• 1 Tata Institute for Fundamental Research

This paper gives a new elementary proof of the version of Siegel's theorem on $L(1,\chi)=\sum_{n=1}^{\infty}\chi(n)n^{-1}$ for a real character $\chi(\!\!\!\!\mod k)$. The main result of this paper is the theorem: If $3\leq k_1\leq k_2$ are integers, $\chi_1(\!\!\!\!\mod k_1)$ and $\chi_2(\!\!\!\!\mod k_2)$ are two real non-principal characters such that there exists an integer $n>0$ for which $\chi_1(n)\cdot\chi_2(n)=-1$ and, moreover, if $L(1,\chi_1)\leq10^{-40}(\log k_1)^{-1}$, then $L(1,\chi_2)>10^{-4} (\log k_2){-1}\cdot(\log k_1)^{-2}k_2^{-40000L(1,\chi_1)}$. From this the result of T. Tatuzawa on Siegel's theorem follows.