In this note, an upper bound for the sum of fractional parts of certain smooth functions is given. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl’s bound for exponential sums and a device used by Popov.

中文翻译：

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某些小数部分的总和

在本笔记中，给出了某些平滑函数的小数部分之和的上限。这样的和自然地出现在解析数论的许多问题中。由于使用了指数和的外尔边界和波波夫使用的设备，这里的主要特征是对主项的改进。