A new bound for the smallest x with \pi(x) > \li(x)

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


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: http://eprints.maths.manchester.ac.uk/id/eprint/1414

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