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Mathematics of Computation
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
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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.


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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/S0025-5718-99-01104-7
PII: S 0025-5718(99)01104-7
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