New upper bounds for Ramanujan primes

Anitha Srinivasan, Pablo Arés

For $n\ge 1$, the $n^{\mathrm{th}}$ Ramanujan prime is defined as the smallest positive integer $R_n$ such that for all $x\ge R_n$, the interval $(\frac{x}{2},x]$ has at least $n$ prim…