Binary quadratic forms and the zeros of Epstein's zeta function

In 1859, Riemann investigated ζ(s) and made the famous hypothesis (RH) that all nontrivial zeros of ζ(s) must lie on the line Re(s)=1/2. Since then, many variations of ζ(s) have been developed. While some of them are believed to satisfy the analog of RH, there are cases where RH is false.

One interesting false case is Epstein's zeta function. In 1936, Davenport and Heilbronn showed that the Epstein's zeta function will have infinitely many nontrivial zeros off the critical line if certain quadratic integer rings fail to be unique factorization domains. In this talk, we will survey the theory of binary quadratic forms and its connection to quadratic fields, which are the key ingredients in the proof of the Davenport-Heilbronn theorem.