Andrew Giddings

Vol. 13 (2020), No. 2, 181–191

DOI: 10.2140/involve.2020.13.181

Abstract

The sieve of Eratosthenes (SoE) is a well-known method of extracting the set of prime numbers $ℙ$ from the set positive integers $ℕ$ . Applying the SoE again to the index of the prime numbers will result in the set of prime-indexed primes $ℙ_{2} = {3, 5, 11, 17, 31, \dots}$ . More generally, the application of the SoE $k$ -times will yield the set $ℙ_{k}$ 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 $ℙ_{k}$ .

Keywords

prime-indexed primes, abstract analytic number theory, Beurling zeta function

Mathematical Subject Classification 2010

Primary: 11A41, 11N37, 11N80

Milestones

Received: 29 August 2018

Revised: 25 July 2019

Accepted: 28 December 2019

Published: 30 March 2020

Communicated by Filip Saidak

Authors

Kyle Czarnecki
University of Wisconsin Platteville, WI United States

Andrew Giddings
University of Wisconsin Platteville, WI United States

Vol. 13, No. 2, 2020