An Explicit Result for Primes Between Cubes

Abstract

We prove that there is a prime between $n^3$ and $(n+1{)}^3$ for all $n\ge \exp (\exp (33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m\ge 4.971\times 10^9$.

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