Improvements to Turing’s method

Trudgian, Timothy

doi:10.1090/S0025-5718-2011-02470-1

Improvements to Turing's method

Author: Timothy Trudgian
Journal: Math. Comp. 80 (2011), 2259-2279
MSC (2010): Primary 11M06, 11R42; Secondary 11M26
DOI: https://doi.org/10.1090/S0025-5718-2011-02470-1
Published electronically: March 1, 2011
MathSciNet review: 2813359
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This article improves the estimate of the size of the definite integral of , the argument of the Riemann zeta-function. The primary application of this improvement is Turing's Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet -functions and of Dedekind zeta-functions.

1. R. J. Backlund.
Sur les zéros de la fonction $\zeta(s)$ de Riemann.
Comptes rendus de l'Académie des sciences, 158:1979-1982, 1914.
2. Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 4, 385–407. MR 2293591
3. Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, https://doi.org/10.1090/S0025-5718-1979-0537983-2
4. Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
5. H. M. Edwards, Riemann’s zeta function, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 58. MR 0466039
6. 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/zetazeros1e13-1e2 4.pdf, 2004.
7. M. N. Huxley, Exponential sums and the Riemann zeta function. V, Proc. London Math. Soc. (3) 90 (2005), no. 1, 1–41. MR 2107036, https://doi.org/10.1112/S0024611504014959
8. A. A. Karatsuba and M. A. Korolev.
Approximation of an exponential sum by a shorter one.
Doklady Mathematics, 75(1):36-38, 2007.
9. Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723
10. R. Sherman Lehman, Separation of zeros of the Riemann zeta-function, Math. Comp. 20 (1966), 523–541. MR 203909, https://doi.org/10.1090/S0025-5718-1966-0203909-5
11. 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
12. Hans Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z 72 (1959/1960), 192–204. MR 0117200, https://doi.org/10.1007/BF01162949
13. Hans Rademacher, Topics in analytic number theory, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman; Die Grundlehren der mathematischen Wissenschaften, Band 169. MR 0364103
14. Robert Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415–440, S17–S23. MR 1195435, https://doi.org/10.1090/S0025-5718-1993-1195435-0
15. Atle Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), no. 5, 89–155. MR 20594
16. E. C. Titchmarsh.
The zeros of the Riemann zeta-function.
Proceedings of the Royal Society Series A, 151:234-255, 1935.
17. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
18. Emmanuel Tollis, Zeros of Dedekind zeta functions in the critical strip, Math. Comp. 66 (1997), no. 219, 1295–1321. MR 1423079, https://doi.org/10.1090/S0025-5718-97-00871-5
19. T. Trudgian.
Gram's Law fails a positive proportion of the time.
arxiv.org/abs/0811.0883, 2008.
20. 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
21. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469