Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

中文翻译：

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数域上二进制形式的广义除数和

估计 Dirichlet 卷积的平均值$1\ast \unicode[STIX]{x1D712}$, 对于一些真正的狄利克雷角色$\unicode[STIX]{x1D712}$的固定模数，在定义的二进制形式的稀疏值集上$\mathbb{Z}$近年来，它一直是广泛研究的焦点，在 Manin 的 Châtelet 曲面猜想中得到了惊人的应用。我们介绍了这个问题的意义深远的概括，特别是替换$\unicode[STIX]{x1D712}$通过 Jacobi 符号，两个参数都有不同的大小，可能趋于无穷大。本文的主要结果提供了相应平均值的渐近估计和预期数量级的下限。所有这些都是通过采用 Daniel 特有的技术在任意数字字段上执行的$1\ast 1$. 这是第一次对二进制形式的值的除数求和在任何数字字段上进行渐近评估，除了$\mathbb{Q}$. 在对 Picard 数的温和假设下，我们的工作是证明马宁猜想在所有数域上的所有 del Pezzo 表面预测的下限的关键步骤，在随后的工作中给出。