## de Bruijn-Newman Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

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. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/dbnwm/dbnwm.html

Newman, C. M. Fourier Transforms with only Real Zeros.'' Proc. Amer. Math. Soc. 61, 245-251, 1976.