New upper bounds for Ramanujan primes

Abstract

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 $n$ primes. We show that for every $\epsilon >0$, there is a positive integer $N$ such that if $\alpha =2n\left(1+\frac{\log 2+\epsilon }{\log n+j(n)}\right)$, then $R_n<p_{[\alpha ]}$ for all $n>N$, where $p_i$ is the $i^{\mathrm{th}}$ prime and $j(n)>0$ is any function that satisfies $j(n)\to \infty$ and $n{j}^{\prime }(n)\to 0$.

References in corpus (1)

• Estimates of Some Functions Over Primes without R.H.