International Journal of Number Theory May 19, 2010 1742.pdfVIP

International Journal of Number Theory Vol. 6, No. 3 (2010) 681–690 c World Scientific Publishing Company DOI: 10.1142/S1793042110003125 A NEW BOUND FOR THE SMALLEST x WITH π (x) li(x) KUOK FAI CHAO∗ and ROGER PLYMEN† School of Mathematics, Alan Turing Building Manchester University, Manchester M13 9PL, UK ∗ kchao@maths.manchester.ac.uk † plymen@manchester.ac.uk Received 5 September 2008 Accepted 3 March 2009 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 × 10151 . There are at least 10154 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. Keywords : Riemann zeros; Skewes number; primes; logarithmic integral; first crossover. Mathematics Subject Classification 2010: 11M26, 11N05, 11Y35 1. Introduction Let π (x) denote the number of primes less than or equal to x, and let li(x) denote the logarithmic integral. The notation f (x) = Ω±g (x) means that lim sup f (x)/g (x) 0, x→∞ lim inf f (x)/g (x) 0. x→∞ There was, in 1914, overwhelming numerical evidence that π (x) li(x) for all x. In spite of this, Littlewood [9] announced that