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Library · docs/05-de-bruijn-newman.md

The de Bruijn–Newman Constant: Running Heat Flow *on* Zeta

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The payoff of docs/02-theta-heat-and-modularity.md. There we watched heat flow produce Xi. Here we run heat flow on Xi and ask how the zeros move.

The short version. The completed zeta function can be written as a cosine transform, Xi(z/2)/8 = integral_0^inf Phi(u) cos(zu) du, where Phi is built directly out of the theta function of docs/02 and its first two derivatives. Now insert a Gaussian factor e^{t u^2} into that integral. The result H_t(z) is an entire function for every real t, it equals Xi (up to normalisation) at t = 0, and it satisfies dH/dt = -d^2H/dz^2 — the backward heat equation. This flow has a beautiful effect on roots: increasing t makes real roots repel and spread out, and once they are all real they stay real forever. So there is a threshold time Lambda = inf{t : H_t has only real zeros}, the de Bruijn–Newman constant. THEOREM (de Bruijn 1950): Lambda <= 1/2. THEOREM (Rodgers–Tao 2018): Lambda >= 0. And RH is exactly the statement Lambda <= 0. Putting these together: RH is equivalent to Lambda = 0. The zeros of zeta are sitting, right now, at the precise instant of criticality — there is no slack. That is what Newman meant when he said RH, if true, is "only barely so".

1. Xi as a Fourier integral, and where Phi comes from

Recall from docs/03-functional-equation.md the completed function

    xi(s) = (1/2) s(s-1) pi^{-s/2} Gamma(s/2) zeta(s),        Xi(z) = xi(1/2 + iz)

Xi is entire, even, real on the real axis, and RH is the statement that all its zeros are real.

Now set (this is the half-theta that docs/02 and docs/03 call omega; Riemann's own memoir wrote psi for it, and it is not the Chebyshev psi of docs/04),

    psi(x) = sum_{n>=1} e^{-pi n^2 x} = (theta(x) - 1)/2

and define

    Phi(u) = 2 e^{9u} psi''(e^{4u}) + 3 e^{5u} psi'(e^{4u})

Differentiating psi term by term (psi' = -sum pi n^2 e^{-pi n^2 x}, psi'' = sum pi^2 n^4 e^{-pi n^2 x}) and substituting x = e^{4u} gives the explicit form:

    Phi(u) = sum_{n>=1} ( 2 pi^2 n^4 e^{9u} - 3 pi n^2 e^{5u} ) exp(-pi n^2 e^{4u})

This is the point worth pausing on. Phi is not some new special function conjured for the occasion. It is the heat kernel of docs/02, differentiated twice and reparametrised by x = e^{4u}. Everything below is still, at bottom, heat on a circle. The series is implemented as zeta.heatflow.Phi and cross-checked there against an independent mpmath.nsum evaluation (relative agreement 1.4e-41 at 40 digits); re-running both forms here, each gives Phi(0) = 0.4466969004671234440869847 and Phi(0.3) = 0.007423456244391188221752 (dps = 40).

THEOREM. With this Phi,

    H_0(z)  :=  integral from 0 to infinity of  Phi(u) cos(zu) du   =   Xi(z/2) / 8

Nothing here is trusted to memory: zeta.heatflow.H0_vs_Xi measures both constants of the ansatz H_0(z) = c * Xi(a z) from the functions themselves — c = H_0(0)/Xi(0) directly, and the rescaling a by two independent routes (curvature at the origin, using H_0''(0) = -integral u^2 Phi(u) du; and matching the first zero of H_0 against the first zeta ordinate). Both routes return a = 0.5 to within 3e-41, c returns 0.125 to 1.1e-42, and the exact rational relation then verifies pointwise to a maximum residual of 7.9e-42 over real z in [0, 100] plus complex probes (dps = 40). Re-run for this document: both sides give 0.06214009727353926373909672 at z = 0, and |H_0(z) - Xi(z/2)/8| <= 4.4e-42 at z = 1, 5, 14.134725, 28.2. (The z/2 is a normalisation convention inherited from the Polymath15 literature; it means the zeros of H_t sit at 2*gamma, where gamma are the ordinates of the zeta zeros — at t = 0 the module's zeros_of_H_t reproduces 2*gamma_n for the first ten zeros to a measured 1.3e-19 at dps = 30. Keep track of that factor of 2 — it matters when you compare gaps.)

