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.

References

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

ACRC, BlueCrystal Phase 2 User Guide, 2009.

Tom M. Apostol, An elementary view of Euler’s summation formula, Amer. Math. Monthly 106 (1999), no. 5, 409–418. MR 1699259, DOI https://doi.org/10.2307/2589145

M. V. Berry and J. P. Keating, A new asymptotic representation for $\zeta (\frac 12+it)$ and quantum spectral determinants, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 151–173. MR 1177749, DOI https://doi.org/10.1098/rspa.1992.0053

Jonathan Bober, Zeros of $\zeta (s)$, 2014, http://www.lmfdb.org/zeros/zeta/.

Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 4, 385–407. MR 2293591

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, DOI https://doi.org/10.1016/0022-247X%2867%2990183-7

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, DOI https://doi.org/10.1090/S0025-5718-2010-02426-3

H.M. Edwards, Riemann’s Zeta Function, Pure and Applied Mathematics, Academic Press Inc., 1974.

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.

D. E. G. Hare, Computing the principal branch of log-Gamma, J. Algorithms 25 (1997), no. 2, 221–236. MR 1478568, DOI https://doi.org/10.1006/jagm.1997.0881

Ghaith Ayesh Hiary, Fast methods to compute the Riemann zeta function, Ann. of Math. (2) 174 (2011), no. 2, 891–946. MR 2831110, DOI https://doi.org/10.4007/annals.2011.174.2.4

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.

J. C. Lagarias and A. M. Odlyzko, Computing $\pi (x)$: an analytic method, J. Algorithms 8 (1987), no. 2, 173–191. MR 890871, DOI https://doi.org/10.1016/0196-6774%2887%2990037-X

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 https://doi.org/10.1112/plms/s3-20.2.303

Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0231516

A.M. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/~odlyzko/zeta_{t}ables/index.html.

---, 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.

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 https://doi.org/10.1090/S0002-9947-1988-0961614-2

David J. Platt, Computing $\pi (x)$ Analytically, Math. Comp. 84 (2015), 1521–1535.

David J. Platt, Numerical computations concerning the GRH, Math. Comp. 85 (2016), no. 302, 3009–3027. MR 3522979, DOI https://doi.org/10.1090/S0025-5718-2016-03077-X

D. J. Platt and T. S. Trudgian, An improved explicit bound on $|\zeta (\frac 12+it)|$, J. Number Theory 147 (2015), 842–851. MR 3276357, DOI https://doi.org/10.1016/j.jnt.2014.08.019

D. J. Platt and T. S. Trudgian, On the first sign change of $\theta (x)-x$, Math. Comp. 85 (2016), no. 299, 1539–1547. MR 3454375, DOI https://doi.org/10.1090/S0025-5718-2015-03021-X

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.

Michael Rubinstein, lcalc – The L-function calculator, http://code.google.com/p/l-calc/.

Yannick Saouter, Timothy Trudgian, and Patrick Demichel, A still sharper region where $\pi (x)-{\rm li}(x)$ is positive, Math. Comp. 84 (2015), no. 295, 2433–2446. MR 3356033, DOI https://doi.org/10.1090/S0025-5718-2015-02930-5

Timothy Trudgian, Improvements to Turing’s method, Math. Comp. 80 (2011), no. 276, 2259–2279. MR 2813359, DOI https://doi.org/10.1090/S0025-5718-2011-02470-1

A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc. (3) 3 (1953), 99–117. MR 55785, DOI https://doi.org/10.1112/plms/s3-3.1.99

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

S. Wedeniwski, ZetaGrid–Computations connected with the Verification of the Riemann Hypothesis, Foundations of Computational Mathematics Conference, Minnesota, USA, 2002.