Prime number distribution remains one of the most intriguing topics in number theory, influencing various domains of mathematical research. This paper aims to elucidate the properties of prime numbers through the study of additive number theoretic functions, which provide insights into the additive structure of integers. Our objectives include deriving new asymptotic formulas and refining existing bounds of these functions. Methods employed involve a combination of analytical techniques and computational simulations to extract patterns and verify conjectures. Our findings indicate significant progress in understanding the correlation between prime density and additive functions, ultimately leading to the validation of several hypotheses regarding their behavior. Notably, the research highlights the potential of these functions in predicting prime occurrences within specified intervals. In conclusion, our work not only bridges gaps in current theoretical knowledge but also sets a foundation for future explorations into complex mathematical structures, offering novel perspectives on prime distribution. The implications extend to advancing computational approaches in number theory and enhancing algorithmic efficiency in related applications.