de Bruijn-Newman Constant
المؤلف:
Csordas, G.; Odlyzko, A.; Smith, W.; and Varga, R. S.
المصدر:
"A New Lehmer Pair of Zeros and a New Lower Bound for the de Bruijn-Newman Constant." Elec. Trans. Numer. Analysis 1
الجزء والصفحة:
...
23-2-2020
865
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 
,
=H(lambda,z).](http://mathworld.wolfram.com/images/equations/deBruijn-NewmanConstant/NumberedEquation3.gif) |
(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.
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