Publications and Preprints

For $n\ge 1$, the \emph{$n$th Ramanujan prime} is defined to be the smallest positive integer $R_n$ with the property that if $x\ge R_n$, then $\pi(x)-\pi(\frac{x}{2})\ge n$ where $\pi(\nu)$ is the number of primes not exceeding $\nu$ for any $\nu>0$ and $\nu\in \R$. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime $\pi$ and $\theta$ functions due to Dusart.