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

Kuok Fai Chao; Roger Plymen

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

Number Theory 2009-03-23 v7

Authors: Kuok Fai Chao , Roger Plymen

Abstract

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson. Entering $2,000,000$ Riemann 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 and Hudson, and is narrower by a factor of about 12.

Keywords

bound upper bounds

Cite

@article{arxiv.math/0509312,
  title  = {A new bound for the smallest $x$ with $\pi(x) > li(x)$},
  author = {Kuok Fai Chao and Roger Plymen},
  journal= {arXiv preprint arXiv:math/0509312},
  year   = {2009}
}

Comments

Final version, to be published in the International Journal of Number Theory [copyright World Scientific Publishing Company][www.worldscinet.com/ijnt]

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

Abstract

Keywords

Cite

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