Are There Any Lower Bounds for Skewes' Number?

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SUMMARY

The discussion centers on the search for lower bounds of Skewes' number in relation to the prime-counting function, pi(x), and the logarithmic integral, li(x). It is established that while upper bounds have been identified, notably around 1.39e316, lower bounds remain less significant, with Tadej Kotnik's 2007 paper indicating a lower bound of 10^14 for the first crossover of li(x) and pi(x). Additionally, Bays and Hudson suggest potential earlier crossover points, with a conservative estimate around 8e175. Direct verification is currently the only known method for establishing these bounds.

PREREQUISITES
  • Understanding of prime-counting functions and their significance in number theory.
  • Familiarity with the logarithmic integral function, li(x).
  • Knowledge of Skewes' number and its historical context in mathematics.
  • Basic comprehension of analytic number theory and its methodologies.
NEXT STEPS
  • Research the implications of Tadej Kotnik's findings in "The prime-counting function and its analytic approximations" (2007).
  • Explore the methodologies used for direct verification of crossover points between li(x) and pi(x).
  • Investigate the historical significance of Skewes' number in relation to prime number distribution.
  • Examine the work of Bays and Hudson for insights on smaller crossover points and their mathematical proofs.
USEFUL FOR

Mathematicians, number theorists, and researchers interested in prime number distribution and the analytical methods used to explore bounds related to Skewes' number.

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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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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