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Approximation to pi(x)

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  • Kermit Rose
    http://primes.utm.edu/howmany.shtml Consequence One: You can Approximate pi(x) with x/(log x - 1) http://en.wikipedia.org/wiki/Prime-counting_function
    Message 1 of 2 , Jan 2, 2011
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      http://primes.utm.edu/howmany.shtml

      Consequence One: You can Approximate pi(x) with x/(log x - 1)

      http://en.wikipedia.org/wiki/Prime-counting_function

      http://en.wikipedia.org/wiki/Logarithmic_integral

      So why do we use integral(dt/log_e(t))
      instead of integral(dt/(log_e(t) - 1)
      as the approximation to pi(x)?

      Kermit
    • 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 2 of 2 , Jan 2, 2011
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        --- 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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