A new form of the reciprocal relation for generalized Dedekind sums
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 genera...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8522 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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| DOI: | 10.3842/umzh.v77i6.8522 |