New Upper Bound Established for de Bruijn–Newman Constant, Λ ≤ 0.1787854

The Riemann Hypothesis and the de Bruijn–Newman Constant

The Riemann hypothesis (RH) is a significant unsolved problem in mathematics, particularly in number theory. It concerns the distribution of prime numbers and states that all non-trivial zeros of the Riemann zeta function lie on the critical line. The de Bruijn–Newman constant (Λ) is a mathematical constant whose value directly determines the truth of the Riemann hypothesis: if Λ ≤ 0, then the Riemann hypothesis is true.

Establishing a New Upper Bound

A new upper bound for the de Bruijn–Newman constant has been established, proving that Λ ≤ 0.1787854. This improves previous bounds and brings mathematicians closer to understanding the constant's true value. The proof involved a series of computational checks and mathematical methods.

Methodology and Verification

The proof relied on three main checks. The first check involved machine verification of the Riemann hypothesis below a certain barrier. The second check certified 3.1 million windows as zero-free, with a lemma covering the remaining range to infinity. The third check established a boundary that no zero could cross. These checks were combined to derive the new upper bound. The proof itself was subjected to a four-layer verification process.

Implications and Limitations

While the new upper bound is a step forward, the method used to achieve it cannot definitively prove that Λ ≤ 0, which is the condition required to confirm the Riemann hypothesis. The research provides a tighter constraint on the constant's value but does not resolve the Riemann hypothesis itself.