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 .
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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
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2003.Newman, C. M. "Fourier Transforms with only Real Zeros."
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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,
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Algorithms25, 293-303, 2000.Rodgers, B. and Tao, T. "The
De Bruijn-Newman Constant Is Non-Negative." Forum Math., Pi8,
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.