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)

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 . The best current lower bound (Csordas et al. 1993, 1994) is . The Riemann hypothesis is equivalent to the conjecture that .

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.; 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.