Summary
Strong numerical evidence is presented for a new lower bound for the so-called de Bruijn-Newman constant. This constant is related to the Riemann hypothesis. The new bound, −5, is suggested by high-precision floatingpoint computations, with a mantissa of 250 decimal digits, of i) the coefficients of a so-called Jensen polynomial of degree 406, ii) the so-called Sturm sequence corresponding to this polynomial which implies that it has two complex zeros, and iii) the two complex zoros of this polynomial. Aproof of the new bound could be given if one would repeat the computations i) and iii) with a floatingpoint accuracy of at least 2600 decimal digits.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
de Bruijn, N.G.: The roots of trigonometric integrals. Duke Math. J.17, 197–226 (1950)
Csordas, G., Norfolk, T.S., Varga, R.S.: A lower bound for the de Bruijn-Newman constant Λ*. Numer. Math.52, 483–497 (1988)
Csordas, G., Norfolk, T.S., Varga, R.S.: The Reemann hypothesis and the Turan inequalities. Trans. Amer. Math. Soc.296, 521–541 (1986)
Henrici, P.: Applied and computational complex analysis. vol. 1. New York: Wiley 1977
Kress, R.: On the general Hermite cardinal interpolation. Math. Comput.26, 925–933 (1972)
Newman, C.M.: Fourier transforms with only real zeros. Proc. Am. Math. Soc.61, 245–251 (1976)
Stoer, J., Bulirsch, R.: Introduction to numerical analysis. Third correct printing. Berlin, Heidelberg, New York: Springer 1983
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
te Riele, H.J.J. A new lower bound for the de Bruijn-Newman constant. Numer. Math. 58, 661–667 (1990). https://doi.org/10.1007/BF01385647
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01385647