In other words, prove the expected asymptotic formula for the number of primes in the interval $[y,y+Cd]$. This is a curious combination of two well-studied problems: find the minimum $h=h(y)$ for which one obtains the expected asymptotic\[\pi(y+h)-\pi(y)\sim \frac{h}{\log y},\]and understand the asymptotic behaviour of $d=\max_{p_n<x}(p_{n+1}-p_n)$.

The conjectured size of $d$ is $\approx (\log x)^2$ which is far below the $h$ we can obtain such an asymptotic for, even assuming the Riemann hypothesis (which delivers an asymptotic for $h=y^{1/2+o(1)}$).

View the LaTeX source

This page was last edited 23 January 2026.