A Novel Criterion for the Riemann Hypothesis

The Riemann Hypothesis, one of the most profound unsolved problems in mathematics, concerns the distribution of the non-trivial zeros of the Riemann zeta function and their connection to prime numbers. Since its formulation in 1859, numerous approaches have sought to establish its validity, often linking it to the asymptotic behavior of arithmetic functions such as Chebyshev's function \ ( (x) \). This work explores a new criterion based on the comparative growth of \ ( (x) \) and primorial numbers. Through this analysis, the Riemann Hypothesis is shown to follow from the intrinsic properties of \ ( (x) \) and its relationship with primorials, confirming the deep connection between prime distribution and the non-trivial zeros of the Riemann zeta function. The result not only resolves this long-standing conjecture but also provides a new perspective on the interplay between multiplicative number theory and analytic inequalities.