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Approximating pi(x) using the zeros of the Riemann zeta function

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  • richard_in_reading
    I m trying to understand how the zeros of the zeta function can be used to approximate pi(x). From Tomás Oliveira e Silva s webpage I understand that if we
    Message 1 of 2 , Apr 28, 2008
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      I'm trying to understand how the zeros of the zeta function can be
      used to approximate pi(x).

      From Tomás Oliveira e Silva's webpage I understand that if we define
      H(x)=2*(li(x)-pi(x))/pi(sqrt(x))
      which can be thought of as way of displaying the normalised
      irregularities in the distribution. It can be approximated by
      H(x)=1+2*sum(sin(t*ln(x))/t) where t are the zeros of the zeta
      function
      I've plotted H(x) and its approximation using the zeroes for primes
      up to 1000000 and the wiggly shape of both functions is the same but
      there's an offset between the two graphs which is neither a constant
      increment or a constant ratio.

      Is this correct or am I missing something?

      Perhaps if I could get hold of a copy of "Zeroes of Dirichlet L-
      functions and irregularities in the distribution of primes" by Carter
      Bays and Richard H. Hudson I could work out where I went wrong. Does
      anyone have a copy?

      Cheers!

      Richard
    • Andrey Kulsha
      ... http://en.wikipedia.org/wiki/Prime-counting_function#Formulas_for_prime-counting_functions
      Message 2 of 2 , Apr 28, 2008
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        > Is this correct or am I missing something?

        http://en.wikipedia.org/wiki/Prime-counting_function#Formulas_for_prime-counting_functions
        http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm

        :-)

        Hope that helps,

        Andrey

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