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

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.