关于Siegel-Tatuzawa定理(英文)

On the Siegel-Tatuzawa Theorem

Siegel-Tatuzawa定理是在研究Gauss关于虚二次域类数的第一个猜想中产生的一个很重要的结论,Hoffstein等人对Siegel-Tatuzawa定理的结果进行了改进,进一步得到了关于L(1,χ)下界的一些结论.本文在前人研究的基础上,利用L(1,χ)的上界以及双二次域的算术理论给出了对于实本原Dirichlet特征χ,L(1,χ)较好的下界.

Siegel-Tatuzawa Theorem is an important result in proving the first one of the Gauss＇ conjectures on the imaginary quadratic number fields. After that, many people improved upon the result of Siegcl-Tatuzawa Theorem. In this paper we use some Lemmas about the upper bound of L（ 1 ,X） and some arithmetic theory, of the biquadratic number field, get a lower bound of real primitive L-function at s = 1 ： Let 0 〈 E 〈 1/（ 6log10）, and X be a real primitive Dirichlet character modulo k which is greater than e^1/3, then with at most one exception, the following expression holds：L（1,x）〉min{1/7.702logk,31.3ε/k^ε}