The De Bruijn–Newman Constant: An In-Depth Analysis

1. Historical Background

1.1 Origins of the Equation

The story of this equation begins with Bernhard Riemann (1859), whose famous memoir "Über die Anzahl der Primzahlen unter einer gegebenen Größe" introduced the xi function Ξ, an entire function whose zeros encode the non-trivial zeros of ζ(s). Riemann himself derived the integral representation:

$$\Xi(z) = \int_0^\infty \Phi(u)\cos(zu)\,du$$

where the kernel Φ(u) is the "super-exponentially decaying" function:

$$\Phi(u) = \sum_{n=1}^{\infty} \left(2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u}\right) \exp\left(-\pi n^2 e^{4u}\right)$$

This kernel arises from Jacobi's theta function θ(x) = Σ e^{−πn²x} after the substitution x = e^{4u}, and it satisfies the crucial symmetry Φ(u) = Φ(−u) (a consequence of the theta functional equation, itself equivalent to the functional equation of ζ).

1.2 Pólya's Idea (1920s)

George Pólya, in a series of papers (notably "Über die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen", 1927), pioneered the strategy of studying Fourier transforms of kernels like Φ and asking: for which multiplier functions does the transform remain real-rooted? Pólya proved that certain "universal factors" preserve real-rootedness. This is the direct ancestor of the heat-flow deformation.

1.3 De Bruijn (1950)

Nicolaas Govert de Bruijn, in his paper:

N. G. de Bruijn, "The roots of trigonometric integrals", Duke Mathematical Journal 17 (1950), 197–226.

studied the deformed family:

$$H_\lambda(z) = \int_0^\infty e^{\lambda u^2}\,\Phi(u)\cos(zu)\,du, \qquad H_0(z) = \tfrac{1}{8}\Xi\!\left(\tfrac{z}{2}\right)$$

De Bruijn proved two landmark results:

1. H_λ has only real zeros for λ ≥ 1/2 (so Λ ≤ 1/2).
2. Monotonicity: if H_{λ₀} has only real zeros, then H_λ has only real zeros for every λ > λ₀.

The monotonicity comes from the fact that H_λ satisfies the backward heat equation:

$$\frac{\partial H}{\partial \lambda} = -\frac{\partial^2 H}{\partial z^2}$$

(verified by differentiating under the integral: ∂_λ e^{λu²}cos(zu) = u²e^{λu²}cos(zu) = −∂²_z e^{λu²}cos(zu)). Under this flow, complex zeros are attracted toward the real axis and, once real, zeros stay real as λ increases.

1.4 Newman (1976)

Charles M. Newman, in:

C. M. Newman, "Fourier transforms with only real zeros", Proceedings of the American Mathematical Society 61 (1976), 245–251.

proved that Λ > −∞: there exists λ for which H_λ has a non-real zero. This established that the quantity

$$\Lambda = \inf\{\lambda \in \mathbb{R} : H_\lambda \text{ has only real zeros}\}$$

is a well-defined real number, now called the de Bruijn–Newman constant. Newman made the famous conjecture Λ ≥ 0, writing:

"This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so."

1.5 The Equivalence with RH

Since H₀(z) = (1/8)Ξ(z/2), the zeros of H₀ are real if and only if all non-trivial zeros of ζ lie on Re(s) = 1/2. Combined with de Bruijn's monotonicity:

$$\boxed{\text{RH} \iff \Lambda \le 0}$$

2. Modern Results

2.1 Lower Bounds (Computational Era)

A long sequence of computational papers pushed the lower bound up from Newman's Λ > −∞:

These works exploited Lehmer pairs — unusually close pairs of zeta zeros (named after D. H. Lehmer, 1956) which, under the backward flow (λ decreasing), would quickly collide and go complex.

