The Effect of the Density of Square-Free ωp-numbers on the Bounds of Beurling Counting Function: Number Theory

Sarah Al-Ebrahimy; Eman  F. Mohommed

doi:10.21271/ZJPAS.37.1.3

Authors

Sarah Al-Ebrahimy Department of mathematic, College of Education of Al Mustansiriyah, Baghdad, Iraq
Eman Mohommed Department of mathematic, College of Education of Al Mustansiriyah, Baghdad, Iraq https://orcid.org/0000-0003-0030-6512

DOI:

https://doi.org/10.21271/ZJPAS.37.1.3

Keywords:

Beurling's prime system, Square-free, Abundant numbers, Deficient numbers and ωp-numbers.

Abstract

Primitive weird numbers are weird numbers which are not a multiple of any smaller weird numbers. The goal of this work is to use a square-free primitive weird number x=ab where b be an increasing sequence of prime numbers such that q₁ is greater than ∏_(j=1)^r▒〖(q ̅_j+1)〗 and a=∏_(j=1)^r▒q ̅_j and a is deficient number with n greater than 1, to enhancing the classic bounds of Beurling counting function on Riemann Hypothesis.

References

Al-Maamori, F. & Hilberdink, T. 2015. An example in Beurling's theory of generalised primes. Acta Arithmetica, 4, 383-395.

Amato, G., Hasler, M. F., Melfi, G. & Parton, M. 2019. Primitive abundant and weird numbers with many prime factors. Journal of number theory, 201, 436-459.

Apostol, T. M. 1976. Introduction to Analytic Number Theory. Springer-Verlag.

Hasan, S., Al-Maamori, F. & Abdulrahman, H. 2018. Restricted the gap between the error terms of Ω-results for (NƷ− τx) and the error terms of O-results for (N− τx) on Riemann Hypothesis. International Journal of Pure and Applied Mathematics, 120, 751.

Hasan, S. S., Al-Maamori, F. & Majeed, L. 2019. A further restricting the gap between (NƷ-τx) and (N-τx) on R. H. by using the sense of ω-numbers and ωp-numbers. Journal of Advanced Research in Dynamical and Control Systems, 11, 2043-2051.

Melfi, G. 2015. On the conditional infiniteness of primitive weird numbers. Journal of Number Theory, 147, 508-514.