Strange definition of microcanonical partition function I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate over all configurations that have an equal or lower energy than some fixed number $E$. Does this make sense to you?
See eq. 3.4 in this reference.
or somewhat faster see at eq 3.1 here in this paper.
 A: There is quite a big controversy these days about the correct definition of the entropy in the microcanonical ensemble (the debate between the Gibbs and Boltzmann entropy), which is closely related to the question.
Everyone agrees that the correct definition of the density matrix is given by 
$$\rho(E)=\frac{\delta(E-H)}{\omega(E)},$$
where $H$ is the Hamiltonian and 
$$ \omega(E)=Tr\,\delta(E-H).$$
Then the question is the correct definition of the entropy. Boltzmann says $S_B=\ln \omega(E)$, whereas Gibbs argued $S_G=\ln \Omega(E)$ where
$$ \Omega(E)=\int_0^E\omega(e) de.$$
In the text quoted by the OP, the partition function corresponds to $\Omega(E)$.
Note that in most cases, in the thermodynamics limit, both entropies gives the same result. The question arises in the case of small systems and special cases with bounded from above spectra. Hilbert et al. (arXiv:1408.5382 and arXiv:1304.2066) argue that only the Gibbs entropy is thermodynamically consistent. I must say that I find their arguments compelling, and that of their opponents, given in at least two comments of their papers, not at all.
A: Boltzmann entropy is the upper limit of Gibbs entropy. I myself never saw any usage of Gibbs entropy. Gibbs entropy is misused as the sum of pLnp over the states instead of over the microstates as it should. The sum over the states is an approximation that yields the canonical distribution. I recommend reading my post "Microcanonical partition function"
