A new bound for the smallest with
Authors:
Carter Bays and Richard H. Hudson
Journal:
Math. Comp. 69 (2000), 12851296
MSC (1991):
Primary 1104, 11A15, 11M26, 11Y11, 11Y35
Published electronically:
May 4, 1999
MathSciNet review:
1752093
Fulltext PDF Free Access
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Abstract: Let denote the number of primes and let denote the usual integral logarithm of . We prove that there are at least integer values of in the vicinity of with . This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of , where appears to exceed by more than . The plots strongly suggest, although upper bounds derived to date for are not sufficient for a proof, that exceeds for at least integers in the vicinity of . If it is possible to improve our bound for by finding a sign change before , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of and find that as departs from the region in the vicinity of , the density is , and that it varies from this by no more than over the next integers. This should be compared to Rubinstein and Sarnak.
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 1.
 Carter Bays and Richard H. Hudson, Zeroes of Dirichlet functions and irregularities in the distribution of primes, Math. Comp., posted on March 10, 1999, PII: S00255718(99)011059 (to appear in print).
 2.
 Carter Bays, Kevin B. Ford, Richard H. Hudson, and Michael Rubinstein, Zeros of Dirichlet functions near the real axis and Chebyshev's bias, submitted for publication.
 3.
 M. Deléglise and J. Rivat, Computing : The Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp. 65 (1996), 235245. MR 96d:11139
 4.
 H. M. Edwards Riemann's zeta function, Academic Press, New York, 1974. MR 57:5922
 5.
 Ian Richards, On the incompatibility of two conjectures concerning primes, Bull. Amer. Math. Soc. 80 (1974), 419438. MR 49:2601
 6.
 Richard H. Hudson, Averaging effects on irregularities in the distribution of primes in arithmetic progressions, Math. Comp. 44 (1985), 561571. MR 86h:11074
 7.
 A. E. Ingham, The distribution of prime numbers, Cambridge University Press, Cambridge, 1932; reprint, 1990. MR 91f:11064
 8.
 Jeff Lagarias, Victor Miller, and Andrew Odlyzko, Computing : The MeisselLehmer method, Math. Comp. 44 (1985), 537560. MR 86h:11111
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 R. Sherman Lehman, On the difference , Acta Arith. 11 (1966), 397410. MR 34:2546
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 J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 18691872.
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 H. L. Montgomery, The zeta function and prime numbers, Proc. Queen's Number Theory Conf., 1979 Queens Papers in Pure and Applied Mathematics, v. 54, Queen's Univ., Kingston, Ont., 1980, pp. 131. MR 82k:10047
 13.
 Herman J. J. te Riele, On the sign of the difference , Math. Comp. 48 (1987), 323328. MR 88a:11135
 14.
 Hans Riesel, Prime numbers and computer methods for factorization, Birkhäuser, Boston, 1985. MR 88k:11002
 15.
 Michael Rubinstein and Peter Sarnak, Chebyshev's bias, Experimental Mathematics 3 (1994), 173197. MR 96d:11099
 16.
 S. Skewes, On the difference , J. London Math. Soc. 8 (1933), 277283.
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Additional Information
Carter Bays
Affiliation:
Department of Computer Science, University of South Carolina, Columbia, South Carolina 29208
Email:
bays@cs.sc.edu
Richard H. Hudson
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email:
hudson@math.sc.edu
DOI:
http://dx.doi.org/10.1090/S0025571899011047
PII:
S 00255718(99)011047
Received by editor(s):
June 30, 1997
Received by editor(s) in revised form:
April 1, 1998, and July 7, 1998
Published electronically:
May 4, 1999
Article copyright:
© Copyright 2000
American Mathematical Society
