Ramanujan prime

Short description: Prime fulfilling an inequality related to the prime-counting function

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:

$\pi (x)-\pi \left(\frac{x}{2}\right)\ge 1,2,3,4,5,\dots \text{ for all }x\ge 2,11,17,29,41,\dots \text{ respectively}$    OEIS: A104272

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 R_n for which $\pi (x)-\pi (x/2)\ge n,$ for all x ≥ R_n. In other words: Ramanujan primes are the least integers R_n for which there are at least n primes between x and x/2 for all x ≥ R_n.

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

Note that the integer R_n is necessarily a prime number: $\pi (x)-\pi (x/2)$ and, hence, $\pi (x)$ must increase by obtaining another prime at x = R_n. 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 $n\ge 1$, the bounds

$2n\ln 2n<R_n<4n\ln 4n$

hold. If $n>1$, then also

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

where p_n is the nth prime number.

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

R_n ~ p_2n (n → ∞).

All these results were proved by Sondow (2009), except for the upper bound R_n < p_3n 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 R_n ≤ c·p_3n since it is an equality for n = 5.