Number/Representation Theory

The form $x^2+y^2$ covers $\tfrac{1}{2}$ of the primes, while the forms $x^2+y^2, x^2+2y^2$ cover $\tfrac{3}{4}$ of them. In this talk, we will show that the proportion of primes covered by the forms $x^2+dy^2$, with $1 \leqslant d \leqslant \Delta$, is $1 - \exp((\alpha(\Delta) + o(1)) \tfrac{\sqrt{\Delta}}{\log \Delta})$ for some $\tfrac{\pi}{2} \leqslant \alpha(\Delta) \leqslant \tfrac{\pi}{2} + \log 4$. Moreover, inspired by a result of Kaplansky, we prove that any prime represented by two of the forms $x^2+17y^2, x^2+65y^2, x^2+1105y^2$ is actually represented by all three. We establish these results by studying how class groups of quadratic fields interact.