Where does the formula come from? Take the Mellin representation of docs/03, xi(s) = 1/2 + (s(s-1)/2) * integral_1^inf (x^{s/2-1} + x^{-(1+s)/2}) psi(x) dx, integrate by parts twice to kill the polynomial prefactor s(s-1), then substitute x = e^{4u}. The e^{9u} and e^{5u} are bookkeeping from the chain rule and the Jacobian; the 2 and the 3 are what the two integrations by parts leave behind.

One gorgeous consequence. Phi is an even function: Phi(-u) = Phi(u). That is not visible in the series at all — under u -> -u the exponent e^{4u} becomes e^{-4u}, which looks like a completely different function. It is true because x -> 1/x is exactly Jacobi's modular identity theta(1/x) = sqrt(x) theta(x): with F(x) = d/dx[ x^{3/2} psi'(x) ] one has Phi(u) = 2 e^{3u} F(e^{4u}), and evenness of Phi is precisely F(1/x) = x^{3/2} F(x), which zeta.heatflow verifies independently to 1.6e-51 at x = 1.5 (dps = 50). On the series itself, Phi_is_even_defect measures the relative defect below 1e-34 for |u| <= 0.5 at dps = 40 — and is scrupulous about where that test loses meaning: past |u| ~ 0.85 the raw series is a catastrophic cancellation (summands of size 1 conspiring to produce a value below 1e-70) and returns pure noise, which is why the module computes Phi by folding through the exact identity Phi(u) = Phi(|u|) first. The evenness of Phi IS the functional equation. Which is why we may equally write H_t(z) = (1/2) integral over all of R of e^{t u^2} Phi(u) e^{izu} du — a genuine Fourier transform of an even, real, rapidly decaying function, hence an even entire function of z, real on the reals.

2. The deformation, and the sign of the heat equation

Define, for real t:

    H_t(z)  =  integral from 0 to infinity of  e^{t u^2} Phi(u) cos(zu) du

This converges for every real t, positive or negative, because Phi(u) ~ exp(-pi e^{4u}) decays doubly exponentially, and a double exponential eats e^{tu^2} for breakfast. So H_t is entire in z for all t, and H_0 = Xi(z/2)/8.

Multiplying by e^{tu^2} on the transform side is running the heat equation on the z side. Differentiate under the integral:

    dH/dt     =  integral  u^2 e^{tu^2} Phi(u) cos(zu) du
    d^2H/dz^2 =  integral  e^{tu^2} Phi(u) * (-u^2) cos(zu) du   =   -dH/dt

so

    dH/dt  =  -d^2H/dz^2                        (BACKWARD heat equation)

Verify the sign; do not trust your memory here. zeta/heatflow.py owns this check as heat_equation_residual, which forms |dH/dt + d2H/dz2| by central differences — the combination that vanishes under the backward convention and is order-1 under the forward one. Re-measured for this document (dps = 30, step h = 1e-6):

    (z, t) = ( 3,  0.0):    |dH/dt + d2H/dz2|  =  1.8e-18
    (z, t) = (10,  0.2):    |dH/dt + d2H/dz2|  =  1.0e-18
    (z, t) = (28, -0.1):    |dH/dt + d2H/dz2|  =  3.5e-19

(the residual floor is the finite-difference step, not the quadrature). The module also compares the two integrands directly, and that check reproduces here too: at (z, t) = (10, 0.2),

    dH/dt      =  -4.85383252228e-5
    d2H/dz2    =  +4.85383252228e-5

exact negatives to every displayed digit. The wrong sign is not a near-miss; it is off by a factor of exactly -1.

A terminological trap worth defusing. "Backward heat equation" sounds bad — it is famously ill-posed, it amplifies high frequencies (that is literally what e^{tu^2} does), and in general you cannot run it. Here you can, because Phi decays fast enough to absorb any amount of amplification. Note the direction bookkeeping:

That second bullet is the one people get backwards. Gaussian smoothing destroys real roots: convolve a function whose sign changes at spacing d with a Gaussian much wider than d and the sign changes are averaged away — they have to go somewhere, and where they go is off the real axis.

3. The constant, and the theorems

Because "all zeros real" is preserved by increasing t (this is the content of de Bruijn's and Newman's work, not an obvious fact), the set of good times is a half-line, and we may define:

    Lambda  =  inf { t in R : H_t has only real zeros }

Corollary: RH is equivalent to Lambda = 0. That is a genuine reformulation — not a weakening, not a heuristic, an if-and-only-if.

