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,

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

Simple example

The number of primes between 13 and its half (6.5) is three. Those primes are 7, 11 and 13 itself. This does not mean that the third Ramanujan prime is 13. The number of primes between 17 and its half (8.5) is also three. Those primes are 11, 13 and 17 itself. The number of primes between 16 and its half (8) is two, (11 and 13), then the third Ramanujan prime could not be 13, but it is 17 : ${\displaystyle R_{3}=17}$.

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