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. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

: $\pi(x) - \pi\left( \frac x 2 \right) \ge 1,2,3,4,5,\ldots \text x \ge 2, 11, 17, 29, 41, \ldots \text$    

where $\pi(x)$ 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 ''n''th Ramanujan prime is the least integer ''R_n'' for which $\pi(x) - \pi(x/2) \ge n,$ for all ''x'' ≥ ''R_n''. In other words: Ramanujan primes are the least integers ''R_n'' for which there are at least ''n'' primes between ''x'' and ''x''/2 for all ''x'' ≥ ''R_n''.

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

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

: $\pi(R_n) - \pi\left( \frac 2 \right) = n.$

Bounds and an asymptotic formula

For all $n \geq 1$, the bounds

:$2n\ln2n < R_n < 4n\ln4n$

hold. If $n > 1$, then also

:$p_ < R_n < p_$

where ''p''_''n'' is the ''n''th prime number.

As ''n'' tends to infinity, ''R''_''n'' is asymptotic to the 2''n''th prime, i.e.,

:''R''_''n'' ~ ''p''_2''n'' (''n'' → ∞).

All these results were proved by Sondow (2009), except for the upper bound ''R''_''n'' < ''p''_3''n'' which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to

:$R_n \le \frac \ p_$

which is the optimal form of ''R''_''n'' ≤ ''c·p''_3''n'' since it is an equality for ''n'' = 5.

References

{{Prime number classes

Srinivasa Ramanujan
Classes of prime numbers