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)
|
The Riemann hypothesis is equivalent to the conjecture that 
(Rodgers and Tao 2020).
de Bruijn (1950) proved that 
has only real zeros for 
.
C. M. Newman (1976) proved that there exists a constant 
such that 
has only real zeros iff 
and conjectured that 
. 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 |
."
Numer. Math. 52, 483-497, 1988.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 
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.Rodgers, B. and Tao, T. "The
De Bruijn-Newman Constant Is Non-Negative." Forum Math., Pi 8,
e6, 62 pp., 2020.Saouter, Y.; Gourdon, X.; and Demichel, P. "An
Improved Lower Bound for the De Bruijn-Newman Constant." Math. Comp. 80,
2281-2287, 2011.te Riele, H. J. J. "A New Lower Bound
for the De Bruijn-Newman Constant." Numer. Math. 58, 661-667,
1991.