论文标题
持续的分数和笨拙的总和
Continued Fractions and Hardy Sums
论文作者
论文摘要
经典的Dedekind总和$ s（d，c）$可以表示为持续分数扩展的部分代表的总和。 hardy总和，在模块化组的子组下的$θ$ functions的对数转换中产生的模拟整数值总和已显示出满足许多反映经典dedekind总和的属性的属性。然而，到目前为止，作为部分商总和的表示。我们定义了非古典的持续分数，并证明hardy总和可以表示为这些持续分数的部分代表的总和。作为一个应用程序，我们给出了一个新的证明，表明Hardy Sums的图在$ \ Mathbf {r} \ times \ Mathbf {z} $中。
The classical Dedekind sums $s(d, c)$ can be represented as sums over the partial quotients of the continued fraction expansion of the rational $\frac{d}{c}$. Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of $θ$-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial quotients of these continued fractions. As an application, we give a new proof that the graph of the Hardy sums is dense in $\mathbf{R} \times \mathbf{Z}$.
