The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function have a real part of . 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 , where is the Chebyshev function, is the Euler-Mascheroni constant, and is the natural logarithm. Defining as the -th primorial, the product of the first primes, we employ Nicolas' criterion to prove that there exists a prime $p_k > 10^8$ and a prime such that and $p_k^{1.907} \ll p_{k'} < p_k^2$, where implies is significantly larger than . This existence leads to , 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.
