  In 1845, Bertrand conjectured that for all integers $x\ge2$, 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 $n\ge1$, there is a
(smallest) prime $R_n$ such that $\pi(x)- \pi(x/2) \ge n$ for all $x \ge R_n$.
In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic
behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present
paper, we generalize the interval of interest by introducing a parameter $c \in
(0,1)$ and defining the $n$th $c$-Ramanujan prime as the smallest integer
$R_{c,n}$ such that for all $x\ge R_{c,n}$, 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 $n\to\infty$, 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 was first observed by Sondow, Nicholson,
and Noe in the case $c = 1/2$. 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.
