Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



A new bound for the smallest $x$ with $\pi(x)>\mathrm{li}(x)$

Authors: Carter Bays and Richard H. Hudson
Journal: Math. Comp. 69 (2000), 1285-1296
MSC (1991): Primary 11-04, 11A15, 11M26, 11Y11, 11Y35
Published electronically: May 4, 1999
MathSciNet review: 1752093
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\pi(x)$ denote the number of primes $\le x$ and let $\mathrm{li}(x)$ denote the usual integral logarithm of $x$. We prove that there are at least $10^{153}$ integer values of $x$ in the vicinity of $1.39822\times 10^{316}$ with $\pi(x)>\mathrm{li}(x)$. This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of $\pi(x)-\mathrm{li}(x)$ 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 $1.617\times 10^{9608}$, where $\pi(x)$ appears to exceed $\mathrm{li}(x)$ by more than $.18x^{\frac 12}/\log x$. The plots strongly suggest, although upper bounds derived to date for $\mathrm{li}(x)-\pi(x)$ are not sufficient for a proof, that $\pi(x)$ exceeds $\mathrm{li}(x)$ for at least $10^{311}$ integers in the vicinity of $1.398\times 10^{316}$. If it is possible to improve our bound for $\pi(x)-\mathrm{li}(x)$ by finding a sign change before $10^{316}$, our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of $\mathrm{li}(x)-\pi(x)$ and find that as $x$ departs from the region in the vicinity of $1.62\times 10^{9608}$, the density is $1-2.7\times 10^{-7}=.99999973$, and that it varies from this by no more than $9\times 10^{-8}$ over the next $10^{30000}$ integers. This should be compared to Rubinstein and Sarnak.

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

  • 1. Carter Bays and Richard H. Hudson, Zeroes of Dirichlet $L$-functions and irregularities in the distribution of primes, Math. Comp., posted on March 10, 1999, PII: S0025-5718(99)01105-9 (to appear in print).
  • 2. Carter Bays, Kevin B. Ford, Richard H. Hudson, and Michael Rubinstein, Zeros of Dirichlet $L$-functions near the real axis and Chebyshev's bias, submitted for publication.
  • 3. M. Deléglise and J. Rivat, Computing $\pi(x)$: The Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp. 65 (1996), 235-245. 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), 419-438. MR 49:2601
  • 6. Richard H. Hudson, Averaging effects on irregularities in the distribution of primes in arithmetic progressions, Math. Comp. 44 (1985), 561-571. 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 $\pi(x)$: The Meissel-Lehmer method, Math. Comp. 44 (1985), 537-560. MR 86h:11111
  • 9. R. Sherman Lehman, On the difference $\pi(x)-\operatorname{li}(x)$, Acta Arith. 11 (1966), 397-410. MR 34:2546
  • 10. J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 1869-1872.
  • 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), 667-681. 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. 1-31. MR 82k:10047
  • 13. Herman J. J. te Riele, On the sign of the difference $\pi(x)-\operatorname{li}(x)$, Math. Comp. 48 (1987), 323-328. 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), 173-197. MR 96d:11099
  • 16. S. Skewes, On the difference $\pi(x)-\operatorname{li}(x)$, J. London Math. Soc. 8 (1933), 277-283.
  • 17. S. Skewes, On the difference $\pi(x)-\operatorname{li}(x)$, II, Proc. London Math. Soc. (3) 5 (1955), 48-70. MR 16:676c

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11-04, 11A15, 11M26, 11Y11, 11Y35

Retrieve articles in all journals with MSC (1991): 11-04, 11A15, 11M26, 11Y11, 11Y35

Additional Information

Carter Bays
Affiliation: Department of Computer Science, University of South Carolina, Columbia, South Carolina 29208

Richard H. Hudson
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

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

American Mathematical Society