Abstract We study the second moment of Dirichlet L-functions to a large prime modulus q twisted by the square of an arbitrary Dirichlet polynomial. We break the 1 2 -barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than q 51 / 101 = q 1 / 2 + 1 / 202 . As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet L-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

中文翻译：

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打破狄利克雷 L 函数扭曲二阶矩的 12 势垒

摘要 我们研究了狄利克雷 L 函数对被任意狄利克雷多项式的平方扭曲的大素模数 q 的二阶矩。我们打破了这个问题中的 1 2 -障碍，得到了一个渐近公式，条件是狄利克雷多项式的长度小于 q 51 / 101 = q 1 / 2 + 1 / 202 。作为应用，我们获得了狄利克雷 L 函数三阶矩的正确数量级的上限。当狄利克雷多项式的系数更专业时，我们会给出进一步的结果。