The squarefree integers are divisible by no square of a prime. It is well known that they have a positive density within the integers.

We consider the number of squarefree integers in a random interval of size $H$: $\# \{n \in [x,x+H] : n \textrm{ squarefree} \}$,

where $x$ is a random number between $1$ and $X$. The variance of this quantity has been studied by R. R. Hall in 1982,

obtaining asymptotics in the range $H < X^{2/9}$, with a proof method that stays in 'physical space'.

Keating and Rudnick recently conjectured that his result persists for the entire range $H < X^{1-\epsilon}$.

We make progress on this conjecture, with properties of Dirichlet polynomials playing a role in our results.

We will show how one can verify the conjecture for H slightly beyond $X^{1/2}$.

This is joint work with Kaisa MatomÃ¤ki, Maks Radziwill and Brad Rodgers.