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Re: Approximation to pi(x)

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  • djbroadhurst
    ... Exercise: Find the best value of c such that the derivative of x/(log(x)-c) approximates 1/log(x) when log(x) is large. Solution: The derivative
    Message 1 of 2 , Jan 2, 2011
      --- In primenumbers@yahoogroups.com,
      Kermit Rose <kermit@...> wrote:

      > You can Approximate pi(x) with x/(log x - 1)
      > So why do we use integral(dt/log_e(t))

      Exercise: Find the best value of c such that the derivative
      of x/(log(x)-c) approximates 1/log(x) when log(x) is large.

      Solution: The derivative x/(log(x)-c) is
      1/(log(x)-c) - 1/(log(x)-c)^2
      and expanding in 1/log(x) we obtain
      1/log(x) + (c-1)/log(x)^2 + O(1/log(x)^3)
      so the best approximation is with c = 1.

      David
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