A New Ceiling for $\Lambda$: The de Bruijn–Newman Constant is at most 0.1787854

By Jude Gomila | August 19, 2026

I am Jude Gomila. Since 2025, I have been privately investigating the properties of the zeta function. This specific article is a component of a broader series detailing discoveries born from the synergy of human intuition human/AI collaboration.

The Core Discovery: $\Lambda \le 0.1787854$

This post focuses on the de Bruijn–Newman constant, denoted as $\Lambda$. This single real number acts as a toggle for one of the greatest mysteries in mathematics: the Riemann hypothesis (RH) is true if and only if $\Lambda \le 0$.

While a proof that $\Lambda \le 0$ remains elusive, we can incrementally lower the known "ceiling" (the upper bound) of this constant. I have developed a computer-assisted proof that unconditionally lowers this bound from $0.2$ to $0.1787854$. This result relies on no unproven conjectures.

Transparency Note: Every claim made here is linked to my [audit repository] and an [independent review record]. I welcome feedback, bug reports, or suggestions for improvement via GitHub issues.

Summary of the Bound

The new bound is expressed as an exact rational number, derived via exact arithmetic from a massive set of machine-verified interval certificates ($3,149,013 + 883 + 1$):

$\Lambda \le \frac{129}{800} + \frac{87677}{5,000,000} = 0.1787854$

Historical Context of $\Lambda$

The quest to pin down $\Lambda$ has seen several milestones:

---

The Proof Architecture

The proof transitions from the complex $\zeta(s)$ function to a specific heat flow analysis. The logic can be visualized as follows:

The Verification Checklist

To arrive at the final bound, the following steps were completed:

• Check One: Machine-verify that RH holds below the critical barrier.
• Check Two (Part A): Certify $3.1$ million individual windows as zero-free.
• Check Two (Part B): Apply a lemma to cover the remaining range to infinity.
• Check Three: Establish a "wall" that prevents any zero from crossing.

---

Prologue: Why the Riemann Hypothesis Matters

To understand $\Lambda$, one must understand the primes. Primes ($2, 3, 5, 7, \dots$) are the "atoms" of arithmetic. While they appear erratic individually, their collective distribution follows a smooth trend known as the Prime Number Theorem.

The real question is the error term: how much does the actual count of primes, $\pi(x)$, deviate from the smooth prediction, $Li(x)$?

Fig 1: The "staircase" of primes vs. the smooth logarithmic integral.

The Role of the Zeta Function

In 1859, Bernhard Riemann linked primes to the zeros of the zeta function:

$\zeta(s) = \sum_{n \ge 1} \frac{1}{n^s} = \prod_{p \text{ prime}} \left(1 - p^{-s}\right)^{-1}$

Riemann discovered that the "wobble" in the prime count is actually a sum of waves. Each zero of $\zeta(s)$ creates a wave; the zero's height determines the frequency, and its horizontal position ($\text{Re } s$) determines the amplitude.

• If RH is true: All zeros lie on the critical line $\text{Re } s = 1/2$. This results in the smallest possible amplitude for the error waves (roughly $\sqrt{x}$).
• If RH is false: A zero at $\beta \neq 1/2$ would cause a wave to grow like $x^\beta$, creating a massive, biased drift in the prime count.

The "Wave" Simulation

Imagine a slider that adds zeros one by one: 0 zeros $\rightarrow$ Just the smooth trend (misses all steps). 30 zeros $\rightarrow$ The curve begins to carve out the prime staircase.

# Conceptual representation of the wave contribution
def wave_contribution(zero_position, x):
    amplitude = x**(zero_position.real)
    frequency = zero_position.imag
    return amplitude * math.cos(frequency * math.log(x))

The Riemann Hypothesis is a Clay Millennium Prize problem because it guarantees that primes are distributed as randomly as possible, without any hidden bias. While computers have checked trillions of zeros and found them all on the line, we need a formal proof—which is why narrowing the bound of $\Lambda$ is so critical.