Ramanujan prime number

Ramanujan prime numbers are prime numbers that satisfy an inequality according to S. Ramanujan , which followed from his generalization of Bertrand's postulate , which Ramanujan proved anew. Bertrand's postulate states that there is at least one prime number for all numbers between and . Ramanujan primes are defined as the smallest numbers, so that there are between and at least prime numbers for all of them . Ramanujan proved that there is this for everyone. The name Ramanujan prime was introduced by Jonathan Sondow in 2005. ${\displaystyle x\geq 1}$ ${\displaystyle x}$ ${\displaystyle 2x}$ ${\displaystyle R_{n}}$ ${\displaystyle x\geq R_{n}}$ ${\displaystyle x}$ ${\displaystyle {\tfrac {x}{2}}}$ ${\displaystyle n}$ ${\displaystyle n}$

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

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

In other words, they are the smallest numbers , so that between and at least prime numbers for all of them . Because the function can only grow at a prime position , it must be a prime number and the following applies: ${\displaystyle R_{n}}$ ${\displaystyle x\geq R_{n}}$ ${\displaystyle {\tfrac {x}{2}}}$ ${\displaystyle x}$ ${\displaystyle n}$ ${\displaystyle x\mapsto \pi (x)-\pi ({\tfrac {x}{2}})}$ ${\displaystyle x}$ ${\displaystyle R_{n}}$

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

The first Ramanujan prime numbers 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 precisely the case (with ). ${\displaystyle n=1}$ ${\displaystyle R_{1}=2}$

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

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

for derived. The right side grows monotonously towards infinity for . ${\displaystyle x>300}$ ${\displaystyle x\to \infty }$

properties

It applies to everyone ${\displaystyle n\geq 1}$

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

where denotes the natural logarithm , as well as ${\displaystyle \ln }$

${\displaystyle p_{2n}<R_{n}<p_{3n}}$for , ${\displaystyle n\geq 2}$

where the -th is prime. ${\displaystyle p_{n}}$ ${\displaystyle n}$

Asymptotically applies

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

from which follows with the prime number theorem:

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

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

example

The first prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, ... (Follow A000040 in OEIS )

We consider the following two properties (where is the number of prime numbers and the -th Ramanujan prime number): ${\displaystyle \pi (x)}$ ${\displaystyle \leq x}$ ${\displaystyle R_{n}}$ ${\displaystyle n}$

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

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

and now examine these for the first : ${\displaystyle x\in \mathbb {N} ,x\leq 228}$

Web links

• J. Sondow: Ramanujan Prime . In: MathWorld (English).

Individual evidence

1. ↑ Ramanujan: A proof of Bertrand's postulate. In: Journal of the Indian Mathematical Society. 11: 181-182 (1919).
2. ^ J. Sondow: Ramanujan primes and Bertrand's postulate. In: American Mathematical Monthly. Volume 116, 2009, pp. 630-635, Arxiv, pdf.
3. ^ The first 1000 and 10000 primes