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

Kuok Fai Chao; Roger Plymen

doi:10.1142/S1793042110003125

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

Research output: Contribution to journal › Article › peer-review

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 π(x) - li(x) > 3.2 × 10 151. There are at least 10 154 successive integers x in this interval for which π(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. © 2010 World Scientific Publishing Company.

Original language	English
Pages (from-to)	681-690
Number of pages	9
Journal	International Journal of Number Theory
Volume	6
Issue number	3
DOIs	https://doi.org/10.1142/S1793042110003125
Publication status	Published - May 2010

Keywords

First crossover
Logarithmic integral
Primes
Riemann zeros
Skewes number

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10.1142/S1793042110003125

http://dx.doi.org/10.1142/S1793042110003125

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title = "A new bound for the smallest x with π(x) > li(x)",

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 π(x) - li(x) > 3.2 × 10 151. There are at least 10 154 successive integers x in this interval for which π(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. {\textcopyright} 2010 World Scientific Publishing Company.",

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T1 - A new bound for the smallest x with π(x) > li(x)

AU - Chao, Kuok Fai

AU - Plymen, Roger

PY - 2010/5

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N2 - 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 π(x) - li(x) > 3.2 × 10 151. There are at least 10 154 successive integers x in this interval for which π(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. © 2010 World Scientific Publishing Company.

AB - 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 π(x) - li(x) > 3.2 × 10 151. There are at least 10 154 successive integers x in this interval for which π(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. © 2010 World Scientific Publishing Company.

KW - First crossover

KW - Logarithmic integral

KW - Primes

KW - Riemann zeros

KW - Skewes number

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DO - 10.1142/S1793042110003125

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EP - 690

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 3

ER -

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

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Keywords

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