Numerical computations concerning the GRH

Platt, David

doi:10.1090/mcom/3077

Numerical computations concerning the GRH
HTML articles powered by AMS MathViewer

by David J. Platt;

Math. Comp. 85 (2016), 3009-3027

DOI: https://doi.org/10.1090/mcom/3077

Published electronically: January 15, 2016

PDF | Request permission

Abstract:

We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus $q\leq 400 000$. We check, to height, $\textrm {max}\left (\frac {10^8}{q},\frac {A\cdot 10^7}{q}+200\right )$ with $A=7.5$ in the case of even characters and $A=3.75$ for odd characters. In addition we confirm that no Dirichlet L-function with a modulus $q\leq 2 000 000$ vanishes at its central point.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 434929
L. Bluestein, A linear filtering approach to the computation of discrete Fourier transform, IEEE Transactions on Audio and Electroacoustics, 18(4):451–455, 1970.
Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 4, 385–407. MR 2293591, DOI 10.1080/10586458.2006.10128976
William L. Briggs and Van Emden Henson, The DFT, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. An owner’s manual for the discrete Fourier transform. MR 1322049
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 10.1016/0022-247X(67)90183-7
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.
H.A. Helfgott, Minor arcs for Goldbach’s problem, arXiv preprint arXiv:1205.5252, 2012.
H.A. Helfgott, Major arcs for Goldbach’s problem, arXiv preprint arXiv:1305.2897, 2013.
IEEE, IEEE Standard for Binary Floating-Point Arithmetic, IEEE Std 754-1985., 1985.
B. Lambov, Reliable Implementation of Real Number Algorithms: Theory and Practice, chapter Interval Arithmetic Using SSE-2, Lecture Notes in Computer Science. Springer, 2008.
Ramon E. Moore, The automatic analysis and control of error in digital computation based on the use of interval numbers, Error in Digital Computation, Vol. 1 (Proc. Advanced Sem. Conducted by Math. Res. Center, U.S. Army, Univ. Wisconsin, Madison, Wis., 1964) Wiley, New York-London-Sydney, 1965, pp. 61–130. MR 176614
Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1966. MR 231516
J.M. Muller, CRlibm. http://lipforge.ens-lyon.fr/www/crlibm/, 2010.
Hans Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z. 72 (1959/60), 192–204. MR 117200, DOI 10.1007/BF01162949
N. Revol and F. Rouillier, A library for arbitrary precision interval arithmetic, In 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, 2002.
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
Timothy Trudgian, Improvements to Turing’s method, Math. Comp. 80 (2011), no. 276, 2259–2279. MR 2813359, DOI 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 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