Zusammenfassung
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 $\pi(x)- \pi(x/2) n$ for all $x R_n$.
In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic
behavior $R_n p_2n$ (where $p_m$ is the $m$th prime). In the present
paper, we generalize the interval of interest by introducing a parameter $c ın
(0,1)$ and defining the $n$th $c$-Ramanujan prime as the smallest integer
$R_c,n$ such that for all $xR_c,n$, 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 $R_c,n
p_n1-c$ as $n\toınfty$, 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.
Nutzer