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:
OEIS: A104272
where
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
for all x ≥ Rn.[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 x ≥ Rn.
The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
Note that the integer Rn is necessarily a prime number:
and, hence,
must increase by obtaining another prime at x = Rn. Since
can increase by at most 1,
![{\displaystyle \pi (R_{n})-\pi \left({\frac {R_{n)){2))\right)=n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132830e84861e54d5e773716ea5ea749b6995f01)
Bounds and an asymptotic formula
For all
, the bounds
![{\displaystyle 2n\ln 2n<R_{n}<4n\ln 4n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c655ac93d4ce111845de5b32bca87f3a3ef6a3)
hold. If
, then also
![{\displaystyle p_{2n}<R_{n}<p_{3n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/45a1c454846d8ccf3201adc04cda2b313d95fae2)
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}\leq {\frac {41}{47))\ p_{3n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/b082edbaf9652aadc04df7429a82f3ccca567e87)
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.