Square-free integers represented by binary quadratic forms of a fixed discriminant (online broadcast in lecture hall)

In this Work, we prove a result concerning the infinitude of square-free integers represented by a class of polynomials in two variables. More precisely, we prove that infinitely many square-free positive integers are represented by a primitive integral positive-definite binary quadratic form of a given discriminant $D$. We obtain our result by deriving an asymptotic formula for the summatory function associated to it using some known analytic properties of $L$-functions. This is a joint work with Manish Kumar Pandey.