The Abdeslam Prime Function: A New Perspective on Prime Gaps

This paper introduces the Abdeslam Prime Function ( 𝐴 𝑝 (Ap(n)), a new approach to understanding prime gaps. Unlike traditional models that predict rapid growth, our findings suggest that prime gaps follow a logarithmic pattern, making them more structured and predictable. Empirical analysis shows that Ap(n) remains remarkably stable across a wide range of primes, challenging established theories such as Cramér’s conjecture. We refine this observation with a stability theorem, demonstrating that fluctuations decrease as numbers grow larger. Additionally, we examine how modular arithmetic influences prime gaps, revealing a deeper structure in prime number distribution. By comparing 𝐴 𝑝 Ap(n) with classical models, we show that it provides a more accurate description of prime gaps. Finally, we explore a potential link between prime gap stability and the Riemann Hypothesis (RH). This connection, if proven, could offer new insights into one of the most important unsolved problems in mathematics. This work introduces a new framework for studying prime gaps and opens the door to further research on their underlying structure.