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Cotangent zeta functions in function fields

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Abstract

We introduce and study an analogue of the cotangent zeta function in the function fields setting. We establish a relation of our newly introduced zeta function to the Apostol–Dedekind sum, and then prove a functional equation for our function. Finally, we compute special values of the cotangent zeta function at quadratic irrationals.

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References

  1. Apostol, T.M.: Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J. 17, 147–157 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apostol, T.M.: Theorems on generalized Dedekind sums. Pac. J. Math. 2, 1–9 (1952)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Berndt, B.C.: Dedekind sums and a paper of G. H. Hardy. J. Lond. Math. Soc. 13, 129–137 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charollois, P., Greenberg, M.: Rationality of secant zeta values. Ann. Math. Québec 38, 1–6 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Y.J., Yu, J.: On class number relations in characteristic two. Math. Z. 259, 197–216 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fukuhara, S., Yui, N.: A generating function of higher-dimensional Apostol-Dedekind sums and its reciprocity law. J. Number Theory 117, 87–105 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. González, C.D.: Class numbers of quadratic function fields and continued fractions. J. Number Theory 40, 38–59 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Goss, D.: Basic Structures of Function Field Arithmetic. Springer, Berlin (1998)

    MATH  Google Scholar 

  10. Hamahata, Y.: Dedekind sums with a parameter in function fields. Contemp. Math. 632, 139–150 (2015)

  11. Hayes, D.R.: Real quadratic function fields. In: CMS Conference Proceedings, vol. 7, pp. 203–236. AMS (1987)

  12. Lalín, M., Rodrigue, F., Rogers, M.: Secant zeta functions. J. Math. Anal. Appl. 409, 197–204 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lerch, M.: Sur une série analogue aux fonctions modulaires. C. R. Acad. Paris 138, 952–954 (1904)

    MATH  Google Scholar 

  14. Zagier, D.: Higher dimensional Dedekind sums. Math. Ann. 202, 149–172 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the referees for careful reading and valuable comments.

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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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Hamahata, Y. Cotangent zeta functions in function fields. Ramanujan J 42, 15–27 (2017). https://doi.org/10.1007/s11139-016-9816-y

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  • DOI: https://doi.org/10.1007/s11139-016-9816-y

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