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de Bruijn-Newman Constant


Let Xi be the xi-function defined by

 Xi(iz)=1/2(z^2-1/4)pi^(-z/2-1/4)Gamma(1/2z+1/4)zeta(z+1/2).
(1)

Xi(z/2)/8 can be viewed as the Fourier transform of the signal

 Phi(t)=sum_(n=1)^infty(2pi^2n^4e^(9t)-3pin^2e^(5t))e^(-pin^2e^(4t))
(2)

for t in R>=0. Then denote the Fourier transform of Phi(t)e^(lambdat^2) as H(lambda,z),

 F_t[Phi(t)e^(lambdat^2)](z)=H(lambda,z).
(3)

The Riemann hypothesis is equivalent to the conjecture that Lambda<=0 (Rodgers and Tao 2020).

de Bruijn (1950) proved that H has only real zeros for lambda>=1/2. C. M. Newman (1976) proved that there exists a constant Lambda such that H has only real zeros iff lambda>=Lambda and conjectured that Lambda>=0. The following table summarizes best known lower bounds on Lambda prior to 2020, when Rodgers and Tao (2020) proved that Lambda>=0.

lower boundreference
-inftyNewman 1976
-50Csordas-Norfolk-Varga 1988
-5te Riele 1991
-0.385Norfolk-Ruttan-Varga 1992
-0.0991Csordas-Ruttan-Varga 1991
-4.379×10^(-6)Csordas-Smith-Varga 1994
-5.895×10^(-9)Csordas-Odlyzko-Smith-Varga 1993
-2.63×10^(-9)Odlyzko 2000
-1.15×10^(-11)Saouter-Gourdon-Demichel 2011

See also

de Bruijn Constant, Xi-Function

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References

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.; Norfolk, T. S.; and Varga, R. S. "A Lower Bound for the De Bruijn-Newman Constant Lambda." 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 Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 203-205, 2003.Newman, C. M. "Fourier Transforms with only Real Zeros." Proc. Amer. Math. Soc. 61, 245-251, 1976.Norfolk, T. S.; Ruttan, A.; and Varga, R. S. "A Lower Bound for the de Bruijn-Newman Constant Lambda 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.

Referenced on Wolfram|Alpha

de Bruijn-Newman Constant

Cite this as:

Weisstein, Eric W. "de Bruijn-Newman Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/deBruijn-NewmanConstant.html

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