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
Abstract 
References 
Similar Articles 
Additional Information
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.
 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 𝜋(𝑥): the
Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp. 65 (1996), no. 213, 235–245. MR 1322888
(96d:11139), http://dx.doi.org/10.1090/S0025571896006746
 4.
H.
M. Edwards, Riemann’s zeta function, Academic Press [A
subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1974. Pure and Applied Mathematics, Vol. 58. MR 0466039
(57 #5922)
 5.
Ian
Richards, On the incompatibility of two
conjectures concerning primes; a discussion of the use of computers in
attacking a theoretical problem, Bull. Amer.
Math. Soc. 80
(1974), 419–438. MR 0337832
(49 #2601), http://dx.doi.org/10.1090/S000299041974134348
 6.
Richard
H. Hudson, Averaging effects on irregularities in
the distribution of primes in arithmetic progressions, Math. Comp. 44 (1985), no. 170, 561–571. MR 777286
(86h:11074), http://dx.doi.org/10.1090/S00255718198507772867
 7.
A.
E. Ingham, The distribution of prime numbers, Cambridge
Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint
of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
(91f:11064)
 8.
J.
C. Lagarias, V.
S. Miller, and A.
M. Odlyzko, Computing 𝜋(𝑥): the
MeisselLehmer method, Math. Comp.
44 (1985), no. 170, 537–560. MR 777285
(86h:11111), http://dx.doi.org/10.1090/S00255718198507772855
 9.
R.
Sherman Lehman, On the difference
𝜋(𝑥)𝑙𝑖(𝑥), Acta Arith.
11 (1966), 397–410. MR 0202686
(34 #2546)
 10.
J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 18691872.
 11.
J.
van de Lune, H.
J. J. te Riele, and D.
T. Winter, On the zeros of the Riemann zeta
function in the critical strip. IV, Math.
Comp. 46 (1986), no. 174, 667–681. MR 829637
(87e:11102), http://dx.doi.org/10.1090/S00255718198608296373
 12.
H.
L. Montgomery, The zeta function and prime numbers,
Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston,
Ont., 1979), Queen’s Papers in Pure and Appl. Math., vol. 54,
Queen’s Univ., Kingston, Ont., 1980, pp. 1–31. MR 634679
(82k:10047)
 13.
Herman
J. J. te Riele, On the sign of the difference
𝜋(𝑥)𝑙𝑖(𝑥), Math. Comp. 48 (1987), no. 177, 323–328. MR 866118
(88a:11135), http://dx.doi.org/10.1090/S00255718198708661186
 14.
Hans
Riesel, Prime numbers and computer methods for factorization,
Progress in Mathematics, vol. 57, Birkhäuser Boston Inc., Boston,
MA, 1985. MR
897531 (88k:11002)
 15.
Michael
Rubinstein and Peter
Sarnak, Chebyshev’s bias, Experiment. Math.
3 (1994), no. 3, 173–197. MR 1329368
(96d:11099)
 16.
S. Skewes, On the difference , J. London Math. Soc. 8 (1933), 277283.
 17.
S.
Skewes, On the difference
𝜋(𝑥)𝑙𝑖𝑥. II, Proc. London
Math. Soc. (3) 5 (1955), 48–70. MR 0067145
(16,676c)
 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
 9.
 R. Sherman Lehman, On the difference , Acta Arith. 11 (1966), 397410. MR 34:2546
 10.
 J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 18691872.
 11.
 J. van de Lune, J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), 667681. MR 87e:11102
 12.
 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.
 17.
 S. Skewes, On the difference , II, Proc. London Math. Soc. (3) 5 (1955), 4870. MR 16:676c
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (1991):
1104,
11A15,
11M26,
11Y11,
11Y35
Retrieve articles in all journals
with MSC (1991):
1104,
11A15,
11M26,
11Y11,
11Y35
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
