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 ∈ [x,x+H] : n is 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 less than X2/9, with a proof method that stays in ‘physical space’. Keating and Rudnick recently conjectured that his result persists for the entire range H less than X1-ε. We make progress on this conjecture, with properties of the Riemann zeta function playing a role in our results. We will show how, on RH, one can verify the conjecture for H up to X2/3.
This is joint work with Kaisa Matomäki, Maks Radziwill and Brad Rodgers.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
