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NOTICE WARNING CONCERNING COPYRIGHT ...shelf2.library.cmu.edu/Tech/6844907.pdfD. H. Lehmer [13,14] performed the first extensive computation of zeros of £( s ) on a digital computer,
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NOTICE WARNING CONCERNING COPYRIGHT RESTRICTIONS: The copyright law of the United States (title 17, U.S. Code) governs the making of photocopies or other reproductions of copyrighted material. Any copying of this document without permission of its author may be prohibited by law.
CMU-CS-78-148
COMPUTATION OF THE ZEROS OF THE RIEMANN ZETA
FUNCTION IN THE CRITICAL STRIP
Richard P. Brent Computing Research Group
Australian National University Canberra, Australia
Department of Computer Science Carnegie-Mellon University
Pittsburgh, Pennsylvania, USA
This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract NOOO14-76-C-0370, NR 044-422.
U N I V E R S I T Y L I B R A R I E S C A J W K I E - M E U O N U N I V E R S I T Y
P I T T S B U R G H . P E N N S Y L V A N I A 1 5 2 1 3
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ABSTRACT
We describe a computation which shows that the Riemann zeta
function £(s) has exactly 70,000,000 zeros a + it in the region
0 < t < 30,549,654. Moreover, all these zeros are simple and lie on
the line a = h. (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 "RosserTs
Let = [g m > g m +j) D e a Gram block which satisfies Rosser's
rule and has length j > 2. We say that B m is of type (j, k) if
1 £ k £ i and [g . , g ,) contains at least two zeros of Z(t). J m+k-l m+k
This is neither an unambiguous nor a complete classification, but it is
sufficient to deal with all nontrivial Gram blocks up to B ^ q q ^ q ,
except for those noted in Table 3. The first occurrences of Gram blocks
of various types are noted in Table 4. No blocks of type (7, 1) or (7, 7)
occur up to B_„ n n n . y 70,000,000
TABLE 4 : First Occurrences of Gram Blocks of Various Types
j
2 2
3 3 3
4 4 4 4
5 5 5 5 5
6 6 6 6 6 6
7 7 7 7 7
22 -
Our program did not explicitly search for pairs of close zeros
of Z(t), but we did detect some such pairs when the program had difficulty
in finding the expected number of sign changes in the Gram block containing
them. For example,
t - t < 0.00053 and max |Z(t)| < 0.00000248 n n te(t ,t )
for n = 41,820,581. This is a more extreme example of the phenomenon
first observed by Lehmer [13,14]. See also Montgomery [17,18,20,21].
Our program regularly printed out the largest value of |Z(g.)|
found so far. For example, Z(g 6 5 3 7 9 3 9 4 ) > 75.6, and the first 42 terms
in the Riemann-Siegel sum (5.1) are positive at this point!
In all cases where an exception to Rosser1s rule was observed,
there was a large local maximum of |Z(t)| nearby. This suggests that
"interesting" regions might be predicted by finding values of t such that
the first few terms in the Riemann-Siegel sum reinforce each other. Pre
liminary computations suggest that this is a promising approach. To verify
the feasibility of such computations for Gram numbers near 1 0 1 0 we ran
our program (slightly modified) from g n^ I + 0 0 to g n + 6 H o o , where n = 10 1 0.
All Gram blocks in this region satisfy Rosser1s rule and, using Theorem 3.2,
we can show that p^, P n + 1> > pn+sooo a r e s i m P l e a n d l i e on the critical
line.
ACKNOWLEDGEMENT
I wish to thank Enrico Bombieri, John Coates, Derrick H. Lehmer,
Hugh Montgomery, Lowell Schoenfeld and Daniel Shanks for their comments and
suggestions; and.the Australian National University for the provision of computer time,
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REFERENCES
1. ANONYMOUS, Sperry Univao 1100 Series Fortran V Library Programmer Reference Manual (UP 7876, rev. 4), Sperry Rand Corp., 1974.
2. R. BACKLUND, MSur les zeros de la fonction £(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 linen, AMS Notices, 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.
