  We consider the determinantal point process with the confluent hypergeometric
kernel. This process is a universal point process in random matrix theory and
describes the distribution of eigenvalues of large random Hermitian matrices
near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we
study the generating function of this process on any given number of intervals.
It can be expressed as the Fredholm determinant of the confluent hypergeometric
kernel with $n$ discontinuities. In this paper, we derive an integral
representation for the determinant by using the Hamiltonian of the coupled
Painlev\'e V system. By evaluating the total integral of the Hamiltonian, we
obtain the asymptotics of the determinant as the $n$ discontinuities tend to
infinity up to and including the constant term. Here the constant term is
expressed in terms of the Barnes $G$-function.
