The large sieve for self-dual Eisenstein series of varying levels

We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is recursive, and has some elements in common with Heath-Brown’s quadratic large sieve, and the asymptotic large sieve of Conrey, Iwaniec, and Soundararajan.