In recent years, maximizing Gál sums regained interest due to a firm link with large values of $L$‐functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the $L^1$‐norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of nonvanishing theta functions associated to Dirichlet characters; and (iii) new lower bounds for low moments of character sums.

中文翻译：

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小盖尔总结和应用

近年来，最大化Gál的总金额重新获得了兴趣，这是由于与价格的较大价值建立了牢固的联系 $\mathrm{大号}$-职能。在本文中，我们将针对Gal类型的少量款项进行调查${\mathrm{大号}}^{\mathrm{1个}}$-规范。我们还考虑了使最小乘积能量的加权形式最小化的问题。我们将估算结果应用于：（i）对Burgess的字符总和的对数改进，以改进Kerr，Shparlinski和Yau的先前结果；（ii）Louboutin和第二作者对与Dirichlet字符相关的不变的theta函数的数目的较早的下界进行了改进；（iii）在字符总和较低的时刻设置新的下限。