Ramanujan prime

Not to be confused with Hardy–Ramanujan number.

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:

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

$R_{n}\leq {\frac {41}{47}}\ p_{3n}$

which is the optimal form of R_n ≤ c·p_3n since it is an equality for n = 5.