#### 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 $ℙ$ 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 ${ℙ}_{2}=\left\{3,5,11,17,31,\dots \right\}$. 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 ${ℙ}_{k}$.

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##### Keywords
prime-indexed primes, abstract analytic number theory, Beurling zeta function
##### Mathematical Subject Classification 2010
Primary: 11A41, 11N37, 11N80