From Chebyshev to Primorials: Establishing the Riemann Hypothesis

The Riemann Hypothesis, one of the most celebrated open problems in mathematics, asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line \( \Re(s) = \frac{1}{2} \) and has profound consequences for the distribution of prime numbers. Since Riemann's original 1859 paper, a vast body of work has attempted to settle the question, frequently by examining the asymptotic behavior of arithmetic functions such as Chebyshev's prime-counting function \( \theta(x) \). In this work we introduce a new criterion that links the Riemann Hypothesis to the comparative growth of \( \theta(x) \) relative to primorial numbers. More precisely, we study the ratio \( R(N_k) = \Psi(N_k)/(N_k\log\log N_k) \), where \( N_k \) is the \( k \)-th primorial and \( \Psi \) is the Dedekind function, and show that the Riemann Hypothesis follows from intrinsic monotonicity properties of this ratio. The argument combines Mertens' theorem, the prime number theorem, and an explicit error analysis of the relevant asymptotic expansions to produce a self-contained proof by contradiction. Beyond its implications for the hypothesis itself, the result offers a fresh framework for understanding how the multiplicative structure of primorials governs the analytic behavior of \( \zeta(s) \), thereby casting new light on one of mathematics' most enduring mysteries.