From Chebyshev to Primorials: Establishing the Riemann Hypothesis
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The Nicolas criterion gives an equivalent formulation of the Riemann Hypothesis as an inequality involving the Euler totient function evaluated at primorial numbers. A natural strategy for establishing this inequality is to prove that a suitable subsequence of the associated ratio sequence is eventually strictly decreasing under the assumption that the Riemann Hypothesis is false. The present work shows that such a subsequence exists. When this monotonicity property is combined with the known limiting behavior of the ratio sequence and the Nicolas equivalence, a contradiction emerges: assuming the Riemann Hypothesis is false forces the subsequence to converge to a limit that is simultaneously equal to $e^{\gamma}$ (by a subsequence argument) and strictly less than $e^{\gamma}$ (by strict monotonicity). The Riemann Hypothesis therefore follows as a direct consequence.