High Quality Content by WIKIPEDIA articles! John Edensor Littlewood, Skewes' teacher, proved (in (Littlewood 1914)) that there is such a number (and so, a first such number); and indeed found that the sign of the difference ?(x) ? li(x) changes infinitely often. All numerical evidence then available seemed to suggest that ?(x) is always less than li(x), though mathematicians familiar with Riemann's work on the Riemann zeta function would probably have realized that occasional exceptions were likely by the argument given below (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however, exhibit a concrete such number x. Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating ?(x) < li(x) below e^{e^{e^{79}}}<10^{10^{10^{34}}}.

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