Distribution of monomial-prime numbers and Mertens sum evaluations

Lin Feng (), Huixi Li () and Biao Wang ()
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Lin Feng: Nankai University
Huixi Li: Nankai University
Biao Wang: Yunnan University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1440-1455

Abstract: Abstract In this paper, we mainly study the monomial-prime numbers, which are of the form $$pn^k$$ p n k for primes p and integers $$k\ge 2$$ k ≥ 2 . First, we give an asymptotic estimate on the number of numbers of a general form pf(n) for arithmetic functions f satisfying certain growth conditions, which generalizes Bhat’s recent result on the Square-Prime Numbers. Then, we prove three Mertens-type theorems related to numbers of a more general form, partially extending the recent work of Qi-Hu, Popa and Tenenbaum on the Mertens sum evaluations. At the end, we evaluate the average and variance of the number of distinct monomial-prime factors of positive integers by applying our Mertens-type theorems.

Keywords: Monomial-Prime Number; Mertens Theorem; Prime Number Theorem; Zeta Function; 11N25; 11N37; 11N80 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00449-4

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