SAMS Subject Classification Number: 22

We consider the Dirichlet L-functions associated with characters defined by the Kronecker’s symbol \(\chi(n)=\big(\frac{\Delta}{n}\big)\). The study of the zeros of such functions has profound consequences in many results in number theory. The Generalized Riemann Hypothesis states that the zeros of the \(L\)-functions with positive real parts lie on the line \(\Re(s)=\frac{1}{2}\). However, it is known that there might exist zeros near to 1 in the interval of the form \((1-C/\log |\Delta|,1)\) where \(C\) is an absolute constant. These hypothetical zeros are called Siegel zeros or Landau-Siegel zeros. In this talk, we first review some elementary definitions and background results, and then give a survey on what we know explicitly on the bounds of Siegel zeros.