In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.

中文翻译：

---

经典 Dedekind 和的扭曲概括

在本文中，我们用广义伯努利数表示三个不同但相关的字符和。其中两个和是 Berndt 和 Arakawa-Ibukiyama-Kaneko 在模形式理论的背景下引入和研究的和的概括。第三个总和概括了 Ramanujan 在 theta 函数恒等式的上下文中已经研究过的总和。我们的方法是基本的，仅依赖于代数和数论的基本事实。