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One more proof of Siegel's theorem

Abstract : 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.
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K Ramachandra. One more proof of Siegel's theorem. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1980, Volume 3 - 1980, pp.25-40. ⟨10.46298/hrj.1980.89⟩. ⟨hal-01103859⟩

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