# Ramanujan Primes: Bounds, Runs, Twins, and Gaps

@article{Sondow2011RamanujanPB, title={Ramanujan Primes: Bounds, Runs, Twins, and Gaps}, author={Jonathan Sondow and John W. Nicholson and Tony D. Noe}, journal={arXiv: Number Theory}, year={2011} }

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below…

## 13 Citations

### On the estimates of the upper and lower bounds of Ramanujan primes

- Mathematics
- 2016

For $$n\ge 1$$n≥1, the nth Ramanujan prime is defined as the least positive integer $$R_{n}$$Rn such that for all $$x\ge R_{n}$$x≥Rn, the interval $$(\frac{x}{2}, x]$$(x2,x] has at least n primes.…

### On the number of primes up to the $n$th Ramanujan prime

- Mathematics
- 2017

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the…

### New upper bounds for Ramanujan primes

- MathematicsGlasnik Matematicki
- 2018

For $n\ge 1$, the $n^{\rm th}$ Ramanujan prime is defined as the smallest positive integer $R_n$ such that for all $x\ge R_n$, the interval $(\frac{x}{2}, x]$ has at least $n$ primes. We show that…

### On generalized Ramanujan primes

- Mathematics
- 2016

In this paper, we establish several results concerning the generalized Ramanujan primes. For $$n\in \mathbb {N}$$n∈N and $$k \in \mathbb {R}_{> 1}$$k∈R>1, we give estimates for the $$n$$nth…

### Generalized Ramanujan Primes

- Mathematics
- 2014

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…

### Derived Ramanujan Primes : R'_{n}

- Mathematics
- 2012

We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity…

### An Upper Bound for Ramanujan Primes

- MathematicsIntegers
- 2014

This work states that for each > 0, there exists an N such that Rn < p[2n(1+ )] for all n > N .

### An Improved Upper Bound for Ramanujan Primes

- Mathematics, Computer ScienceIntegers
- 2015

If α = 2n ( 1 + 3 log n+ log2 n− 4 ) , then it is shown that Rn < p[α] for all n > 241, where pi is the i prime.

### Bounds on the Number of Primes in Ramanujan Interval

- Mathematics
- 2021

The Ramanujan primes are the least positive integers Rn having the property that if m ≥ Rn, then πm − π(m/2) ≥ n. This document develops several bounds related to the Ramanujan primes, sharpening the…

### Why should one expect to find long runs of (non)-Ramanjuan primes ?

- Mathematics
- 2012

Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are…

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