Generalized Ramanujan Primes

Mathematics – Number Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

13 pages, 2 tables, to appear in the CANT 2011 Conference Proceedings. This is version 2.0. Changes: fixed typos, added refere

Scientific paper

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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Generalized Ramanujan Primes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized Ramanujan Primes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized Ramanujan Primes will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-508376

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.