2.2 Rodgers–Tao (2018): Λ ≥ 0

Brad Rodgers and Terence Tao proved Newman's conjecture:

B. Rodgers, T. Tao, "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, Pi, Vol. 8 (2020), e6.

arXiv link: https://arxiv.org/abs/1801.05914

Published version: https://doi.org/10.1017/fmp.2020.6

Heuristic of the proof: If Λ < 0, then at time λ = 0 the zeros would have already been flowing for positive time, and the heat flow drives zero statistics toward perfect local equidistribution (an arithmetic-progression-like "super-rigid" structure — the flow's attracting fixed point). But this contradicts known results on the pair correlation of zeta zeros (Montgomery's theorem) and numerical GUE statistics: the actual zeros exhibit genuine random-matrix fluctuations incompatible with such rigidity. Hence Λ ≥ 0 — "RH, if true, is barely true."

2.3 Polymath 15 (2019): Λ ≤ 0.22

The collaborative Polymath 15 project, led by Tao, improved de Bruijn's 70-year-old upper bound:

D.H.J. Polymath, "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn–Newman constant", Research in the Mathematical Sciences 6, 31 (2019).

arXiv link: https://arxiv.org/abs/1904.12438

Published version: https://doi.org/10.1007/s40687-019-0193-1

Project blog: https://terrytao.wordpress.com/tag/polymath15/

They proved Λ ≤ 0.22 by combining:

• Rigorous asymptotic approximations of H_λ (an "effective Riemann–Siegel formula" for the flow),
• Numerical verification that zeros of H_{0.2} up to a large height are real,
• A barrier argument controlling zeros at height ~ 3 × 10¹⁰ and beyond.

Also relevant earlier improvement: Ki, Kim, Lee (2009), "On the de Bruijn–Newman constant", Advances in Mathematics 222, showed Λ < 1/2 strictly (H_λ has at most finitely many complex zeros for λ > 0).

Current state of knowledge:

$$0 \le \Lambda \le 0.22$$

and Λ = 0 is exactly the Riemann Hypothesis (given Λ ≥ 0). For your paper: any new technique lowering the upper bound is publishable research; reaching Λ ≤ 0 would prove RH.

3. Key References for Your Paper

1. N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197–226.
2. C. M. Newman, Fourier transforms with only real zeros, Proc. AMS 61 (1976), 245–251.
3. G. Csordas, T. S. Norfolk, R. S. Varga, The Riemann Hypothesis and the Turán inequalities, Trans. AMS 296 (1986).
4. G. Csordas, W. Smith, R. S. Varga, Lehmer pairs of zeros, the de Bruijn–Newman constant Λ, and the Riemann Hypothesis, Constr. Approx. 10 (1994), 107–129.
5. A. M. Odlyzko, An improved bound for the de Bruijn–Newman constant, Numerical Algorithms 25 (2000), 293–303. https://www-users.cse.umn.edu/~odlyzko/doc/debruijn.newman.pdf
6. H. Ki, Y.-O. Kim, J. Lee, On the de Bruijn–Newman constant, Adv. Math. 222 (2009), 281–306.
7. B. Rodgers, T. Tao (2018/2020): https://arxiv.org/abs/1801.05914
8. D.H.J. Polymath (2019): https://arxiv.org/abs/1904.12438
9. Tao's expository blog post: https://terrytao.wordpress.com/2018/01/19/the-de-bruijn-newman-constant-is-non-negative/

4. Computational Examples in Python

4.1 Computing Φ(u), H_λ(z), and verifying H₀ = (1/8)Ξ(z/2)

import mpmath as mp

mp.mp.dps = 30  # 30 digits of working precision

def Phi(u, N=60):
    """
    The super-exponentially decaying kernel
    Phi(u) = sum_{n>=1} (2 pi^2 n^4 e^{9u} - 3 pi n^2 e^{5u}) exp(-pi n^2 e^{4u}).
    Converges extremely fast; N=60 terms is far more than enough.
    """
    u = mp.mpf(u)
    total = mp.mpf(0)
    for n in range(1, N + 1):
        n2 = mp.mpf(n)**2
        x = mp.exp(4*u)
        term = (2*mp.pi**2*n2**2*mp.exp(9*u)
                - 3*mp.pi*n2*mp.exp(5*u)) * mp.exp(-mp.pi*n2*x)
        total += term
        if abs(term) < mp.mpf(10)**(-mp.mp.dps - 10):
            break
    return total

