A goal which I have been pursuing is to understand how number fields are distributed with respects to their invariants. To be more precise I was captivated by the following question:
Let $N(X,n,G)$ be the number of number fields of dimension $n$ where $G$ is the Galois group of its Galois closure, and their discriminants is bounded by $|X|$ up to isomorphism. This question might be natural to ask: For which group $G\leq S\_n$ one might get a positive proportion of number fields when $X\to \infty$. 


From Class Field Theory or one can show it more elementary, using Delone-Faddeev correspondence, for $n=3$, $C\_3$ has a density 0. And also when $G$ is an abelian group then the answer would be same as $C\_3$ by using class field theory. Prof. Manjul Bhargava proved for $n=4$ , $D\_4$ has a positive density, and also he showed for $n=5$ no other groups, except $S\_5$, could have positive density. 


I think, once Manjul told me, one might expect for $n=p$, $p$ is a prime number, the only group which can contribute with positive density is $S\_p$. But I could not even find a heuristic that why this should be true.


Is there a heuristic or even a theorem which can support the above expectation? Or more generally what do we know about $N(X,n,G)$? 



Statistics of Number fields
There are general conjectures and heuristics; see for example a [recent survey](http://math.stanford.edu/~akshay/research/evicm.pdf) by Ellenberg and Venkatesh on this and the function field analog. 


Roughly, there is a (modified) conjecture of Malle asserting that given $G$ the number of respective extensions of discriminant at most $X$ is asymptotically
$$c x^{1/a} (\log x)^b $$
with a specific dependence of $a,b$ on $G$; and there are some heuristics of Bhargava for $c$.
The 'modified' is due to the fact that nowadays conjecture is not exactly the original version of Malle (Klüners gave a counter example to the original some years ago).


This circle of ideas is also linked to the Cohen--Lenstra heuristics on class groups; this is also discussed in this survey. 


