# 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.