Mertens conjecture proof

This paper addresses the disproof and reinterpretation of the Mertens Conjecture, which originally claimed that the summatory Möbius function M(n) satisfies |M(n)| < √n for all n. While the original conjecture was disproven in 1985, this paper presents a refined inequality that aligns with analytic number theory and modern bounds. We propose that M(n) remains constrained by a modified bound: |M(n)| < c·√n·log n, for some constant c. This formulation respects the known counterexamples while preserving the core intuition behind the original claim, offering a new lens on Möbius oscillation and zeta function behavior.