Ramanujan prime

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:

π ( x ) − π ( x / 2 ) ≥ 1 , 2 , 3 , 4 , 5 , …  for all  x ≥ 2 , 11 , 17 , 29 , 41 , …  respectively  A104272

where π ( 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 π ( x ) − π ( x / 2 ) ≥ n , for all x ≥ R_n. 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. Equivalently.

Note that the integer R_n is necessarily a prime number: π ( x ) − π ( x / 2 ) and, hence, π ( x ) must increase by obtaining another prime at x = R_n. Since π ( x ) − π ( x / 2 ) can increase by at most 1,

π ( R n ) − π ( R n 2 ) = n .

Bounds and an asymptotic formula

For all n ≥ 1 , the bounds

2 n ln ⁡ 2 n < R n < 4 n ln ⁡ 4 n

hold. If n > 1 , then also

p 2 n < R n < p 3 n

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), except for the upper bound R_n < p_3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to

R n ≤ 41 47   p 3 n

which is the optimal form of R_n ≤ c·p_3n since it is an equality for n = 5.

In a different direction, Axler showed that

R n < p ⌈ t ⋅ n ⌉

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

α = 1 + 3 ln ⁡ n + ln ⁡ ln ⁡ n − 4

then R n < p ⌊ 2 n α ⌋ for all n > 241 , where ⌊ ⋅ ⌋ is the floor function. For large n, the bound is smaller and thus better than p ⌊ 2 n c ⌋ for any fixed constant c > 1 .

In 2016, Shichun Yang and Alain Togbe establish the estimates of the upper and lower bounds of Ramanujan primes R n when n is big: if n > 10 300 and R n = p s , then

β < s < α ,

where

α = 2 n ( 1 + ln ⁡ 2 ln ⁡ n − ln ⁡ 2 ln ⁡ ln ⁡ n − ln 2 ⁡ 2 − ln ⁡ 2 − 0.13 ln 2 ⁡ n ) , β = 2 n ( 1 + ln ⁡ 2 ln ⁡ n − ln ⁡ 2 ln ⁡ ln ⁡ n − ln 2 ⁡ 2 − ln ⁡ 2 + 0.11 ln 2 ⁡ n ) .

Generalized Ramanujan primes

Given a constant c between 0 and 1, the nth c-Ramanujan prime is defined as the smallest integer R_c,n with the property that for any integer x ≥ R_c,n there are at least n primes between cx and x, that is, π ( x ) − π ( c x ) ≥ n . In particular, when c = 1/2, the nth 1/2-Ramanujan prime is equal to the nth Ramanujan prime: R_0.5,n = R_n.

For c = 1/4 and 3/4, the sequence of c-Ramanujan primes begins

R_0.25,n = 2, 3, 5, 13, 17, ...  A193761,R_0.75,n = 11, 29, 59, 67, 101, ...  A193880.

It is known that, for all n and c, the nth c-Ramanujan prime R_c,n exists and is indeed prime. Also, as n tends to infinity, R_c,n is asymptotic to p_{n/(1 − c)}

R_c,n ~ p_{n/(1 − c)} (n → ∞)

where p_{n/(1 − c)} is the ⌊ n/(1 − c) ⌋ th prime and ⌊ . ⌋ is the floor function.

Ramanujan prime corollary

2 p i − n > p i  for  i > k  where  k = π ( p k ) = π ( R n ) ,

i.e. p_k is the kth prime and the nth Ramanujan prime.

This is very useful in showing the number of primes in the range [p_k, 2p_i−n] is greater than or equal to 1. By taking into account the size of the gaps between primes in [p_i−n,p_k], one can see that the average prime gap is about ln(p_k) using the following R_n/(2n) ~ ln(R_n).

Proof of Corollary:

If p_i > R_n, then p_i is odd and p_i − 1 ≥ R_n, and hence π(p_i − 1) − π(p_i/2) = π(p_i − 1) − π((p_i − 1)/2) ≥ n. Thus p_i − 1 ≥ p_i−1 > p_i−2 > p_i−3 > ... > p_i−n > p_i/2, and so 2p_i−n > p_i.

An example of this corollary:

With n = 1000, R_n = p_k = 19403, and k = 2197, therefore i ≥ 2198 and i−n ≥ 1198. The smallest i − n prime is p_i−n = 9719, therefore 2p_i−n = 2 × 9719 = 19438. The 2198th prime, p_i, is between p_k = 19403 and 2p_i−n = 19438 and is 19417.

The left side of the Ramanujan Prime Corollary is the  A168421; the smallest prime on the right side is  A168425. The sequence  A165959 is the range of the smallest prime greater than p_k. The values of π ( R n ) are in the  A179196.

The Ramanujan Prime Corollary is due to John Nicholson.

Srinivasan's Lemma states that p_k−n < p_k/2 if R_n = p_k and n > 1. Proof: By the minimality of R_n, the interval (p_k/2,p_k] contains exactly n primes and hence p_k−n < p_k/2.