7. J. GRAM, "Sur les zeros de la fonction £(s) de Riemann" Acta Math.,v. 27, 1903, pp. 289-304.
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 22 #8679.
9. J. I. HUTCHINSON, "On the roots of the Riemann zeta-function", Trans. Amer. Math. Soc, v. 27, 1925, pp. 49-60.
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.)
11. R. S. LEHMAN, "Separation of zeros of the Riemann zeta-function", Math. Comp., v. 20, 1966, pp. 523-541.
12. R. S. LEHMAN, "On the difference "rr(x) - li(x)", Acta Arithnetica, v.ll, 1966, pp. 397-410.
13. D. H. LEHMER, "Extended computation of the Riemann zeta-function", Mathematika, v. 3, 1956, pp. 102-108; ROT 108, MTAC, v. 11, 1957, p. 273. MR 19, p. 121.
- 24 -
14. D. H. LEHMER, "On the roots of the Riemann zeta-function", Acta Math., v. 95, 1956, pp. 291-298; RMT 52, M!4C, v.ll, 1957, pp. 107-108. MR 19, p. 121.
15. J. E. LITTLEWOOD, "On the zeros of the Riemann zeta-function", Proc. Cambridge Phil. Soc, v. 22, 1924, pp. 295-318.
16. N. A. MELLER, "Computations connected with the'check of Riemannfs hypothesis", Dokl. Akad. Nauk SSSR, v. 123, 1958, pp. 246-248 (in Russian). MR 20#6396.
17. H. L. MONTGOMERY, "The pair correlation of zeros of the zeta function", Proa. Sympos. Pure Math., v.24, Amer. Math. Soc, Providence, Rhode Island, 1973, pp. 181-193.
18. H. L. MONTGOMERY, "Distribution of the zeros of the Riemann zeta function", Proo. International Congress of Mathematicians, Vancouver, 1974, pp. 379-381.
19. H. L. MONTGOMERY, "Extreme values of the Riemann zeta function", Comment. Math. Eelvetici, v.52, 1977, pp. 511-518.
20. H. L. MONTGOMERY, "Problems concerning prime numbers", Proo. Sympos. Pure Math., v. 28, Amer. Math. Soc, Providence, Rhode Island, 1976, pp. 307-310.
21. H. L. MONTGOMERY § P. J. WEINBERGER, "Notes on small class numbers", Acta Arith., v. 24, 1974, pp. 529-542.
22. B. RIEMANN, Gesarrmelte Werke, Teubner, Leipzig, 1892. (Reprinted by Dover Books, New York, 1953.)
23. J. B. ROSSER, J. M. YOHE 5 L. SCHOENFELD, "Rigorous computation and the zeros of the Riemann zeta-function", Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), v. 1, North-Holland, Amsterdam, 1969, pp. 70-76. MR 41 #2892. Errata: Math. Corrvp., v.29, 1975, p.243.
24. A. SELBERG, "Contributions to the theory of the Riemann zeta-function", Arch. Math. Naturvid., v. 48, 1946, pp. 99-155.
25. C. L. SIEGEL, "Uber Riemanns Nachlass zur analytischen Zahlen-theorie", Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2, 1932, pp. 45-48. (Also in Gesarrmelte Abhandltxngen, v. 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.
- 25 -
27. E. C. TITCHMARSH, The Theory of the Riemann Zeta-Function, Oxford Univ. Press, New York, 1951. MR 13, 741.
28. A. M. TURING, ?lSome calculations of the Riemann zeta-function1
Proc. London Math. Soc, ser. 3, v. 3, 1953, pp. 99-117.
29. J. H. WILKINSON, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, New Jersey, 1963.
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4. TITLE (and Subtitle)
COMPUTATION OF THE ZEROS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP
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4. TITLE (and Subtitle)
COMPUTATION OF THE ZEROS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP 6. PERFORMING ORG. REPORT NUMBER I
7. A U T H O R ^ ;
Richard P. Brent
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N00014-76-Cx0370
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