Determinants as sums of two squares

A classical theorem due independently to Landau and Ramanujan 
gives an asymptotic formula for the number of integers which can be 
written as a sum of two squares. We prove an analogous result for the 
determinant of a matrix using the spectral theory of automorphic forms. 
This is a special case of a more general result on a problem of Serre 
concerning specialisations of Brauer group elements on semisimple 
algebraic groups. This is joint work with Sho Tanimoto and Ramin 
Takloo-Bighash.