Disproving the Riemann Hypothesis with Primorial Bounds

The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function have a real part of $\frac{1}{2}$. As a pivotal conjecture in pure mathematics, it remains unproven and is equivalent to various statements, including one by Nicolas in 1983 asserting that the hypothesis holds if and only if $\prod_{p \leq x} \frac{p}{p - 1} > e^{\gamma} \cdot \log \theta(x)$ for all $x \geq 2$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, and $\log$ is the natural logarithm. Defining $N_n = 2 \cdot \ldots \cdot p_n$ as the $n$-th primorial, the product of the first $n$ primes, we employ Nicolas' criterion to prove that there exists a prime $p_k > 10^8$ and a prime $p_{k'}$ such that $\theta(p_{k'}) \leq \theta(p_k)^2$ and $p_k^{1.907} \ll p_{k'} < p_k^2$, where $p_k^{1.907} \ll p_{k'}$ implies $p_{k'}$ is significantly larger than $p_k^{1.907}$. This existence leads to $\frac{N_k}{\varphi(N_k)} \leq e^{\gamma} \cdot \log \log N_k$, contradicting Nicolas' condition and confirming the falsity of the Riemann Hypothesis. This result decisively refutes the conjecture, enhancing our insight into prime distribution and the behavior of the zeta function's zeros through analytic number theory.