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Thursday, November 14, 2013

**Abstract:** We provide a possible method to eliminate the Siegel zero of $L(s,\chi)$, or, equivalently, to establish an effective lower bound for $L(1,\chi)$, where $\chi$ is a real primitive character $(\mod D)$. There are two major ingredients. The first is to relate the lower bound for $L(1,\chi)$ to the distribution of zeros of $L(s,\chi)L(s,\chi\psi)$, with $\psi$ belonging to a large set of primitive characters $\Psi$, in a region $\Omega$. It is shown that if $$L(1,\chi) < (\log D)^{-B}$$, where $B$ is a large constant, then for most $\psi \in \Psi$, not only all the zeros of $L(s,\chi)L(s,\chi\psi)$ in $\Omega$ are critical and simple, but also all the gaps between consecutive zeros are near to the average gap. The second is, with the aim of deriving a contradiction from the gap assertion, to reduce the problem to evaluating a discrete mean. Eventually, the problem is reduced to proving a lower bound for the norm of a self adjoint operator on a Hilbert space.