In my book about Lattice basis reduction I found the following:

Gaussian Heuristic: for big enough, there are approximately lattice points in of norm , where denotes the volume of the m-dimensional unit ball.

Corollary: for a lattice of rank the shortest nonzero lattice vector has norm at most approximately .

Then they give this example:

Given , and > with large. Small values for the linear form with corresponds to lattices

How large should be taken to compute with and minimal?

Their answer is:

The Gaussian heuristic with expects solutions of size approximately , with approximately .

How do they come up with ?

Using the above Corallary I have that the shortest nonzero lattice vector has norm at most approximately

Then since we have that the norm of .

So and thus is approximately ?

Am I doing this right? I hope someone can explain where the is coming from.

Thank you!