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Hadamard product and density of zeros inside the critical strip

  1. Jun 1, 2006 #1
    Hadamard product and "density" of zeros inside the critical strip..

    If we take the "Hadamard Product" for the Riemann (Xi-function) zeta function:

    [tex] \xi(s)=\xi(0)\prod_{\rho} (1-s/\rho)e^{s/\rho} [/tex]

    taking log at both sides we would have:

    [tex] log\xi(s)-log\xi(0)=\sum_{n=0}^{\infty}D(n)[log(1-s/n)+(s/n)] [/tex]

    Where D(n) is the "density of non trivial zeros" for the Riemann zeta function if we define [tex] N(x)=\sum_{n}^{x}D(n) [/tex] we have the integral equation:

    [tex] log\xi(s)-log\xi(0)=\int_{-\infty}^{\infty}dxN(x)[1/(x-s)+(1/x)-(s/x^{2})] [/tex]

    from the study of this equation we could obtain a formula for N(x) and how the "Non-trivial zeros are distributed along the critical strip.
  2. jcsd
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