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## 2006.94: A new bound for the smallest x with \pi(x) > \li(x)

2006.94: Kuok Fai Chao and Roger Plymen (2009) A new bound for the smallest x with \pi(x) > \li(x). International Journal of Number Theory. pp. 1-12.

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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 Number of primes up to x. Logarithmic integral. Zeta zeros. A bound for the first crossover. MSC 2000 > 11 Number theory 2006.94 Professor Roger Plymen 23 February 2010