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 ?(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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