Ramanujan prime

In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.

Origins and definition
In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:


 * $$\pi(x) - \pi\left( \frac x 2 \right) \ge 1,2,3,4,5,\ldots \text{ for all } x \ge 2, 11, 17, 29, 41, \ldots \text{ respectively}$$

where $$\pi(x)$$ is the prime-counting function, equal to the number of primes less than or equal to x.

The converse of this result is the definition of Ramanujan primes:


 * The nth Ramanujan prime is the least integer Rn for which $$\pi(x) - \pi(x/2) \ge n,$$ for all x ≥ Rn. In other words: Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all x ≥ Rn.

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.

Note that the integer Rn is necessarily a prime number: $$\pi(x) - \pi(x/2)$$ and, hence, $$\pi(x)$$ must increase by obtaining another prime at x = Rn. Since $$\pi(x) - \pi(x/2)$$ can increase by at most 1,


 * $$ \pi(R_n) - \pi\left( \frac{R_n} 2 \right) = n. $$

Bounds and an asymptotic formula
For all $$n \geq 1$$, the bounds


 * $$2n\ln2n < R_n < 4n\ln4n$$

hold. If $$n > 1$$, then also


 * $$p_{2n} < R_n < p_{3n}$$

where pn is the nth prime number.

As n tends to infinity, Rn is asymptotic to the 2nth prime, i.e.,


 * Rn ~ p2n   (n → ∞).

All these results were proved by Sondow (2009), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to


 * $$R_n \le \frac{41}{47} \ p_{3n}$$

which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.