Martin Cech

A double sum of Jacobi symbols and multiple Dirichlet series

We will study the double sum of Jacobi symbols
\sum_{n\leq X}\sum_{m\leq Y}\left(\frac{m}{n}\right).
This sum was studied by Conrey, Farmer, and Soundararajan, who gave a surprising asymptotic formula. We will show how to obtain the same asymptotic with an improved error term by transforming the double sum into a double integral that involves a double Dirichlet series, and study the analytic properties of the double Dirichlet series.