Chao, Kuok Fai and Plymen, Roger (2009) A new bound for the smallest x with \pi(x) > \li(x). International Journal of Number Theory. pp. 1-12. (In Press)
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Official URL: http://arxiv.org/abs/math.NT/0509312
Abstract
We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays & Hudson[2]. Entering 2,000,000 zeta zeros, we prove that there exists x in the interval [exp(727.951858), exp(727.952178)] for which \pi(x) - li(x) > 3.2 \times 10^151. There are at least 10^154 successive integers x in this interval for which \pi(x) > li(x). This interval is strictly a sub-interval of the interval in Bays & Hudson, and is narrower by a factor of about 12.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Number of primes up to x. Logarithmic integral. Zeta zeros. A bound for the first crossover. |
| Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 11 Number theory |
| Depositing User: | Professor Roger Plymen |
| Date Deposited: | 23 Feb 2010 |
| Last Modified: | 20 Oct 2017 14:12 |
| URI: | https://eprints.maths.manchester.ac.uk/id/eprint/1414 |
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A new bound for the smallest x with \pi(x) > \li(x). (deposited 17 May 2006)
- A new bound for the smallest x with \pi(x) > \li(x). (deposited 23 Feb 2010) [Currently Displayed]
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