The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at most $\frac{(c−1+\epsilon)n}{\log n}$ for every $c>1$.
What is the precise value of $n_\epsilon$ as a function of $\epsilon$ known unconditionally and conditionally on reasonable conjectures?
