An improved lower bound for the de Bruijn-Newman constant

Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick

doi:10.1090/S0025-5718-2011-02472-5

An improved lower bound for the de Bruijn-Newman constant
HTML articles powered by AMS MathViewer

by Yannick Saouter, Xavier Gourdon and Patrick Demichel PDF

Math. Comp. 80 (2011), 2281-2287 Request permission

Abstract:

In this article, we report on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann $\zeta$ function. Given this new close pair of zeros, we improve the known lower bound for de Bruijn-Newman constant $\Lambda$. The Riemann hypothesis is equivalent to the assertion $\Lambda \leq 0$. In this article, we establish that in fact we have $\Lambda > -1.14541 \times 10^{-11}$. This new bound confirms the belief that if the Riemann hypothesis is true, it is barely true.

References

Jonathan M. Borwein, David M. Bradley, and Richard E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math. 121 (2000), no. 1-2, 247–296. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780051, DOI 10.1016/S0377-0427(00)00336-8
G. Csordas, T. S. Norfolk, and R. S. Varga, A lower bound for the de Bruijn-Newman constant $\Lambda$, Numer. Math. 52 (1988), no. 5, 483–497. MR 945095, DOI 10.1007/BF01400887
G. Csordas, A. M. Odlyzko, W. Smith, and R. S. Varga, A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant $\Lambda$, Electron. Trans. Numer. Anal. 1 (1993), no. Dec., 104–111 (electronic only). MR 1253639
G. Csordas, A. Ruttan, and R. S. Varga, The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis, Numer. Algorithms 1 (1991), no. 3, 305–329. MR 1135299, DOI 10.1007/BF02142328
George Csordas, Wayne Smith, and Richard S. Varga, Lehmer pairs of zeros, the de Bruijn-Newman constant $\Lambda$, and the Riemann hypothesis, Constr. Approx. 10 (1994), no. 1, 107–129. MR 1260363, DOI 10.1007/BF01205170
N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197–226. MR 37351, DOI 10.1215/S0012-7094-50-01720-0
P. Dusart. Autour de la fonction qui compte le nombre de nombres premiers. PhD thesis, Université de Limoges, 1998.
Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul Zimmermann, MPFR: a multiple-precision binary floating-point library with correct rounding, ACM Trans. Math. Software 33 (2007), no. 2, Art. 13, 15. MR 2326955, DOI 10.1145/1236463.1236468
X. Gourdon and P. Demichel. The first $10^{13}$ zeros of the Riemann Zeta function, and zeros computation at very large height. Available at http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e131e24.pdf, 2004.
Haseo Ki, Young-One Kim, and Jungseob Lee, On the de Bruijn-Newman constant, Adv. Math. 222 (2009), no. 1, 281–306. MR 2531375, DOI 10.1016/j.aim.2009.04.003
R. Sherman Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 20 (1970), 303–320. MR 258768, DOI 10.1112/plms/s3-20.2.303
Charles M. Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61 (1976), no. 2, 245–251 (1977). MR 434982, DOI 10.1090/S0002-9939-1976-0434982-5
T. S. Norfolk, A. Ruttan, and R. S. Varga, A lower bound for the de Bruijn-Newman constant $\Lambda$. II, Progress in approximation theory (Tampa, FL, 1990) Springer Ser. Comput. Math., vol. 19, Springer, New York, 1992, pp. 403–418. MR 1240792, DOI 10.1007/978-1-4612-2966-7_{1}7
A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308. MR 866115, DOI 10.1090/S0025-5718-1987-0866115-0
A. M. Odlyzko, An improved bound for the de Bruijn-Newman constant, Numer. Algorithms 25 (2000), no. 1-4, 293–303. Mathematical journey through analysis, matrix theory and scientific computation (Kent, OH, 1999). MR 1827160, DOI 10.1023/A:1016677511798
A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), no. 2, 797–809. MR 961614, DOI 10.1090/S0002-9947-1988-0961614-2
Robert Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415–440, S17–S23. MR 1195435, DOI 10.1090/S0025-5718-1993-1195435-0
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
H. J. J. te Riele, A new lower bound for the de Bruijn-Newman constant, Numer. Math. 58 (1991), no. 6, 661–667. MR 1083527, DOI 10.1007/BF01385647
T. Trudgian. Improvements to Turing’s method. http://arxiv.org/abs/0903.1885v1, March 2009.
A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc. (3) 3 (1953), 99–117. MR 55785, DOI 10.1112/plms/s3-3.1.99
J. van de Lune. Unpublished, 2001.
S. Wedeniwski. Zetagrid home page. http://www.zetagrid.net/, 2005.