Ramanujan prime

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:

$\pi(x) - \pi(x/2)$ ≥ 1, 2, 3, 4, 5, ... for all x ≥ 2, 11, 17, 29, 41, ... respectively,

where $\pi(x)$ 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 Rn for which $\pi(x) - \pi(x/2)$n, for all xRn.[2]

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently,

Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all xRn.

Note that the integer Rn is necessarily a prime number: $\pi(x) - \pi(x/2)$ and, hence, $\pi(x)$ must increase by obtaining another prime at x = Rn. Since $\pi(x) - \pi(x/2)$ can increase by at most 1,

$\pi($Rn$) - \pi($Rn$/2) = n$.

Bounds and an asymptotic formula

For all n ≥ 1, the bounds

2n ln 2n < Rn < 4n ln 4n

hold. If n > 1, then also

p2n < Rn < p3n

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

$R_n \le \frac{41}{47} \ p_{3n}$

which is the optimal form of Rn < c·p3n since it is an equality for n = 5.

In a different direction, Axler[6] showed that

$R_n \le p_{\lceil t\cdot n \rceil}$

is optimal for t = 48/19, where $\lceil\cdot \rceil$ is the ceiling function.

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, $\pi(x) - \pi(cx) \ge n$. 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

R0.25,n = 2, 3, 5, 13, 17, ... ,
R0.75,n = 11, 29, 59, 67, 101, ... .

It is known[7] 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 $\lfloor$n/(1 − c)$\rfloor$th prime and $\lfloor .\rfloor$ is the floor function.

Ramanujan prime corollary

$2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,$

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, 2*pi-n] is greater than or equal to 1. By taking into account the size of the gaps between primes in [pin,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 > ... > pin > pi/2, and so 2pin > pi.

An example of this corollary:

With n = 1000, Rn = pk = 19403, and k = 2197, therefore i ≥ 2198 and in ≥ 1198. The smallest i − n prime is pin = 9719, therefore 2pin = 2 × 9719 = 19438. The 2198th prime, pi, is between pk = 19403 and 2pin = 19438 and is 19417.

The left side of the Ramanujan Prime Corollary is the ; the right side is the . The values of $\pi(R_n)\,$ are in the .

The Ramanujan Prime Corollary is due to John Nicholson.

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: 630–635, arXiv:0907.5232
4. ^ Laishram, S. (2010), "On a conjecture on Ramanujan primes", International Journal of Number Theory 6: 1869–1873.
5. ^ Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps", Journal of Integer Sequences 14: 11.6.2
6. ^ Axler, Christian. "On generalized Ramanujan primes". Retrieved 16 April 2014.
7. ^ Amersi, N.; Beckwith, O.; Miller, S.J.; Ronan, R.; Sondow, J. (2011), Generalized Ramanujan primes, arXiv:1108.0475