Primes in short intervals assuming the Riemann hypothesis

Source abstract

In 2001 Baker, Harman, and Pintz proved, for $x$ sufficiently large, the existence of at least one prime in the sub-linear interval $[x,x+x^{\frac{1}{2} +\frac{1}{40}}]$. But just how large is sufficiently large? In the current note I shall make the Baker--Harman--Pintz theorem effective, but at an extremely high price. Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for $x \geq 3.5\times 10^{90}$. Furthermore, again assuming RH, I shall also show that for any $0< m\leq15.831$ and $x \geq 1$ there is at least one prime in the interval $[x,x+x^{\frac{1}{2} +\frac{1}{m}}]$, and that for $M\geq 2.29$ and $y \geq 1$ there is at least one prime in the interval $[y^M, (y+1)^M]$.