Ramanujan prime

Not to be confused with Hardy–Ramanujan number.

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:

$\pi (x)-\pi (x/2)\geq 1,2,3,4,5,\ldots {\text{ for all }}x\geq 2,11,17,29,41,\ldots {\text{ respectively}}$OEIS: A104272

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 $\pi (x)-\pi (x/2)\geq n,$ 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.

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,

$\pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.$

Bounds and an asymptotic formula

For all $n\geq 1$, 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 the optimal form of R_n ≤ c·p_3n since it is an equality for n = 5.

In a different direction, Axler showed that

$R_{n}<p_{\lceil t\cdot n\rceil }$

is optimal for t > 48/19, where $\lceil \cdot \rceil$ 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

$\alpha =1+{\frac {3}{\ln n+\ln \ln n-4}}$

then $R_{n}<p_{\lfloor 2n\alpha \rfloor }$ for all $n>241$, where $\lfloor \cdot \rfloor$ is the floor function. For large n, the bound is smaller and thus better than $p_{\lfloor 2nc\rfloor }$ 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

$\beta <s<\alpha ,$

where

$\alpha =2n{\Big (}1+{\frac {\ln 2}{\ln n}}-{\frac {\ln 2\ln \ln n-\ln ^{2}2-\ln 2-0.13}{\ln ^{2}n}}{\Big )},$

$\beta =2n{\Big (}1+{\frac {\ln 2}{\ln n}}-{\frac {\ln 2\ln \ln n-\ln ^{2}2-\ln 2+0.11}{\ln ^{2}n}}{\Big )}.$

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, ... OEIS: A193761,

R_0.75,n = 11, 29, 59, 67, 101, ... OEIS: 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 $\lfloor$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, 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 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 p_k. The values of $\pi (R_{n})\,$ are in the OEIS: 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.