Ramanujan Primes: Bounds, Runs, Twins, and Gaps

Jonathan Sondow, John W. Nicholson, Tony D. Noe

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x\ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem t…