On generalized Ramanujan primes

Abstract

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 et al. ( arXiv:0906.0104v1 , 2009) concerning the number of primes in certain intervals holds for every sufficiently large positive integer.