The de Bruijn-Newman constant 
is the real number such
that the function 
defined below has only real
zeros iff 
. Let 
be the xi-function defined
by
 |
(1)
|
can be viewed as the Fourier transform of the
signal
 |
(2)
|
for 
,
.
Then denote the Fourier transform of 
as 
,
=H(lambda,z).](/images/equations/deBruijn-NewmanConstant/NumberedEquation3.svg) |
(3)
|
Newman (1976) proved that such a constant exists and conjectured that 
. The Riemann
hypothesis is equivalent to the conjecture that 
, so Newman's conjecture asserts that the Riemann
hypothesis, if true, is only barely true. The following table summarizes best
known lower bounds on 
prior to 2020, when Rodgers and Tao (2020) proved that
.
| lower bound | reference |
 | Newman (1976) |
 | Csordas-Norfolk-Varga (1988) |
 | te Riele (1991) |
 | Norfolk-Ruttan-Varga (1992) |
 | Csordas-Ruttan-Varga (1991) |
 | Csordas-Smith-Varga (1994) |
 | Csordas-Odlyzko-Smith-Varga (1993) |
 | Odlyzko (2000) |
 | Saouter-Gourdon-Demichel (2011) |
Upper bounds on 
are summarized below. The final three rows are computer-assisted
bounds reported in 2026 that had not undergone external peer review as of Sep. 15,
2026.
| upper bound | reference | status |
 | de Bruijn (1950) | published |
 | Ki-Kim-Lee (2009) | published |
 | Polymath (2019) | published |
 | Platt-Trudgian (2021) | published |
 | Mosaic Intelligence (2026) | unreviewed |
 | Gomila (2026) | unreviewed |
 | Gordon (2026) | unreviewed |
Mosaic Intelligence (2026) describes its bound as AI-generated. Gomila (2026) reports human-directed work using Claude and ChatGPT/Codex, while Gordon (2026) reports assistance from GPT-6 Astra and Claude Code. The Mosaic Intelligence and Gomila projects provide numerical certificates. Gordon's partial Lean formalization is conditional on stated analytic inputs.
Hakimoglu-Brown (2026) reported the conditional bound 
, assuming that the Riemann
zeta function has no nontrivial zero with real part
greater than 
.
The zero-free assertion is an external, unreviewed input. The bound is therefore
not an unconditional replacement for the values above. Hakimoglu-Brown (2026) acknowledges
ChatGPT assistance with analytic derivations and exact verification calculations,
and states that the argument has not been independently refereed.
For Dirichlet L-series, a generalized de Bruijn-Newman constant 
is similarly defined by the heat deformation of the
completed function. Freire et al. (2026) proved
 |
(4)
|
for every primitive real-valued number theoretic character of conductor between 3 and 400. The bound is unconditional and uses rigorous numerical checks together with finite-height zero verification. It is not a bound for all conductors or all complex-valued number theoretic characters. Freire et al. (2026) report AI assistance in refining analytic error estimates and numerical methods.
." Numer. Math. 52, 483-497, 1988.Csordas,
G.; Odlyzko, A.; Smith, W.; and Varga, R. S. "A New Lehmer Pair of Zeros
and a New Lower Bound for the de Bruijn-Newman Constant." Elec. Trans. Numer.
Analysis 1, 104-111, 1993.Csordas, G.; Ruttan, A.; and Varga,
R. S. "The Laguerre Inequalities with Applications to a Problem Associated
with the Riemann Hypothesis." Numer. Algorithms 1, 305-329, 1991.Csordas,
G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of Zeros, the de Bruijn-Newman
Constant and the Riemann Hypothesis." Constr. Approx. 10, 107-129,
1994.de Bruijn, N. G. "The Roots of Trigonometric Integrals."
Duke Math. J. 17, 197-226, 1950.Finch, S. R. "De
Bruijn-Newman Constant." §2.3 2 in 
-Functions
with Primitive Real-Valued Characters." 19 Oct 2026. 
for the de Bruijn-Newman
Constant." Aug. 20, 2026. 
for the de Bruijn-Newman Constant."
8 Oct 2026. 
for the de Bruijn-Newman Constant." Jul. 3,
2026. 
II." In Progress in Approximation Theory
(Ed. A. A. Gonchar and E. B. Saff). New York: Springer, pp. 403-418,
1992.Odlyzko, A. M. "An Improved Bound for the De Bruijn-Newman
Constant." Numer. Algorithms 25, 293-303, 2000.Platt,
D. and Trudgian, T. "The Riemann Hypothesis Is True up to 
." Bull. London Math. Soc. 53,
792-797, 2021.