def H(lam, z, U=6):
    """
    H_lambda(z) = int_0^U exp(lam*u^2) * Phi(u) * cos(z*u) du.
    Phi decays like exp(-pi e^{4u}), so U=6 truncation error is negligible.
    """
    lam, z = mp.mpf(lam), mp.mpc(z)
    f = lambda u: mp.exp(lam*u**2) * Phi(u) * mp.cos(z*u)
    return mp.quad(f, [0, 1, 2, 3, U])

def Xi(z):
    """Riemann Xi function: Xi(z) = xi(1/2 + i z) with
       xi(s) = (s(s-1)/2) pi^{-s/2} Gamma(s/2) zeta(s)."""
    s = mp.mpf('0.5') + 1j*mp.mpc(z)
    return (s*(s-1)/2) * mp.pi**(-s/2) * mp.gamma(s/2) * mp.zeta(s)

# --- Verification: H_0(z) should equal (1/8) * Xi(z/2) ---
print("Verify H_0(z) = (1/8) Xi(z/2):")
for z in [0, 5, 10, 20]:
    lhs = H(0, z)
    rhs = Xi(mp.mpf(z)/2) / 8
    print(f"  z={z:>3}:  H_0 = {mp.nstr(lhs, 12):>20}"
          f"   Xi/8 = {mp.nstr(rhs, 12):>20}"
          f"   |diff| = {mp.nstr(abs(lhs-rhs), 3)}")

Expected output shows agreement to ~25+ digits, confirming the identity H₀(z) = (1/8)Ξ(z/2).

4.2 Verifying the Backward Heat Equation ∂H/∂λ = −∂²H/∂z²

def verify_heat_equation(lam, z, h=mp.mpf('1e-6')):
    """Numerically check dH/dlambda = -d^2H/dz^2 via central differences."""
    dH_dlam = (H(lam + h, z) - H(lam - h, z)) / (2*h)
    d2H_dz2 = (H(lam, z + h) - 2*H(lam, z) + H(lam, z - h)) / h**2
    print(f"lambda={lam}, z={z}:")
    print(f"   dH/dlambda  = {mp.nstr(dH_dlam, 10)}")
    print(f"  -d2H/dz2     = {mp.nstr(-d2H_dz2, 10)}")
    print(f"   |difference| = {mp.nstr(abs(dH_dlam + d2H_dz2), 3)}\n")

verify_heat_equation(0.0, 10.0)
verify_heat_equation(0.1, 15.0)

4.3 Watching Zeros Flow: Zeros of H_λ for Different λ

The first zero of H₀ sits at z = 2γ₁ ≈ 28.2695 (twice the first zeta zero γ₁ ≈ 14.1347, because of the z/2 rescaling). We track how zeros drift under the flow:

def find_real_zeros(lam, zmax=80, step=0.5):
    """Locate real zeros of H_lambda in (0, zmax) by sign changes + bisection."""
    zeros = []
    prev_z, prev_v = mp.mpf('0.1'), H(lam, mp.mpf('0.1')).real
    z = mp.mpf('0.1') + step
    while z < zmax:
        v = H(lam, z).real
        if prev_v * v < 0:
            root = mp.findroot(lambda t: H(lam, t).real, (prev_z, z),
                               solver='bisect')
            zeros.append(root)
        prev_z, prev_v = z, v
        z += step
    return zeros

print("First zeta zero gamma_1 =", mp.nstr(mp.zetazero(1).imag, 12))
print()
for lam in [0.0, 0.1, 0.3]:
    zs = find_real_zeros(lam, zmax=70)
    print(f"lambda = {lam}: first zeros of H_lambda:",
          [mp.nstr(r, 10) for r in zs[:4]])

Sample behavior: at λ = 0 the first zero is ≈ 28.2695 (= 2×14.1347...); as λ increases the zeros spread apart (repel), illustrating the flow toward rigid, well-separated configurations. Running the flow backward (λ < 0), close pairs (Lehmer pairs) would collide and escape into the complex plane — which is exactly the mechanism behind the lower-bound computations.

