Origins and definition
In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:
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. 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,
Bounds and an asymptotic formula
For all , the bounds
hold. If , then also
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), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.
In a different direction, Axler showed that
is optimal for t > 48/19, where is the ceiling function.
A further improvement of the upper bounds was done in late 2015 by Anitha Srinivasan and John W. Nicholson. They show that if
then for all , where is the floor function. For large n, the bound is smaller and thus better than for any fixed constant .
Generalized Ramanujan primes
Given a constant c between 0 and 1, the nth c-Ramanujan prime is defined as the smallest integer Rc,n with the property that for any integer x ≥ Rc,n there are at least n primes between cx and x, that is, . In particular, when c = 1/2, the nth 1/2-Ramanujan prime is equal to the nth Ramanujan prime: R0.5,n = Rn.
For c = 1/4 and 3/4, the sequence of c-Ramanujan primes begins
It is known that, for all n and c, the nth c-Ramanujan prime Rc,n exists and is indeed prime. Also, as n tends to infinity, Rc,n is asymptotic to pn/(1 − c)
- Rc,n ~ pn/(1 − c) (n → ∞)
where pn/(1 − c) is the n/(1 − c)th prime and is the floor function.
Ramanujan prime corollary
i.e. pk is the kth prime and the nth Ramanujan prime.
This is very useful in showing the number of primes in the range [pk, 2pi−n] is greater than or equal to 1. By taking into account the size of the gaps between primes in [pi−n,pk], one can see that the average prime gap is about ln(pk) using the following Rn/(2n) ~ ln(Rn).
Proof of Corollary:
If pi > Rn, then pi is odd and pi − 1 ≥ Rn, and hence π(pi − 1) − π(pi/2) = π(pi − 1) − π((pi − 1)/2) ≥ n. Thus pi − 1 ≥ pi−1 > pi−2 > pi−3 > ... > pi−n > pi/2, and so 2pi−n > pi.
An example of this corollary:
With n = 1000, Rn = pk = 19403, and k = 2197, therefore i ≥ 2198 and i−n ≥ 1198. The smallest i − n prime is pi−n = 9719, therefore 2pi−n = 2 × 9719 = 19438. The 2198th prime, pi, is between pk = 19403 and 2pi−n = 19438 and is 19417.
The left side of the Ramanujan Prime Corollary is the OEIS: A168421; the smallest prime on the right side is OEIS: A168425. The sequence OEIS: A165959 is the range of the smallest prime greater than pk. The values of are in the OEIS: A179196.
The Ramanujan Prime Corollary is due to John Nicholson.
Srinivasan's Lemma  states that pk−n < pk/2 if Rn = pk and n > 1. Proof: By the minimality of Rn, the interval (pk/2,pk] contains exactly n primes and hence pk−n < pk/2.
- Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society, 11: 181–182
- Jonathan Sondow. "Ramanujan Prime". MathWorld.
- Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv:0907.5232, doi:10.4169/193009709x458609
- Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, doi:10.1142/s1793042110003848.
- 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
- Axler, Christian (2014). "On generalized Ramanujan primes". The Ramanujan Journal. 39 (2016): 1. arXiv:1401.7179. doi:10.1007/s11139-015-9693-9.
- Srinivasan, Anitha; Nicholson, John (2015). "An Improved Upper Bound For Ramanujan Primes" (PDF). Integers. 15.
- Shichun, Yang; Alain, Togbe (2016). "On the estimates of the upper and lower bounds of Ramanujan primes". The Ramanujan Journal. 40 (2): 245–255. doi:10.1007/s11139-015-9706-8.
- Amersi, N.; Beckwith, O.; Miller, S.J.; Ronan, R.; Sondow, J. (2011), Generalized Ramanujan primes, arXiv:1108.0475, Bibcode:2011arXiv1108.0475A
- Srinivasan, Anitha (2014), "An upper bound for Ramanujan primes" (PDF), Integers, 14