A new form of the reciprocal relation for generalized Dedekind sums

Authors

  • Shiyuan Luo Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an, Shaanxi, China
  • Zhefeng Xu Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an, Shaanxi, China

DOI:

https://doi.org/10.3842/umzh.v77i6.8522

Keywords:

generalized Dedekind sums, generalized Hardy sums, Bernoulli polynomial, Fourier expansion, reciprocity formula, Dirichlet L-functions.

Abstract

UDC 517.5

We mainly generalize a reciprocal relation in the inverse form, namely,  $$S(2\overline{h}, m, n, k)+S(2\overline{k}, m, n, h)$$ for the generalized Dedekind sums by using the Fourier expansions of the Bernoulli polynomials and  analytic methods. Further, the reciprocal relation for generalized Hardy sums $s_5(h,m,k)$ is also derived. Moreover, we unexpectedly obtain a computational formula for one kind of the mean value of Dirichlet $L$-functions with the weight given by even character sums.

References

The full version of this paper will be published in Ukrainian Mathematical Journal, Vol. 77, No. 6, 2025.

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Published

01.08.2025

Issue

Vol. 77 No. 6 (2025)

Section

Research articles

How to Cite

Luo, Shiyuan, and Zhefeng Xu. “A New Form of the Reciprocal Relation for Generalized Dedekind Sums”. Ukrains’kyi Matematychnyi Zhurnal, vol. 77, no. 6, Aug. 2025, pp. 451–452, https://doi.org/10.3842/umzh.v77i6.8522.