# 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.[1] At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

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

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 Rn for which $\displaystyle{ \pi(x) - \pi(x/2) \ge n, }$ for all xRn.[2] 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 xRn.

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

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

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

## Bounds and an asymptotic formula

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

$\displaystyle{ 2n\ln2n \lt R_n \lt 4n\ln4n }$

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

$\displaystyle{ p_{2n} \lt R_n \lt p_{3n} }$

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),[3] except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010).[4] The bound was improved by Sondow, Nicholson, and Noe (2011)[5] to

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

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

## References

1. Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society 11: 181–182
2. Jonathan Sondow. "Ramanujan Prime".
3. Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly 116 (7): 630–635, doi:10.4169/193009709x458609
4. Laishram, S. (2010), "On a conjecture on Ramanujan primes", International Journal of Number Theory 6 (8): 1869–1873, doi:10.1142/s1793042110003848 .
5. Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps", Journal of Integer Sequences 14: 11.6.2, Bibcode2011arXiv1105.2249S