标题： 有限三角函数和的评价与关系 显示英文标题

标题： Evaluations and relations for finite trigonometric sums

摘要： 几种方法用于评估有限三角级数。 在每种情况下，要么该和之前尚未被评估，要么已经被评估，但仅通过解析方法，例如复分析或模变换公式。 我们建立了三角级数的互易性和三个求和关系。 受我们已经评估过的某些和的启发，我们在被加数中添加了互质条件，从而定义了拉马努金和的类似物，我们进而对它们进行了评估。 其中一种类似物导致了一个类似于弗兰埃尔-兰道准则的黎曼假设的判据。 显示英文摘要

摘要： Several methods are used to evaluate finite trigonometric sums. In each case, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation formulas. We establish both reciprocity and three sum relations for trigonometric sums. Motivated by certain sums that we have evaluated, we add coprime conditions to the summands and thereby define analogues of Ramanujan sums, which we in turn evaluate. One of these analogues leads to a criterion for the Riemann Hypothesis, analogous to the Franel-Landau criterion.