On generalized Ramanujan primes

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.