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Fractional parts of Dedekind sums in function fields

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Abstract

In this paper, we determine the fractional part of a given Dedekind sum in a rational function field \(\mathbb {F}_q(T)\). Using this result, we prove two theorems. The first theorem states that any element of \(\mathbb {F}_q(T)\) is the fractional part of a certain Dedekind sum. In the second theorem, we introduce an analog of the Rademacher function and establish the composition law.

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References

  1. Bayad, A., Hamahata, Y.: Higher dimensional Dedekind sums in function fields. Acta Arithmetica 152, 71–80 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dedekind, R.: Erläuterungen zu den Fragmenten xxviii. In: Collected Works of Bernhard Riemann, pp. 466–478. Dover Publ., New York (1953)

  3. Carlitz, L.: On certain functions connected with polynomials in a Galois field. Duke Math. J. 1, 137–168 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dieter, U.: Beziehungen zwischen Dedekindschen Summen. Abh. Math. Sem. Univ. Hamburg 21, 109–125 (1957)

  5. Freitag, E.: Siegelsche Modulfunktionen, Springer (1983)

  6. Girstmair, K.: A criterion for the equality of Dedekind sums mod \(\mathbb{Z}\). Int. J. Number Theory 10, 565–568 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Girstmair, K.: On the fractional parts of Dedekind sums. Int J. Num Theory 11, 29–38 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Goss, D.: Basic structures of function field arithmetic, Springer (1998)

  9. Hamahata, Y.: Denominators of Dedekind sums in function fields. Int J. Num Theory 9, 1423–1430 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamahata, Y.: Continued fractions and Dedekind sums for function fields. In: Number Theory and Related Fields. In Memory of Alf van der Poorten, pp. 187–197, Springer (2013)

  11. Hamahata, Y.: Values of Dedekind sums for function fields. Functiones et Approximatio Commentarii Mathematici 52, 29–35 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hickerson, D.: Continued fractions and density results for Dedekind sums. J. Reine Angew. Math. 290, 113–116 (1977)

    MathSciNet  MATH  Google Scholar 

  13. Kirby, R., Melvin, P.: Dedekind sums, \(\mu \)-invariants and the signature cocycle. Math. Ann. 299, 231–267 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lang, S.: Introduction to modular forms, Springer (1976)

  15. Rademacher, H.: Zur Theorie der Dedekindschen Summen. Math. Z. 63, 445–463 (1956)

    Article  MathSciNet  Google Scholar 

  16. Rademacher, H., Grosswald, E.: Dedekind sums. The mathematical association of America, Washington, D.C. (1972)

  17. Rosen, M.: Number theory in function fields, Springer (2002)

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Authors and Affiliations

  1. Department of Applied Mathematics, Okayama University of Science, Ridai-cho 1-1, Okayama, 700-0005, Japan

    Yoshinori Hamahata

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  1. Yoshinori Hamahata
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Correspondence to Yoshinori Hamahata.

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Communicated by J. Schoißengeier.

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Hamahata, Y. Fractional parts of Dedekind sums in function fields. Monatsh Math 180, 549–562 (2016). https://doi.org/10.1007/s00605-015-0781-0

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  • DOI: https://doi.org/10.1007/s00605-015-0781-0

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