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<p>We mainly generalize a reciprocal relation in the inverse form, namely,</p><p><InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S\left(2\overline{h }, m, n, k\right)+ S\left(2\overline{k }, m, n, h\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mfenced close=")" open="("> <mn>2</mn> <mover> <mi>h</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mfenced> <mo>+</mo> <mi>S</mi> <mfenced close=")" open="("> <mn>2</mn> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>h</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation></p><p>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 <i>s</i><sub>5</sub>(<i>h, m, k</i>) is also derived. Moreover, we unexpectedly obtain a computational formula for one kind of the mean value of Dirichlet <i>L</i>-functions with the weight given by even character sums.</p>

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A New Form of the Reciprocal Relation for Generalized Dedekind Sums

  • 期刊论文

  • 发表日期：

    2025年11月7日

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  • Shiyuan Luo,
  • Zhefeng Xu

摘要

We mainly generalize a reciprocal relation in the inverse form, namely,

\(S\left(2\overline{h }, m, n, k\right)+ S\left(2\overline{k }, m, n, h\right)\) S 2 h ¯ , m , n , k + S 2 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 s5(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.