Joint IAS/Princeton University Number Theory Seminar

Let \(\chi\) be a primitive real character. We first establish a relationship between the existence of the Landau-Siegel zero of \(L(s,\chi)\) and the distribution of zeros of the Dirichlet \(L\)-function \(L(s,\psi)\), with \(\psi\) belonging to a set \(\Psi\) of primitive characters, in a region \(\Omega\). It is shown that if the Landau-Siegel zero exists (equivalently, \(L(1,\chi)\) is small), then, for most \(\psi \in \Psi\), not only all the zeros of \(L(s,\psi)\) in \(\Omega\) are simple and lie on the critical line, but also the gaps between consecutive zeros are close to integral multiples of the half of the average gap. In comparison with certain conjectures on the vertical distribution of zeros of \(\zeta(s)\), it is reasonable to believe that the gap assertion would fail to hold. In order to derive a contradiction from the gap assertion, we attempt to reduce the problem to evaluating a certain discrete mean; the idea is motivated by the work of Conrey, Ghosh and Conek on the simple zeros of \(\zeta(s)\). We shall describe the coefficient of the main term and provide some numerical evidences. In some special cases, the problem is further reduced to calculating small positive eigenvalues of linear integral equations with Hermitian kernels.