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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?

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    $\begingroup$ You can easily extract explicit bounds from Dussart's results mentioned in mathoverflow.net/a/208622 . They should be tight up to a multiplicative factor. $\endgroup$ – Emil Jeřábek May 23 '16 at 7:50

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