4.4 Toy Model: The Heat Flow on a Polynomial (Zero Collision Demo)

The full H_λ is expensive; the mechanics of the flow are perfectly illustrated by finite "toy" versions. Under the backward heat flow, a polynomial p evolves as p ↦ e^{−λ∂²_z} p:

import numpy as np
import matplotlib.pyplot as plt
from numpy.polynomial import polynomial as P
from math import factorial

def heat_flow_poly(coeffs, lam):
    """
    Apply exp(-lam * d^2/dz^2) to a polynomial (finite sum since
    derivatives terminate). lam > 0 = de Bruijn direction (zeros -> real);
    lam < 0 = backward direction (real zeros can collide and go complex).
    """
    c = np.array(coeffs, dtype=float)
    result = c.copy()
    deriv = c.copy()
    k = 1
    while len(deriv) > 2:
        deriv = P.polyder(deriv, 2)          # take two derivatives
        result[:len(deriv)] += ((-lam)**k / factorial(k)) * deriv
        k += 1
    return result

# Toy "Lehmer pair": zeros at +/-1, +/-1.05  (a very close pair near z=+/-1)
p = P.polyfromroots([-1.05, -1.0, 1.0, 1.05])

lams = np.linspace(-0.05, 0.05, 11)
print(f"{'lambda':>8} | zeros of e^(-lam D^2) p")
print("-" * 60)
for lam in lams:
    q = heat_flow_poly(p, lam)
    roots = np.sort_complex(np.roots(q[::-1]))
    formatted = ", ".join(f"{r:.4f}" for r in roots)
    print(f"{lam:>8.3f} | {formatted}")

# Visualize trajectories in the complex plane
plt.figure(figsize=(8, 5))
for lam in np.linspace(-0.08, 0.08, 60):
    q = heat_flow_poly(p, lam)
    roots = np.roots(q[::-1])
    color = 'red' if lam < 0 else 'blue'
    plt.scatter(roots.real, roots.imag, s=8, c=color, alpha=0.5)
plt.axhline(0, color='gray', lw=0.5)
plt.xlabel('Re(z)'); plt.ylabel('Im(z)')
plt.title('Heat flow on zeros: red = backward (λ<0) → collision → complex;\n'
          'blue = forward (λ>0) → real & repelling')
plt.tight_layout()
plt.savefig('debruijn_newman_flow.png', dpi=150)
plt.show()

What you will observe: for λ > 0 (blue) the close pair of real zeros repels and stays real; for λ < 0 (red) the pair at ±(1, 1.05) collides and splits off the real axis into complex conjugates. This is a miniature of exactly why Lehmer pairs of zeta zeros force Λ to be very close to 0 from below, and why Rodgers–Tao's rigidity argument forces Λ ≥ 0.

4.5 Notes on the Computation

• Precision: Φ(u) decays like exp(−πe^{4u}), so integrals truncated at U = 6 have error below 10⁻¹⁰⁰⁰. However, H_λ(z) itself decays rapidly in z, so for large z (say z > 100) you need higher mp.mp.dps to avoid catastrophic cancellation — this is precisely why Polymath 15 needed a Riemann–Siegel-type effective approximation instead of naive quadrature.
• Cross-check: mp.zetazero(n) gives the n-th zeta zero; the n-th real zero of H₀ should be exactly 2·Im(ρₙ).

5. Summary for Your Paper

This equation is one of the most active modern reformulations of RH because it converts a static statement (zeros on a line) into a dynamical one (stability of zeros under a PDE flow), opening the door to techniques from PDE, random matrix theory, and statistical mechanics.