Ramanujan prime

Ramanujan primes are primes that satisfy an inequality according to S. Ramanujan, which followed from his generalisation of Bertrand's postulate, which Ramanujan proved anew in the process. Bertrand's postulate states that for all numbers is at least one prime {\displaystyle x\geq 1}between xand 2xRamanujan primes R_{n}are defined as smallest numbers such that for all {\displaystyle x\geq R_{n}}between xand {\displaystyle {\tfrac {x}{2}}}at least nprimes. That these exist for every nwas proved by Ramanujan. The name Ramanujan prime was introduced by Jonathan Sondow in 2005.

Let be {\displaystyle x\mapsto \pi (x)}the prime function, that is, π \pi (x)is the number of primes not greater than x. Then the n‑th Ramanujan prime is the smallest number R_{n}for which holds:

{\displaystyle \pi (x)-\pi \left({\frac {x}{2}}\right)\geq n}for all {\displaystyle x\geq R_{n}}

In other words, they are the smallest numbers R_{n}such that for all {\displaystyle x\geq R_{n}}between {\displaystyle {\tfrac {x}{2}}}and are xat least nprimes. Because the function can {\displaystyle x\mapsto \pi (x)-\pi ({\tfrac {x}{2}})}only xgrow at a prime , R_{n}must be a prime and it holds:

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

The first Ramanujan primes are:

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, ... (sequence A104272 in OEIS).

Bertrand's postulate is just the case n=1(with {\displaystyle R_{1}=2}).

Ramanujan proved the existence of these prime numbers by using the inequality

{\displaystyle \pi (x)-\pi \left({\frac {x}{2}}\right)>{\frac {1}{\log x}}\left({\frac {x}{6}}-3{\sqrt {x}}\right)}

for{\displaystyle x>300} . The right-hand side grows monotonically towards infinity for x\to \infty .

Properties

It holds for any n\geq 1

{\displaystyle 2n\,\ln(2n)<R_{n}<4n\,\ln(4n)},

where \ln denotes the natural logarithm, and

{\displaystyle p_{2n}<R_{n}<p_{3n}}{\displaystyle

where p_{n}is the nn}th prime number.

Asymptotic applies

{\displaystyle R_{n}\sim p_{2n}}for {\displaystyle n\to \infty ,}

from which it follows with the prime number theorem:

{\displaystyle R_{n}\sim 2n\,\ln(2n)}

The above results are from Jonathan Sondow except for the inequality {\displaystyle R_{n}<p_{3n}}which Sondow conjectured and which Shanta Laishram proved.


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