Isolating some non-trivial zeros of zeta

Platt, David

doi:10.1090/mcom/3198

Isolating some non-trivial zeros of zeta

Author: David J. Platt
Journal: Math. Comp. 86 (2017), 2449-2467
MSC (2010): Primary 11Y35, 11M26
DOI: https://doi.org/10.1090/mcom/3198
Published electronically: February 13, 2017
MathSciNet review: 3647966
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Abstract: We describe a rigorous algorithm to compute Riemann's zeta function on the half line and its use to isolate the non-trivial zeros of zeta with imaginary part $\leq 30,610,046,000$ to an absolute precision of $\pm 2^{-102}$ . In the process, we provide an independent verification of the Riemann Hypothesis to this height.

[1] Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
[2] ACRC, BlueCrystal Phase 2 User Guide, 2009.
[3] Tom M. Apostol, An elementary view of Euler’s summation formula, Amer. Math. Monthly 106 (1999), no. 5, 409–418. MR 1699259, https://doi.org/10.2307/2589145
[4] M. V. Berry and J. P. Keating, A new asymptotic representation for 𝜁(\frac12+𝑖𝑡) and quantum spectral determinants, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 151–173. MR 1177749, https://doi.org/10.1098/rspa.1992.0053
[5] Jonathan Bober, Zeros of $\zeta (s)$ , 2014, http://www.lmfdb.org/zeros/zeta/.
[6] Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 4, 385–407. MR 2293591
[7] J. L. Brown Jr., On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem, J. Math. Anal. Appl. 18 (1967), 75–84. MR 204952, https://doi.org/10.1016/0022-247X(67)90183-7
[8] J. Arias de Reyna, High precision computation of Riemann’s zeta function by the Riemann-Siegel formula, I, Math. Comp. 80 (2011), no. 274, 995–1009. MR 2772105, https://doi.org/10.1090/S0025-5718-2010-02426-3
[9] H.M. Edwards, Riemann's Zeta Function, Pure and Applied Mathematics, Academic Press Inc., 1974.
[10] X. Gourdon, The $10^{13}$ First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height, http://numbers.computation.free.fr/Constants/Miscellaneous/ zetazeros1e13-1e24.pdf, 2010.
[11] D. E. G. Hare, Computing the principal branch of log-Gamma, J. Algorithms 25 (1997), no. 2, 221–236. MR 1478568, https://doi.org/10.1006/jagm.1997.0881
[12] Ghaith Ayesh Hiary, Fast methods to compute the Riemann zeta function, Ann. of Math. (2) 174 (2011), no. 2, 891–946. MR 2831110, https://doi.org/10.4007/annals.2011.174.2.4
[13] Fredrik Johansson et al., mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.19), 2014, http://www.mpmath.org/doc/0.19/index.html.
[14] J. C. Lagarias and A. M. Odlyzko, Computing 𝜋(𝑥): an analytic method, J. Algorithms 8 (1987), no. 2, 173–191. MR 890871, https://doi.org/10.1016/0196-6774(87)90037-X
[15] R. Sherman Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 20 (1970), 303–320. MR 0258768, https://doi.org/10.1112/plms/s3-20.2.303
[16] Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0231516
[17] A.M. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html.
[18] -, The $10^{20}$ -th zero of the Riemann zeta function and 175 million of its neighbors, http://www.dtc.umn.edu/~odlyzko/unpublished/index.html, 1992.
[19] 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, https://doi.org/10.1090/S0002-9947-1988-0961614-2
[20] David J. Platt, Computing $\pi (x)$ Analytically, Math. Comp. 84 (2015), 1521-1535.
[21] David J. Platt, Numerical computations concerning the GRH, Math. Comp. 85 (2016), no. 302, 3009–3027. MR 3522979, https://doi.org/10.1090/S0025-5718-2016-03077-X
[22] D. J. Platt and T. S. Trudgian, An improved explicit bound on |𝜁(\frac12+𝑖𝑡)|, J. Number Theory 147 (2015), 842–851. MR 3276357, https://doi.org/10.1016/j.jnt.2014.08.019
[23] D. J. Platt and T. S. Trudgian, On the first sign change of 𝜃(𝑥)-𝑥, Math. Comp. 85 (2016), no. 299, 1539–1547. MR 3454375, https://doi.org/10.1090/S0025-5718-2015-03021-X
[24] N. Revol and F. Rouillier, A library for arbitrary precision interval arithmetic, 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, 2002.
[25] Michael Rubinstein, lcalc - The L-function calculator, http://code.google.com/p/l-calc/.
[26] Yannick Saouter, Timothy Trudgian, and Patrick Demichel, A still sharper region where 𝜋(𝑥)-𝑙𝑖(𝑥) is positive, Math. Comp. 84 (2015), no. 295, 2433–2446. MR 3356033, https://doi.org/10.1090/S0025-5718-2015-02930-5
[27] Timothy Trudgian, Improvements to Turing’s method, Math. Comp. 80 (2011), no. 276, 2259–2279. MR 2813359, https://doi.org/10.1090/S0025-5718-2011-02470-1
[28] A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc. (3) 3 (1953), 99–117. MR 0055785, https://doi.org/10.1112/plms/s3-3.1.99
[29] James S. Walker, Fast Fourier transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. With 1 IBM-PC floppy disk (5.25 inch; DD). MR 1169681
[30] S. Wedeniwski, ZetaGrid-Computations connected with the Verification of the Riemann Hypothesis, Foundations of Computational Mathematics Conference, Minnesota, USA, 2002.