%0 Generic %1 amersi2011generalized %A Amersi, Nadine %A Beckwith, Olivia %A Miller, Steven J. %A Ronan, Ryan %A Sondow, Jonathan %D 2011 %K combinatorial generalized ramanujan mathematics %T Generalized Ramanujan Primes %U http://arxiv.org/abs/1108.0475 %X 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.

@misc{amersi2011generalized, abstract = {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.}, added-at = {2013-12-23T06:59:58.000+0100}, author = {Amersi, Nadine and Beckwith, Olivia and Miller, Steven J. and Ronan, Ryan and Sondow, Jonathan}, biburl = {https://www.bibsonomy.org/bibtex/2513e3949c9d704b7de0557740b91e548/aeu_research}, description = {Generalized Ramanujan Primes}, interhash = {6d801136f0c730f687360fb897025add}, intrahash = {513e3949c9d704b7de0557740b91e548}, keywords = {combinatorial generalized ramanujan mathematics}, note = {cite arxiv:1108.0475Comment: 13 pages, 2 tables, to appear in the CANT 2011 Conference Proceedings. This is version 2.0. Changes: fixed typos, added references to OEIS sequences, and cited Shevelev's preprint}, timestamp = {2013-12-23T08:22:34.000+0100}, title = {Generalized Ramanujan Primes}, url = {http://arxiv.org/abs/1108.0475}, year = 2011 }