Λ ≤ 0.1787854 — a new bound for the de Bruijn–Newman constant
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A complete walkthrough of the proofA new ceiling for Λ: the de Bruijn–Newman constant is at most 0.1787854I'm Jude Gomila and I've been exploring the zeta function in private since 2025. This post is part of a series of posts on discoveries obtained from human/ai collaboration.
This post is about the de Bruijn–Newman constant Λ — a single real number with this property: the Riemann hypothesis holds exactly when Λ ≤ 0. Nobody can prove that yet, but its known ceiling can be lowered, and this is my computer-assisted proof taking it from 0.2 to 0.1787854, unconditionally, with no unproved conjecture anywhere in the chain. I'll walk you through the whole proof, step by step. Every claim links back to my audit repository and the independent review record.
Feedback, bugs and upgrade comments are welcome as GitHub issues.Previous boundΛ ≤ 0.2This resultΛ ≤ 0.1787854MethodPolymath 15 criterion + interval certificatesSpecial thanksDan Romik, Max AtkinΛ ≤ 0.1787854= 129/800 + 87677/5,000,000: an exact rational, obtained by exact arithmetic from 3,149,013 + 883 + 1 machine-checked interval certificates.00Why the Riemann hypothesis matters01Λ, the constant whose value decides the Riemann hypothesis02Heating the function pulls its zeros onto the real axis03The bounds on Λ, and how the methods work04Three finite checks that prove an upper bound on Λ05Check one: RH is already machine-verified below the barrier06Check two: 3.1 million windows certified zero-free07Check two, continued: one lemma covers the rest to infinity08Check three: a wall no zero can cross09Combining the checks gives Λ ≤ 0.178785410How the proof was checked, four layers deep11Why this method cannot reach Λ ≤ 012Provenance & linksPrologueWhy the Riemann hypothesis mattersThe primes 2, 3, 5, 7, 11, 13, … are the atoms of arithmetic: every whole number factors into primes in exactly one way, so facts about primes become facts about all numbers.
Individually they are irregular — no known rule produces the next prime from the ones before it. Counted in bulk, they obey a law: the number of primes up to x stays close to a single smooth curve (the prime number theorem, proved in 1896). The open question is the size of the error — how far the true count can stray from the curve.
That error term is what the Riemann hypothesis governs, and it is why RH matters: sharpen the error term and you sharpen hundreds of results in number theory that depend on it.0204050100150200250each vermillion tick: a primethe smooth prediction Li(x)π(x): how many primes ≤ xFig.
1The prime-counting staircase π(x) (the dark stepped line) climbs one step at each prime. The dashed blue curve is the smooth prediction Li(x). The gap between them is a sum of waves, one wave per zeta zero, and the Riemann hypothesis says every wave has the smallest possible amplitude.In 1859 Bernhard Riemann explained where that hidden order comes from.
He took Euler's identity, which connects the primes to a single function of one complex variable,ζ(s) = ∑n≥11ns = ∏p prime(1−p−s)−1\htmlData{term=zeta, tc=v}{\zeta(s)}\;=\;\htmlData{term=sum, tc=s}{\sum_{n\ge1}\frac{1}{n^{s}}}\;=\;\htmlData{term=prod, tc=b}{\prod_{p\ \mathrm{prime}}\left(1-p^{-s}\right)^{-1}}hover or tap a colored term for what it doesextended it to the whole complex plane, and discovered that the wobble of the prime count around its smooth curve is governed — exactly, via an explicit formula — by the locations of the zeros of this function.
Each zero contributes one wave to the error; the zero's height sets the wave's frequency and, crucially, its horizontal position sets the wave's amplitude. Riemann observed that every zero he could examine sat on one vertical line, Re s = ½, now called the critical line — the position giving the smallest possible amplitude — and remarked it was “very probable” all of them do. That remark is the Riemann hypothesis. Its concrete content: the prime-count error up to x never exceeds roughly √x, the same size as the wobble of a fair coin flipped x times. The primes are allowed to look random; RH says they are never allowed to drift with a bias.The wave description is an actual formula, and you can run it below. The slate staircase counts prime powers (a cousin of the staircase above, weighted so the mathematics is exact), and the vermillion curve is Riemann's formula built from the smooth trend plus one wave per zeta zero.
Drag the slider and watch thirty zeros carve the primes:Try it — build the primes out of zeta zeros, one wave at a time04080120x = 20x = 60x = 100the primes: ψ(x), one step per prime powerRiemann's formula with 0 zero-waveszero-waves included 0 / 30no zeros: just the smooth trend x, which misses every stepevery one of these zeros has real part exactly ½, which makes its wave swell like √x as x grows — the slowest growth the explicit formula allows.
