In the present note, we prove new lower bounds on large values of character sums \(\varDelta (x,q):=\max _{\chi \ne \chi _0} \big \vert \sum _{n\le x} \chi (n)\big \vert \) in certain ranges of x. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of L-functions, we improve some results of Hough in the range \(\log x = o\big (\sqrt{\log q}\big )\). Our results are expressed using the counting function of y-friable integers less than x, where we improve the level of smoothness y for short intervals.

中文翻译：

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短字符总和的最大大小

在本注释中，我们证明了字符和的大值\（\ varDelta（x，q）：= \ max _ {\ chi \ ne \ chi _0} \ big \ vert \ sum _ {n \ le x} \ chi（n）\ big \ vert \）在x的特定范围内。使用涉及作者的工作中开发的共振方法的实现，以展现大的L-函数，我们改善了Hough在\（\ log x = o \ big（\ sqrt {\ log q } \ big）\）。我们的结果使用小于x的y-易碎整数的计数函数表示，在短间隔内我们提高了平滑度y的水平。