Hadamard product and density of zeros inside the critical strip

1. Jun 1, 2006

eljose

Hadamard product and "density" of zeros inside the critical strip..

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

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

taking log at both sides we would have:

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

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

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

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.

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