Let $p$ be a prime number, $C\in \mathbb{N}$ and C is not a square. Then define $$F=\prod_{|z| \leq \sqrt{\frac{x}{2}} \atop |y|\leq \sqrt{\frac{x}{2D}}}{|z^2-Cy^2|}.$$ Note that we omit the term with $z=0=y$. (So the product is not zero.) Now my question: Why does the following inequality hold for $x>x_0$ (i.e. the inequality holds if x is "large enough")?
$$\log(F)>\frac{x}{\sqrt{C}}\log(x)$$
Edit: Something in Selbergs paper was not clear and I think there was also a little mistake. In the product above we have $|y|\leq \sqrt{\frac{x}{2D}}$ and not $|y|\leq \sqrt{\frac{x}{2}p}$. I changed that.