Vol. 13, No. 2, 2020

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Arithmetic functions of higher-order primes

Kyle Czarnecki and Andrew Giddings

Vol. 13 (2020), No. 2, 181–191
Abstract

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