Ramanujan prime

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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[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:

OEISA104272

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 nth Ramanujan prime is the least integer Rn for which 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. Equivalently.

Note that the integer Rn is necessarily a prime number: and, hence, must increase by obtaining another prime at x = Rn. Since can increase by at most 1,

Bounds and an asymptotic formula[edit]

For all , the bounds

hold. If , then also

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

which is the optimal form of Rnc·p3n 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 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, . 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, ... OEISA193761,
R0.75,n = 11, 29, 59, 67, 101, ... OEISA193880.

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 n/(1 − c)th prime and is the floor function.

Ramanujan prime corollary[edit]

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 OEISA168421; the smallest prime on the right side is OEISA168425. The sequence OEISA165959 is the range of the smallest prime greater than pk. The values of are in the OEISA179196.

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[edit]

  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:0907.5232Freely accessible, 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:1105.2249Freely accessible, Bibcode:2011arXiv1105.2249S 
  6. ^ Axler, Christian (2014). "On generalized Ramanujan primes". The Ramanujan Journal. 39 (2016): 1. arXiv:1401.7179Freely accessible. 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:1108.0475Freely accessible 
  10. ^ Srinivasan, Anitha (2014), "An upper bound for Ramanujan primes" (PDF), Integers, 14