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.[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,

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

Bounds and an asymptotic formula

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.

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". MathWorld.
3. Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv:0907.5232, doi:10.4169/193009709x458609
4. Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, CiteSeerX 10.1.1.639.4934, doi:10.1142/s1793042110003848.
5. Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps" (PDF), Journal of Integer Sequences, 14: 11.6.2, arXiv:1105.2249, Bibcode:2011arXiv1105.2249S