Theoretical Statistics and Mathematics Unit, ISI Delhi

On a conjecture on Ramanujan primes

by Shanta Laishram

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.

isid/ms/2011/10 [fulltext]

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