A new form of the reciprocal relation for generalized Dedekind sums

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.