A new bound for the smallest with

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 | 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.

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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:
https://doi.org/10.1090/S0025-5718-99-01104-7

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