Escribe una PREreview

The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function have a real part of 12​. 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 ∏p≤xpp−1>eγ⋅log⁡θ(x)for all x≥2,where θ(x) is the Chebyshev function, γ≈0.57721 is the Euler–Mascheroni constant, and log is the natural logarithm. Defining Nn=2⋅…⋅pn​ as the n-th primorial, the product of the first n primes, we employ Nicolas’ criterion to prove that there exists a prime pk>108 and a prime pk′ such that θ(pk′)≤θ(pk)2andpk1.907≪pk′<pk′2,where pk1.907≪pk′ implies pk′​ is significantly larger than pk1.907. This existence leads to Nkφ(Nk)≤eγ⋅log⁡log⁡Nk,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.