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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| Дата: | 2026 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2026
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/8522 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1865794698053943296 |
|---|---|
| author | Luo, Shiyuan Xu, Zhefeng Luo, Shiyuan Xu, Zhefeng |
| author_facet | Luo, Shiyuan Xu, Zhefeng Luo, Shiyuan Xu, Zhefeng |
| author_institution_txt_mv | [
{
"author": "Shiyuan Luo",
"institution": "Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an, Shaanxi, China"
},
{
"author": "Zhefeng Xu",
"institution": "Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an, Shaanxi, China"
}
] |
| author_sort | Luo, Shiyuan |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T13:39:54Z |
| description | 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. |
| doi_str_mv | 10.3842/umzh.v77i6.8522 |
| first_indexed | 2026-03-24T03:40:06Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8522 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:40:06Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-85222026-03-21T13:39:54Z A new form of the reciprocal relation for generalized Dedekind sums A new form of the reciprocal relation for generalized Dedekind sums Luo, Shiyuan Xu, Zhefeng Luo, Shiyuan Xu, Zhefeng generalized Dedekind sums, generalized Hardy sums, Bernoulli polynomial, Fourier expansion, reciprocity formula, Dirichlet L-functions. 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. УДК 517.5 Нова форма оберненого співвідношення для узагальнених сум Дедекінда Основну увагу приділено узагальненню оберненого співвідношення у вигляді \[S(2\overline{h}, m, n, k) + S(2\overline{k}, m, n, h)\] для узагальнених сум Дедекінда, що здійснюється з використанням ряду Фур’є для многочленів Бернуллі та аналітичних методів. Далі, виведено обернене співвідношення для узагальнених сум Гарді $s_5(h, m, k)$. Крім того, несподівано отримано обчислювальну формулу для одного з типів середнього значення $L$-функцій Діріхле з вагою, заданою сумами парних символів. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8522 10.3842/umzh.v77i6.8522 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 6 (2025); 451–452 Український математичний журнал; Том 77 № 6 (2025); 451–452 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8522/10540 Copyright (c) 2025 Shiyuan Luo, Zhefeng Xu |
| spellingShingle | Luo, Shiyuan Xu, Zhefeng Luo, Shiyuan Xu, Zhefeng A new form of the reciprocal relation for generalized Dedekind sums |
| title | A new form of the reciprocal relation for generalized Dedekind sums |
| title_alt | A new form of the reciprocal relation for generalized Dedekind sums |
| title_full | A new form of the reciprocal relation for generalized Dedekind sums |
| title_fullStr | A new form of the reciprocal relation for generalized Dedekind sums |
| title_full_unstemmed | A new form of the reciprocal relation for generalized Dedekind sums |
| title_short | A new form of the reciprocal relation for generalized Dedekind sums |
| title_sort | new form of the reciprocal relation for generalized dedekind sums |
| topic_facet | generalized Dedekind sums generalized Hardy sums Bernoulli polynomial Fourier expansion reciprocity formula Dirichlet L-functions. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8522 |
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