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Distribution of monomial-prime numbers and Mertens sum evaluations

Author

Listed:
  • Lin Feng

    (Nankai University)

  • Huixi Li

    (Nankai University)

  • Biao Wang

    (Yunnan University)

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.

Suggested Citation

  • Lin Feng & Huixi Li & Biao Wang, 2024. "Distribution of monomial-prime numbers and Mertens sum evaluations," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(4), pages 1440-1455, December.
    • Handle: RePEc:spr:indpam:v:55:y:2024:i:4:d:10.1007_s13226-023-00449-4
    DOI: 10.1007/s13226-023-00449-4
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