This paper investigates the counting function of semiprimes, numbers that are the product of exactly two prime factors. Semiprimes play a crucial role in various areas of number theory and cryptography due to their structural properties and computational complexity. The main focus of this study is to analyze the asymptotic behavior of the semiprime counting function, denoted by Q(x)Q(x)Q(x), which counts the number of semiprimes less than or equal to a given number xxx. We explore existing analytical bounds and approximations for Q(x)Q(x)Q(x), drawing comparisons with related functions such as the prime counting function π(x)\pi(x)π(x). Additionally, we derive new estimates for the error terms associated with known asymptotic formulas and analyze the distribution of semiprimes across different intervals. Numerical computations are employed to verify theoretical predictions and highlight deviations from asymptotic trends at smaller values of xxx. The findings contribute to a deeper understanding of the density and distribution of semiprimes, offering potential insights into applications in cryptography, where the factorization of semiprimes underpins many encryption protocols. This study provides both theoretical results and empirical data, demonstrating the complexity and richness of semiprimebehavior in number theory.

This paper investigates the counting function of semiprimes,