A zero off the line at real part β ≠ ½ would make its wave grow like x^β instead, out of step with all the others, and the prime count would drift off course.It has now been open for 167 years. It is part of the eighth of Hilbert's problems (1900) and is one of the Clay Millennium Prize problems today; hundreds of theorems across number theory and beyond are proved conditionally, “assuming RH.” Its zeros have been checked by computer into the trillions — every one on the line — but a check is not a proof. The way forward is to turn the question into a number that can be moved — and that is exactly what Λ is.Λ (defined properly in the next chapter) repackages the Riemann hypothesis as a statement about one real number: RH holds if and only if Λ ≤ 0 (a proved equivalence, established in Chapter 1). That reformulation has three consequences. First, progress becomes measurable: a yes/no conjecture has no partial credit, but an upper bound on Λ can shrink: ½ → 0.22 → 0.2 → and now 0.1787854. Second, since 2018 we know Λ ≥ 0, so Λ is confined to the interval from 0 to the current ceiling, and RH is the statement that Λ sits at the left endpoint; every improvement to the ceiling is measured distance toward the answer. Third, bounds on Λ are unconditional — nothing in them assumes RH itself.
Lowering Λ is one of the few rigorous, quantifiable ways to make progress on the Riemann hypothesis.Chapter 1Λ, the constant whose value decides the Riemann hypothesisA single real number whose sign settles the question: the Riemann hypothesis holds exactly when Λ ≤ 0.Start with Riemann's xi function, a repackaging of the zeta function: ξ(s)=12s(s−1)π−s/2Γ(s/2)ζ(s)\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). The Riemann hypothesis says all its zeros lie on the critical line Re s=12\mathrm{Re}\,s=\tfrac12. Rotate coordinates so that line becomes the real axis (this proof uses the Polymath 15 normalization H0(z)=18 ξ (12+iz2)H_0(z)=\tfrac18\,\xi\!\left(\tfrac12+\tfrac{iz}{2}\right)) and RH becomes a single sentence:RH, restatedEvery zero of the entire function H0H_0 is a real number.In 1950 de Bruijn had the idea of deforming this function with a one-parameter flow — mathematically, running the heat equation on it:Ht(z) = ∫0∞etu2 Φ(u) cos(zu) du\htmlData{term=ht, tc=v}{H_t(z)}\;=\;\int_0^\infty \htmlData{term=heat, tc=s}{e^{tu^2}}\,\htmlData{term=phi, tc=b}{\Phi(u)}\,\htmlData{term=cos, tc=b}{\cos(zu)}\,duwhere Φ\Phi is the fixed super-exponentially decaying kernel with H0=18ξH_0=\tfrac18\xi.
Positive t smooths the function and, as we'll see, herds its zeros toward the real axis; negative t roughens it and pushes zeros off. De Bruijn proved that once all zeros are real they stay real at every later time.
So there is a single threshold, made precise by Newman in 1976:Λ = inf{ t: Ht has only real zeros }\Lambda\;=\;\htmlData{term=inf, tc=v}{\inf}\{\,t:\ \htmlData{term=real, tc=s}{H_t\ \text{has only real zeros}}\,\}That threshold is the de Bruijn–Newman constant, and it converts the Riemann hypothesis from a statement about infinitely many zeros into a statement about one real number:RH ⟺ Λ≤0\mathrm{RH}\iff \Lambda\le 0Analogy.
Λ is a thermostat reading. The xi function is a room full of particles (its zeros), and t is heat: warm the room and the particles settle onto the floor (the real axis); chill it and some lift off. Λ is the exact temperature at which the last airborne particle lands. The Riemann hypothesis says the room as built — at temperature zero — already has everything on the floor.
Since we can't yet check every particle, we do the next best thing: prove the landing temperature is low.-0.200.20.40.6Rodgers–Tao 2018Λ ≥ 0: “RH, if true, is only barely so”Λ lives hereRH ⟺ Λ = 00.1787854Fig. 2The state of knowledge about Λ. The Riemann hypothesis is equivalent to Λ ≤ 0; the equivalence is itself a theorem. Rodgers–Tao (2018) proved Λ ≥ 0, so RH, if true, is true with nothing to spare.
This work moves the other wall: Λ is now known to be at most 0.1787854. The truth lives somewhere in the vermillion interval, and RH says it lives at its left endpoint.One direction is now settled.
Newman conjectured Λ ≥ 0, famously adding that if RH is true, it is “only barely so” — and Rodgers and Tao proved this in 2018. So Λ is confined: 0≤Λ0\le\Lambda, and progress can now come only from the upper side. This proof moves the ceiling to Λ≤0.1787854\Lambda\le 0.1787854.
With the floor at 0, this removes just over 10% of the interval that remained.Chapter 2Heating the function pulls its zeros onto the real axisComplex zero pairs sink toward the real axis at a computable rate; the proof is a schedule for when the last of them arrives.Under the flow, the zeros of HtH_t move like interacting particles: real zeros repel each other along the axis, and each complex-conjugate pair gets pulled toward the axis.