Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

On the zeros of the Riemann zeta function in the critical strip


Author: Richard P. Brent
Journal: Math. Comp. 33 (1979), 1361-1372
MSC: Primary 10H05
MathSciNet review: 537983
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe a computation which shows that the Riemann zeta function $ \zeta (s)$ has exactly 75,000,000 zeros of the form $ \sigma + it$ it in the region $ 0 < t < 32,585,736.4$; all these zeros are simple and lie on the line $ \sigma = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Rosser's rule" are given.


References [Enhancements On Off] (What's this?)

  • [1] ANONYMOUS, Sperry Univac 1100 Series Fortran V Library Programmer Reference Manual (UP 7876, rev. 4), Sperry Rand Corp., 1974.
  • [2] R. BACKLUND, "Sur les zéros de la fonction $ \zeta (s)$ de Riemann," C. R. Acad. Sci. Paris, v. 158, 1914, pp. 1979-1982.
  • [3] R. P. BRENT, "The first 40,000,000 zeros of the Riemann zeta function lie on the critical line," Notices Amer. Math. Soc., v. 24, 1977, p. A-417.
  • [4] R. P. BRENT, "A Fortran multiple precision arithmetic package," ACM Trans. Math. Software, v. 4, 1978, pp. 57-70.
  • [5] F. D. CRARY & J. B. ROSSER, High Precision Coefficients Related to the Zeta Function, MRC Technical Summary Report #1344, Univ. of Wisconsin, Madison, May 1975, 171 pp.; RMT 11, Math. Comp., v. 31, 1977, pp. 803-804.
  • [6] H. M. EDWARDS, Riemann's Zeta Function, Academic Press, New York, 1974. MR 0466039 (57:5922)
  • [7] J. GRAM, "Sur les zéros de la fonction $ \zeta (s)$ de Riemann," Acta Math., v. 27, 1903, pp. 289-304. MR 1554986
  • [8] C. B. HASELGROVE in collaboration with J. C. P. MILLER, Tables of the Riemann Zeta Function, Roy. Soc. Math. Tables No. 6, Cambridge Univ. Press, New York, 1960; RMT 6, Math. Comp., v. 15, 1961, pp. 84-86. MR 0117905 (22:8679)
  • [9] J. I. HUTCHINSON, "On the roots of the Riemann zeta-function," Trans. Amer. Math. Soc., v. 27, 1925, pp. 49-60. MR 1501297
  • [10] A. E. INGHAM, The Distribution of Prime Numbers, Cambridge Tracts in Math. and Math. Phys., No. 30, Cambridge Univ. Press, London and New York, 1932. (Republished by Stechert-Hafner, New York and London, 1964.) MR 0184920 (32:2391)
  • [11] R. S. LEHMAN, "Separation of zeros of the Riemann zeta-function," Math. Comp., v. 20, 1966, pp. 523-541. MR 0203909 (34:3756)
  • [12] R. S. LEHMAN, "On the distribution of zeros of the Riemann zeta-function," Proc. London Math. Soc., (3), v. 20, 1970, pp. 303-320. MR 0258768 (41:3414)
  • [13] D. H. LEHMER, "Extended computation of the Riemann zeta-function," Mathematika, v. 3, 1956, pp. 102-108; RMT 108, MTAC, v. 11, 1957, p. 273. MR 19, 121. MR 0086083 (19:121b)
  • [14] D. H. LEHMER, "On the roots of the Riemann zeta-function," Acta Math., v. 95, 1956, pp. 291-298; RMT 52, MTAC, v. 11, 1957, pp. 107-108. MR 19, 121. MR 0086082 (19:121a)
  • [15] J. E. LITTLEWOOD, "On the zeros of the Riemann zeta-function," Proc. Cambridge Philos. Soc., v. 22, 1924, pp. 295-318.
  • [16] N. A. MELLER, "Computations connected with the check of Riemann's hypothesis," Dokl Akad. Nauk SSSR, v. 123, 1958, pp. 246-248. (Russian) MR 20 #6396. MR 0099960 (20:6396)
  • [17] H. L. MONTGOMERY, "The pair correlation of zeros of the zeta function," Proc. Sympos. Pure. Math., vol. 24, Amer. Math. Soc., Providence, R. I., 1973, pp. 181-193. MR 0337821 (49:2590)
  • [18] H. L. MONTGOMERY, "Distribution of the zeros of the Riemann zeta function," Proc. Internat. Congr. Math., Vancouver, 1974, pp. 379-381. MR 0419378 (54:7399)
  • [19] H. L. MONTGOMERY, "Extreme values of the Riemann zeta function," Comment. Math. Helv., v. 52, 1977, pp. 511-518. MR 0460255 (57:249)
  • [20] H. L. MONTGOMERY, "Problems concerning prime numbers," Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 307-310. MR 0427249 (55:284)
  • [21] H. L. MONTGOMERY & P. J. WEINBERGER, "Notes on small class numbers," Acta Arith., v. 24, 1974, pp. 529-542. MR 0357373 (50:9841)
  • [22] B. RIEMANN, Gesammelte Werke, Teubner, Leipzig, 1876. (Reprinted by Dover, New York, 1953.) MR 0052364 (14:610a)
  • [23] J. B. ROSSER, J. M. YOHE & L. SCHOENFELD, "Rigorous computation and the zeros of the Riemann zeta-function," Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), vol. 1, North-Holland, Amsterdam, 1969, pp. 70-76. MR 41 #2892. Errata: Math. Comp., v. 29, 1975, p. 243. MR 0258245 (41:2892)
  • [24] A. SELBERG, "Contributions to the theory of the Riemann zeta-function," Arch. Math. Naturvid., v. 48, 1946, pp. 99-155. MR 0020594 (8:567e)
  • [25] C. L. SIEGEL, "Über Riemanns Nachlass zur analytischen Zahlentheorie," Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2, 1932, pp. 45-48. (Also in Gesammelte Abhandlungen, Vol. 1, Springer-Verlag, New York, 1966.)
  • [26] E. C. TITCHMARSH, "The zeros of the Riemann zeta-function," Proc. Roy. Soc. London, v. 151, 1935, pp. 234-255; also ibid., v. 157, 1936, pp. 261-263.
  • [27] E. C. TITCHMARSH, The Theory of the Riemann Zeta-Function, Oxford Univ. Press, New York, 1951. MR 13, 741. MR 0046485 (13:741c)
  • [28] A. M. TURING, "Some calculations of the Riemann zeta-function," Proc. London Math. Soc., (3), v. 3, 1953, pp. 99-117. MR 0055785 (14:1126e)
  • [29] J. H. WILKINSON, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, N. J., 1963. MR 0161456 (28:4661)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10H05

Retrieve articles in all journals with MSC: 10H05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0537983-2
Keywords: Gram blocks, Riemann hypothesis, Riemann zeta function, Riemann-Siegel formula, Rosser's rule, Turing's theorem, zeta functions
Article copyright: © Copyright 1979 American Mathematical Society