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:
π ( x ) − π ( x 2 ) ≥ 1 , 2 , 3 , 4 , 5 , … for all x ≥ 2 , 11 , 17 , 29 , 41 , … respectively {\displaystyle \pi (x)-\pi \left({\frac {x}{2}}\right)\geq 1,2,3,4,5,\ldots {\text{ for all }}x\geq 2,11,17,29,41,\ldots {\text{ respectively}}} OEIS: A104272 where π ( x ) {\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 π ( x ) − π ( x / 2 ) ≥ n , {\displaystyle \pi (x)-\pi (x/2)\geq n,} for all x ≥ Rn. 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 x ≥ Rn. The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Note that the integer Rn is necessarily a prime number: π ( x ) − π ( x / 2 ) {\displaystyle \pi (x)-\pi (x/2)} and, hence, π ( x ) {\displaystyle \pi (x)} must increase by obtaining another prime at x = Rn. Since π ( x ) − π ( x / 2 ) {\displaystyle \pi (x)-\pi (x/2)} can increase by at most 1,
π ( R n ) − π ( R n 2 ) = n . {\displaystyle \pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.}
Bounds and an asymptotic formula For all n ≥ 1 {\displaystyle n\geq 1} , the bounds
2 n ln 2 n < R n < 4 n ln 4 n {\displaystyle 2n\ln 2n<R_{n}<4n\ln 4n}
hold. If n > 1 {\displaystyle n>1} , then also
p 2 n < R n < p 3 n {\displaystyle p_{2n}<R_{n}<p_{3n}}
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), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to
R n ≤ 41 47 p 3 n {\displaystyle R_{n}\leq {\frac {41}{47}}\ p_{3n}}
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.
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