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# de Bruijn-Newman Constant

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 ,

 (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

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 ." 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 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.

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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