  In this paper we establish several results concerning the generalized
Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give
estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations
and to improvements of the results presently in the literature. Moreover, we
obtain results about the distribution of $k$-Ramanujan primes. In addition, we
find explicit formulae for certain $n$th $k$-Ramanujan primes. As an
application, we prove that a conjecture of Mitra, Paul and Sarkar concerning
the number of primes in certain intervals holds for every sufficiently large
positive integer.
