Advertisement

Effective approximation of heat flow evolution of the Riemann \(\xi \) function, and a new upper bound for the de Bruijn–Newman constant

  • D. H. J. PolymathEmail author
Research
  • 32 Downloads

Abstract

For each \(t \in \mathbb {R}\), define the entire function
$$\begin{aligned} H_t(z){:=}\,\int _0^\infty e^{tu^2} \varPhi (u) \cos (zu)\ \mathrm{d}u, \end{aligned}$$
where \(\varPhi \) is the super-exponentially decaying function
$$\begin{aligned} \varPhi (u){:=}\,\sum _{n=1}^\infty (2\pi ^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp (-\pi n^2 e^{4u}). \end{aligned}$$
This is essentially the heat flow evolution of the Riemann \(\xi \) function. From the work of de Bruijn and Newman, there exists a finite constant \(\varLambda \) (the de Bruijn–Newman constant) such that the zeroes of \(H_t\) are all real precisely when \(t \ge \varLambda \). The Riemann hypothesis is equivalent to the assertion \(\varLambda \le 0\); recently, Rodgers and Tao established the matching lower bound \(\varLambda \ge 0\). Ki, and Kim and Lee established the upper bound \(\varLambda < \frac{1}{2}\). In this paper, we establish several effective estimates on \(H_t(x+iy)\) for \(t \ge 0\), including some that are accurate for small or medium values of x. By combining these estimates with numerical computations, we are able to obtain a new upper bound \(\varLambda \le 0.22\) unconditionally, as well as improvements conditional on further numerical verification of the Riemann hypothesis. We also obtain some new estimates controlling the asymptotic behavior of zeroes of \(H_t(x+iy)\) as \(x \rightarrow \infty \).

Notes

References

  1. 1.
    Anderson, D.P.: BOINC: a system for public-resource computing and storage. In: GRID ’04: Proceedings of the Fifth IEEE/ACM International Workshop on Grid Computing, pp. 4–10 (2004)Google Scholar
  2. 2.
    Arias de Reyna, J.: High-precision computation of Riemann’s zeta function by the Riemann–Siegel asymptotic formula, I. Math. Comput. 80, 995–1009 (2011)CrossRefGoogle Scholar
  3. 3.
    Arias de Reyna, J., Van de lune, J.: On the exact location of the non-trivial zeroes of Riemann’s zeta function. Acta Arith. 163, 215–245 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boyd, W.G.C.: Gamma function asymptotics by an extension of the method of steepest descents. Proc. Math. Phys. Sci. 447(1931), 609–630 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Csordas, G., Norfolk, T.S., Varga, R.S.: A lower bound for the de Bruijn–Newman constant \(\Lambda \). Numer. Math. 52, 483–497 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Csordas, G., Odlyzko, A.M., Smith, W., Varga, R.S.: A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda. Electron. Trans. Numer. Anal. 1, 104–111 (1993)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Csordas, G., Ruttan, A., Varga, R.S.: The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis. Numer. Algorithms 1, 305–329 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Csordas, G., Smith, W., Varga, R.S.: Lehmer pairs of zeros, the de Bruijn–Newman constant \(\Lambda \), and the Riemann hypothesis. Constr. Approx. 10(1), 107–129 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Bruijn, N.C.: The roots of trigonometric integrals. Duke J. Math. 17, 197–226 (1950)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ki, H., Kim, Y.O., Lee, J.: On the de Bruijn–Newman constant. Adv. Math. 22, 281–306 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Newman, C.M.: Fourier transforms with only real zeroes. Proc. Am. Math. Soc. 61, 246–251 (1976)CrossRefGoogle Scholar
  12. 12.
    Norfolk, T.S., Ruttan, A., Varga, R.S.: A lower bound for the de Bruijn–Newman constant \(\Lambda \) II. In: Gonchar, A.A., Saff, E.B. (eds.) Progress in Approximation Theory, pp. 403–418. Springer, Berlin (1992)CrossRefGoogle Scholar
  13. 13.
    Odlyzko, A.M.: An improved bound for the de Bruijn–Newman constant. Numer. Algorithms 25, 293–303 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Platt, D.J.: Isolating some non-trivial zeros of zeta. Math. Comput. 86, 2449–2467 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Polya, G.: Über trigonometrische Integrale mit nur reelen Nullstellen. J. Reine Angew. Math. 58, 6–18 (1927)zbMATHGoogle Scholar
  16. 16.
  17. 17.
    Polymath, D.H.J.: Zeroes of the heat flow evolution of the Riemann \(\xi \) function at negative times: numerical experiments and heuristic justifications, http://github.com/km-git-acc/dbn_upper_bound/blob/master/Writeup/Sharkfin/sharkfin.pdf
  18. 18.
    Rodgers, B., Tao, T.: The De Bruijn–Newman constant is nonnegative, preprint. arXiv:1801.05914
  19. 19.
    Saouter, Y., Gourdon, X., Demichel, P.: An improved lower bound for the de Bruijn–Newman constant. Math. Comput. 80, 2281–2287 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.

Personalised recommendations