Spectral L²-bounds on Eisenstein series and Diophantine properties of quadratic forms

We give L²-bounds with respect to the height of the spectral parameter for Epstein zeta functions ZQ(s) and non-holomorphic Eisenstein series Ec(s,z) attached to the Hecke congruence subgroup Γ₀ (p) of prime level p. Our bounds are completely explicit in the level and, additionally, constitute an almost sure logarithmic improvement over previously established bounds for the level-one Eisenstein series. Following the approach of Kelmer--Kontorovich--Lutsko, we combine the approximate functional equation for ZQ(s) with estimates for the number of pairs of lattice vectors at which Q takes nearly equal values. Under a Diophantine hypothesis on Q, we sharpen these counting estimates using results of Eskin--Margulis--Mozes on the distribution of values of quadratic forms of signature $(2,2)$.