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).
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Acknowledgements
The author greatly thanks the referee for helpful comments and suggestions which led to improvements of this paper. The author would like to dedicate this paper to memory of A.J. van der Poorten.
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Hamahata, Y. (2013). Continued Fractions and Dedekind Sums for Function Fields. In: Borwein, J., Shparlinski, I., Zudilin, W. (eds) Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6642-0_9
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DOI: https://doi.org/10.1007/978-1-4614-6642-0_9
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