Ramanujan Primes

<strong>Mathematics Seminar of the School of Physical Sciences ---------------------------------------------------------</strong> <strong>Ramanujan Primes</strong> <strong>Anitha Srinivasan</strong> (St Louis University, Madrid Campus) Date: <strong>August 5, 2016 </strong> <strong>Abstract: </strong>Bertrand's Postulate is well known and says that there is at least one prime between x and 2x for any natural number x. Ramanujan generalized this result to the following beautiful one: Let \pi(x) be the number of primes less than or equal to x. Then \pi(x) - \pi(x/2) \geq 1, 2, 3, 4, 5, . . . for x \geq 2, 11, 17, 29, 41, . . . respectively. Note that \pi(x) - \pi(x/2) \geq 1 for x \geq 2 is Bertrand's Postulate. The n th Ramanujan prime is defined as the least positive integer R_n such that for all x \geq R_n , the interval ( x/2 , x] has at least n primes. We will look at some interesting facts surrounding these primes. Specifically we will look at bounds on these primes. One of the first upper bounds for R_n was given by Laishram, who showed that R_n &lt; p_{3n} for all n, where p_i is the i th prime. We present various improvements on this upper bound and also explore some other questions concerning these intriguing primes.