On the upper bound: de Bruijn's 1/2 stood essentially alone for half a century (Ki–Kim–Lee made it strict in 2009). The Polymath15 collaboration (led by Tao) then pushed it to Lambda <= 0.22 (D. H. J. Polymath, Effective approximation of heat flow evolution of the Riemann xi function, and a new upper bound for the de Bruijn–Newman constant, Res. Math. Sci. 6 (2019), paper 31), and feeding Platt–Trudgian's verification of RH up to height 3 x 10^12 back into the same machinery gives the current record Lambda <= 0.2 (D. J. Platt, T. S. Trudgian, The Riemann hypothesis is true up to 3·10^12, Bull. Lond. Math. Soc. 53 (2021)). The full bound history, with citations checked against the primary sources, is catalogued in code as zeta.heatflow.lambda_facts().

From below, before Rodgers–Tao, the record bounds came from "Lehmer pairs" of unusually close zeros — a chain of improvements through the 1990s (the Csordas–Smith–Varga Lehmer-pair criterion, Constr. Approx. 10 (1994); Lambda > -5.895e-9, Csordas–Odlyzko–Smith–Varga 1993; Lambda > -2.7e-9, Odlyzko 2000) ending with Saouter–Gourdon–Demichel's Lambda > -1.14541e-11 (Math. Comp. 80 (2011)). Rodgers–Tao made all of it moot — though the mechanism those bounds exploited is exactly the collision heuristic of the next section.

4. The intuition: roots as repelling particles

Here is the mechanism, and it is elementary enough to verify in ten lines of numpy.

Let p(z, t) be a monic polynomial with roots r_1(t), ..., r_N(t), evolving by the same equation dp/dt = -p''. Differentiate p = prod_i (z - r_i) and evaluate at z = r_k. Using p''(r_k) = 2 p'(r_k) sum_{j != k} 1/(r_k - r_j), the factor p'(r_k) cancels and you get:

    dr_k/dt  =  +2 * sum_{j != k}  1 / (r_k - r_j)              (under  dp/dt = -p'')
    dr_k/dt  =  -2 * sum_{j != k}  1 / (r_k - r_j)              (under  dp/dt = +p'')

Verify the sign; this is the second place memory betrays people. zeta/heatflow.py owns this as polynomial_heat_flow, and it does the verification the brutal way: it evolves the coefficients exactly (for a polynomial, exp(-t D^2) p = sum_k (-t)^k/k! p^{(2k)} is a finite sum), takes the roots of the evolved polynomial, and compares them against RK4 integration of the root ODE — for all four sign combinations, with roots [-3, -1, 0.5, 2, 4.5] over t in [0, 0.35]:

    exp(-t D^2)  with const = +2   ->   max |ODE - exact| = 9.3e-15    OK
    exp(-t D^2)  with const = -2   ->   max |ODE - exact| = 1.6e+00    WRONG
    exp(+t D^2)  with const = -2   ->   max |ODE - exact| = 2.7e-14    OK
    exp(+t D^2)  with const = +2   ->   max |ODE - exact| = 1.6e+00    WRONG

The pairing is rigid: exp(-t D^2) — the de Bruijn–Newman convention — goes with +2 and repulsion; the classical forward heat equation goes with -2 and attraction. Beware: the root ODE is often quoted with the -2, and quoting it with -2 while also claiming "roots repel as t increases" is self-contradictory; the table above is the antidote. The + sign means repulsion because if r_k sits to the right of r_j, the term 1/(r_k - r_j) is positive and pushes r_k further right. Closer neighbours push harder — the force goes like 1/distance.

Integrating for a while makes the picture visceral. Re-running polynomial_heat_flow on those same five roots for this document (RK4 agreeing with the exact coefficient evolution to 8.9e-15):

    t:              -1.0    -0.5    -0.2     0      0.35
    spread:         2.16    5.12    6.62    7.50    8.85
    min gap:         --      --     1.13    1.50    1.92
    max |Im r|:     2.25    0.57    0       0       0

Real roots stay real and fly apart as t increases; run the flow backwards and the gaps shrink until roots crash into each other and shoot off the axis as conjugate pairs (by the t = -0.5 sample two have already left; -- marks times where a real gap no longer exists). The same monotone gap growth is visible on the genuine H_t: zero_gap_statistics on the window [20, 105] (exactly the first ten zeros) measures a minimum gap of 3.28939 at t = -0.3, 3.53736 at t = 0, and 3.94962 at t = +0.6 — re-verified here at all three values of t.

