Ramanujan prime

In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev.^[1] At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

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

where ${\displaystyle \pi (x)}$ 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 ${\displaystyle \pi (x)-\pi (x/2)\geq n,}$ for all x ≥ R_n.^[2] 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: ${\displaystyle \pi (x)-\pi (x/2)}$ and, hence, ${\displaystyle \pi (x)}$ must increase by obtaining another prime at x = R_n. Since ${\displaystyle \pi (x)-\pi (x/2)}$ can increase by at most 1,

${\displaystyle \pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.}$

For all ${\displaystyle n\geq 1}$, the bounds

${\displaystyle 2n\ln 2n<R_{n}<4n\ln 4n}$

hold. If ${\displaystyle n>1}$, then also

${\displaystyle p_{2n}<R_{n}<p_{3n}}$

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),^[3] except for the upper bound R_n < p_3n which was conjectured by him and proved by Laishram (2010).^[4] The bound was improved by Sondow, Nicholson, and Noe (2011)^[5] to

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

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