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Skewes' number (sort of)

  1. Sep 23, 2007 #1

    CRGreathouse

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    I think everyone knows the story of the staggeringly huge number Skewes found as an upper bound for the first time that li(x) > pi(x), pi the prime-counting function. Further, it's well-known that less-astronomical bounds have since been found, around 1.39e316.

    I was wondering if good lower bounds are known for this problem. Bays/Hudson in their paper giving the above bound suggest several smaller points where perhaps there are earlier crossovers, the lest of which is around 1e176. Is it known that below this (or rather, a more conservative bound like 8e175, based on the illustration) there are no crossovers?
     
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  3. Sep 27, 2007 #2

    CRGreathouse

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    I found an answer, although it's not nearly as large as I'd hoped for -- I guess this means that direct verification is the only method known. Tadej Kotnik, "The prime-counting function and its analytic approximations" (2007) shows a lower bound of 10^14 for the first crossing of li(x) and pi(x). I found this mentioned on Thomas R. Nicely's site,
    http://www.trnicely.net/pi/tabpi.html#Skewes
     
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