Is this Dusart's result depends on the Riemann hypothesis

Question

Dusart proved in 2010 that there's at least one prime between $x$ and $\left(1 + \frac{1}{25\ln^2x}\right)x$ for $x \geq 396738$

My question is: Is this Dusart's result depends on the Riemann hypothesis.

Answer 1

No, Dusart's result is unconditional.

Stronger bounds can be proved using the Riemann hypothesis; Cramér proved conditionally that there exists a constant $C>0$ such that there's always at least one prime between $x$ and $x + C\sqrt x\log x$.

Answer 2

Actually it depends on what you mean by "depends on Riemann Hypothesis" because in order the calculate the constants and minimum values of $x$ in Dusart's result, Dusart did not need the Riemann hypothesis to be true but he used the precise location of the first 15 billion imaginary zeroes of the Riemann zeta function i.e. if the Riemann hypothesis were known to be false and there was a relatively small imaginary zero then the expression of his result would still be true but perhaps with slightly different constants.