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

A new bound for the smallest x with ?(x) > li(x)
A new bound for the smallest x with ?(x) > li(x)
We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson [2]. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [exp (727.951858), exp (727.952178)] for which ?(x) - li(x) > 3.2 × 10151. There are at least 10154 successive integers x in this interval for which ?(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.
1793-0421
681-690
Chao, Kuok Fai
3f2c9398-bfbb-4430-85b6-af9e7644fe58
Plymen, Roger
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Chao, Kuok Fai
3f2c9398-bfbb-4430-85b6-af9e7644fe58
Plymen, Roger
76de3dd0-ddcb-4a34-98e1-257dddb731f5

Chao, Kuok Fai and Plymen, Roger (2010) A new bound for the smallest x with ?(x) > li(x). International Journal of Number Theory, 6 (3), 681-690. (doi:10.1142/S1793042110003125).

Record type: Article

Abstract

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson [2]. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [exp (727.951858), exp (727.952178)] for which ?(x) - li(x) > 3.2 × 10151. There are at least 10154 successive integers x in this interval for which ?(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.

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Published date: 2010

Identifiers

Local EPrints ID: 173715
URI: https://eprints.soton.ac.uk/id/eprint/173715
ISSN: 1793-0421
PURE UUID: 0419309b-4446-41a3-bf8f-47fd97c98a98

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Date deposited: 09 Mar 2011 15:09
Last modified: 19 Jul 2019 23:11

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