Generalized Ramanujan Primes

In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x∕2, x]. This was proved by Chebyshev in 1860 and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R _n such that \(\pi (x) -\pi (x/2) \geq n\) for all x ≥ R _n . In 2009 Sondow called R _n the nth Ramanujan prime and proved the asymptotic behavior R _n  ∼ p _2n (where p _m is the mth prime). He and Laishram proved the bounds p _2n  < R _n  < p _3n , respectively, for n > 1. In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R _c, n such that for all x ≥ R _c, n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R _c, n exists for all n and all c, that \(R_{c,n} \sim p_{ \frac{n} {1-c} }\) as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales but has been shown to fail in some instances on smaller scales.