I was reading about some of the zero density Theorems for my Analytic Number Theory Topics course. While looking over some more complicated results and proofs a few simple questions came up:

First, let $$N(\sigma,T)=|\{ \rho=\beta+i \gamma \text{ }:\text{ } \zeta(\rho)=0, \text{ } 0< \gamma < T, \text{ } \sigma\leq\beta<1 \}|$$ be the cardinality of the set of zeros of the zeta function real part greater than $\sigma$ and imaginary part between 0 and $T$.

We call theorems pertaining to the size of $N(\sigma,T)$ *zero density theorems.* (Complicated bounds on the size of the empty set) These estimates are usually written in the form $N(\sigma , T)\ll T^{A(\sigma)(1-\sigma)+\epsilon}$ where the $\ll$-constant is uniform in $\sigma$. (So the smaller $A(\sigma)$ is, the better, and all such theorems concern the size of $A(\sigma)$)

My questions are:

1) Why do we have a factor of $(1-\sigma)$ in the exponent $T^{A(\sigma)(1-\sigma)}$? It is clear to me what this implies about the density of the zeros, but why does it arise so naturally?

2) The so called density hypothesis is that $A(\sigma) \leq 2$. The best known bound is $A(\sigma) \leq 2.4$ (or possibly 2.3) What is so special about $A(\sigma)\leq 2$? Are there links between this value and certain theorems one may hope to prove? Why does this in particular deserve such a name as "the density hypothesis?"

Thanks a lot!!

(Also, if you have a good reference book or paper that talks about these issues I would be happy to know about it!)