# 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:

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

where ${\displaystyle \pi (x)}$ 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 ${\displaystyle \pi (x)-\pi (x/2)\geq n,}$ for all xRn.[2] In other words: Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all xRn.

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

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

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

## Bounds and an asymptotic formula

For all ${\displaystyle n\geq 1}$, the bounds

${\displaystyle 2n\ln 2n

hold. If ${\displaystyle n>1}$, then also

${\displaystyle p_{2n}

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

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

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

In a different direction, Axler[6] showed that

${\displaystyle R_{n}

is optimal for t > 48/19, where ${\displaystyle \lceil \cdot \rceil }$ is the ceiling function.

A further improvement of the upper bounds was done in late 2015 by Anitha Srinivasan and John W. Nicholson.[7] They show that if

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

then ${\displaystyle R_{n} for all ${\displaystyle n>241}$, where ${\displaystyle \lfloor \cdot \rfloor }$ is the floor function. For large n, the bound is smaller and thus better than ${\displaystyle p_{\lfloor 2nc\rfloor }}$ for any fixed constant ${\displaystyle c>1}$.

In 2016, Shichun Yang and Alain Togbe[8] establish the estimates of the upper and lower bounds of Ramanujan primes ${\displaystyle R_{n}}$ when n is big: if ${\displaystyle n>10^{300}}$ and ${\displaystyle R_{n}=p_{s}}$, then

${\displaystyle \beta

where

${\displaystyle \alpha =2n{\Big (}1+{\frac {\ln 2}{\ln n}}-{\frac {\ln 2\ln \ln n-\ln ^{2}2-\ln 2-0.13}{\ln ^{2}n}}{\Big )},}$
${\displaystyle \beta =2n{\Big (}1+{\frac {\ln 2}{\ln n}}-{\frac {\ln 2\ln \ln n-\ln ^{2}2-\ln 2+0.11}{\ln ^{2}n}}{\Big )}.}$

## Generalized Ramanujan primes

Given a constant c between 0 and 1, the nth c-Ramanujan prime is defined as the smallest integer Rc,n with the property that for any integer x ≥ Rc,n there are at least n primes between cx and x, that is, ${\displaystyle \pi (x)-\pi (cx)\geq n}$. In particular, when c = 1/2, the nth 1/2-Ramanujan prime is equal to the nth Ramanujan prime: 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[9] 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 ${\displaystyle \lfloor }$n/(1 − c)${\displaystyle \rfloor }$th prime and ${\displaystyle \lfloor .\rfloor }$ is the floor function.

## Ramanujan prime corollary

${\displaystyle 2p_{i-n}>p_{i}{\text{ for }}i>k{\text{ where }}k=\pi (p_{k})=\pi (R_{n})\,,}$

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, 2pin] 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 smallest prime on the right side is . The sequence is the range of the smallest prime greater than pk. The values of ${\displaystyle \pi (R_{n})\,}$ are in the .

The Ramanujan Prime Corollary is due to John Nicholson.

Srinivasan's Lemma [10] states that pkn < pk/2 if Rnpk and n > 1. Proof: By the minimality of Rn, the interval (pk/2,pk] contains exactly n primes and hence pkn < pk/2.

## 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 (7): 630–635, arXiv:, doi:10.4169/193009709x458609
4. ^ Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, doi:10.1142/s1793042110003848.
5. ^ 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:, Bibcode:2011arXiv1105.2249S
6. ^ Axler, Christian (2014). "On generalized Ramanujan primes". The Ramanujan Journal. 39 (2016): 1. arXiv:. doi:10.1007/s11139-015-9693-9.
7. ^ Srinivasan, Anitha; Nicholson, John (2015). "An Improved Upper Bound For Ramanujan Primes" (PDF). Integers. 15.
8. ^ 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.
9. ^ Amersi, N.; Beckwith, O.; Miller, S.J.; Ronan, R.; Sondow, J. (2011), Generalized Ramanujan primes, arXiv:, Bibcode:2011arXiv1108.0475A
10. ^ Srinivasan, Anitha (2014), "An upper bound for Ramanujan primes" (PDF), Integers, 14