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

Chao, Kuok Fai and Plymen, Roger (2006) A new bound for the smallest x with \pi(x) > \li(x). math.NT/0509312. pp. 1-16. (Submitted)

There is a more recent version of this item available.
[thumbnail of new.pdf] PDF

Download (155kB)


We reduce the dominant term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays & Hudson. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [1.39792101 \times 10^316, 1.39847603 \times 10^316] for which \pi(x) > \li(x). This interval is strictly a sub-interval of the interval in Bays & Hudson [1], and is narrower by a factor of about 10.

Item Type: Article
Uncontrolled Keywords: Number of primes up to x. Logarithmic integral. Riemann 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: 17 May 2006
Last Modified: 20 Oct 2017 14:12
URI: https://eprints.maths.manchester.ac.uk/id/eprint/106

Available Versions of this Item

Actions (login required)

View Item View Item