Continued Fractions and Dedekind Sums for Function Fields

Abstract

For the classical Dedekind sum d(a, c), Rademacher and Grosswald raised two questions: (1) Is \(\{(a/c,d(a,c))\ \vert \ a/c \in {\mathbf{Q}}^{{\ast}}\}\) dense in R 2? (2) Is \(\{d(a,c)\ \vert \ a/c \in {\mathbf{Q}}^{{\ast}}\}\) dense in R? Using the theory of continued fractions, Hickerson answered these questions affirmatively. In function fields, there exists a Dedekind sum s(a, c) (see Sect. 4) similar to d(a, c). Using continued fractions, we answer the analogous problems for s(a, c).