# 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.

## Contents

## Origins and definition[edit]

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:

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
*n*th Ramanujan prime is the least integer*R*for which for all_{n}*x*≥*R*._{n}^{[2]}In other words: Ramanujan primes are the least integers*R*for which there are at least_{n}*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: and, hence, must increase by obtaining another prime at

*x*=

*R*. Since can increase by at most 1,

_{n}## Bounds and an asymptotic formula[edit]

For all , the bounds

hold. If , then also

where *p*_{n} is the *n*th prime number.

As *n* tends to infinity, *R*_{n} is asymptotic to the 2*n*th prime, i.e.,

*R*_{n}~*p*_{2n}(*n*→ ∞).

All these results were proved by Sondow (2009),^{[3]} except for the upper bound *R*_{n} < *p*_{3n} which was conjectured by him and proved by Laishram (2010).^{[4]} The bound was improved by Sondow, Nicholson, and Noe (2011)^{[5]} to

which is the optimal form of *R*_{n} ≤ *c·p*_{3n} since it is an equality for *n* = 5.

In a different direction, Axler^{[6]} 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.^{[7]} 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 .

In 2016, Shichun Yang and Alain Togbe^{[8]} establish the estimates of the upper and lower bounds of Ramanujan primes when *n* is big: if and , then

where

## Generalized Ramanujan primes[edit]

Given a constant *c* between 0 and 1, the *n*th *c*-Ramanujan prime is defined as the
smallest integer *R _{c,n}* with the property that for any integer

*x ≥ R*there are at least

_{c,n}*n*primes between

*cx*and

*x*, that is, . In particular, when

*c*= 1/2, the

*n*th 1/2-Ramanujan prime is equal to the

*n*th 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^{[9]} that, for all *n* and *c*, the *n*th *c*-Ramanujan prime *R _{c,n}* exists and is indeed prime. Also, as

*n*tends to infinity,

*R*is asymptotic to

_{c,n}*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[edit]

i.e. *p*_{k} is the *k*th prime and the *n*th 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}/(2*n*) ~ 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 2*p*_{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 2*p*_{i−n} = 2 × 9719 = 19438. The 2198th prime, *p*_{i}, is between *p*_{k} = 19403 and 2*p*_{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 are in the A179196.

The Ramanujan Prime Corollary is due to John Nicholson.

Srinivasan's Lemma ^{[10]} 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.

## References[edit]

**^**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**