Symplectic filling

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. 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, ... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. 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 R_n for which $\pi (x)-\pi (x/2)$ ≥ n, for all x ≥ R_n.

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

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.

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

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

Bounds and an asymptotic formula

For all n ≥ 1, the bounds

2n ln 2n < R_n < 4n ln 4n

hold. If n > 1, then also

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), 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

$R_{n}\leq {\frac {41}{47}}\ p_{3n}$

which is optimal since it is an equality for n = 5.

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, $\pi (x)-\pi (cx)\geq n$. 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, ... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.,

R_0.75,n = 11, 29, 59, 67, 101, ... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

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 $\lfloor$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. 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, 2*p_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 Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.; the right side is the Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.. The values of $\pi (R_{n})\,$ are in the Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

The Ramanujan Prime Corollary is due to John Nicholson.

References

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