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 Matomäki, Maks Radziwill and Brad Rodgers.