The two-particle law. Isolate two roots at +-delta/2. Then d(delta)/dt = 4/delta, so

    delta(t)^2  =  delta_0^2  +  8t

For an isolated pair this is not merely a heuristic — it is exact, because exp(-t D^2)(z^2 - delta_0^2/4) = z^2 - delta_0^2/4 - 2t, and polynomial_heat_flow on roots +-0.25 at t = 0.1 confirms it: delta agrees with sqrt(0.25 + 0.8) = 1.0246950766 to machine precision (difference below 1e-15). Two facts fall out of that one line:

  1. Going forward (t up), delta grows like sqrt(8t) — the pair can never collide. Real roots are permanent.
  2. Going backward, delta hits zero at t = -delta_0^2/8, and past that delta^2 < 0, i.e. delta is imaginary: the pair has left the real axis as a complex-conjugate pair. A collision is exactly the event "two real zeros become two non-real zeros".

So Lambda is the moment of the first collision as you run the film backwards. And Rodgers–Tao's theorem Lambda >= 0 says that moment is now. Not "some time in the past" — now. Run the zeta zeros backwards by any positive amount of heat and, by their theorem, somewhere, at some height, a pair has already collided. RH, which says Lambda <= 0, would say the collisions have not started yet. Both together: the zeros of zeta sit at the exact critical instant, with zero margin. That is the precise content of "barely true".

Where is the tightness? Not down at gamma = 14.13. It is out at large height, where the mean spacing 2 pi / log(gamma / 2 pi) shrinks and unusually close pairs occur. Rodgers–Tao's proof is statistical: it shows that Lambda < 0 would force the zeros at large height to be implausibly evenly spaced — more rigid than they can be, given what is known about their pair correlation. The obstruction is a statement about the distribution of gaps, not about any individual zero. The module's track_zeros says the same thing in code, as an honest caveat in its docstring: over |t| <= 1 the first ten zeros stay real and comfortably separated — what the laboratory exhibits is the repulsion mechanism, not the location of Lambda.

Lehmer's famous close pair makes the numbers concrete. Recomputed here (mpmath.zetazero, 20 digits):

    gamma_6709 = 7005.062866174921
    gamma_6710 = 7005.100564672647
    gap        = 0.037698498
    mean gap at this height = 2 pi / log(gamma / 2 pi) = 0.895486
    ratio      = 0.0421          (about 4% of normal spacing)

HEURISTIC. In the H_t variable the zeros sit at 2*gamma, so delta_0 = 0.0753970, and the two-particle law predicts collision at t = -delta_0^2/8 = -7.1 x 10^{-4}. If those two zeros were alone in the world, that would show Lambda > -7.1 x 10^{-4}. They are not alone — every other zero pushes on them — so this is a heuristic, not a proof. The rigorous Lehmer-pair criteria of Csordas– Smith–Varga are the honest version, and applied to far closer pairs found at far greater heights they gave the pre-2018 record lower bounds. But the mechanism is exactly this one.

5. Heat flow, level repulsion, and GUE are one phenomenon

The ODE dr_k/dt = 2 sum_{j != k} 1/(r_k - r_j) is not an accident of this problem. It is the deterministic core of Dyson Brownian motion: add independent Brownian noise to each coordinate and you get exactly the eigenvalue process of a random Hermitian matrix undergoing Gaussian diffusion, whose stationary measure is the GUE eigenvalue density with its prod_{i<j} |r_i - r_j|^2 Vandermonde factor. The same 1/(r_i - r_j) force is the Calogero–Moser interaction in integrable systems.

So three things you might have thought were separate are one thing:

docs/06 covers the statistics; the thing to carry over is that the "zeros repel" you meet there and the "zeros repel" here are literally the same 1/(r_i - r_j). This is a genuine structural coincidence and one of the better reasons to believe a spectral interpretation of the zeros exists.

6. Honest status

This is a research frontier and a genuine reformulation. It is not a road to a proof, and it is worth being blunt about why.

(De Branges's approach — Hilbert spaces of entire functions and positivity conditions — is sometimes mentioned in the same breath. It is a different programme, and its published proof attempts have not been accepted. I mention it only so you know the two are not the same thing.)

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