Lilian Matthiesen, Université de Bristol (Royaume Uni)

Counting rational points on the intersection of certain quadrics

The representation function of a positive definite binary quadratic
form f is defined as r_f(n) = | {(x,y) : f(x,y) = n} |. Building on the first
talk, we will show how to obtain asymptotics for linear correlations
of such representation functions r_f. These correlations correspond
to the divisor problems of the first talk, that is, they include for ins-
tance averages of the form Σ_n,r&leN r_f1(n) r_f2(n+r) r_f3(n+2r)... r_fk-1(n+kr).
As an immediate consequence we obtain asymptotic counts of integer
solutions of certain systems of quadratic equations.