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In a previous paper, we defined the generalized Dedekind-Rademacher sum attached to three Dirichlet characters \(\begin{aligned} \begin{aligned} & \varPhi _{r,s}\left( \begin{matrix} a & b & c \\ x & y & z \\ \psi & \delta & \chi \end{matrix} \right) = \sum _{h\,(\text {mod} \,cm_{\chi })} \chi (h) \mathscr {B}_{r,\psi }\left( am_{\psi }\frac{h+z}{cm_{\chi }}+x\right) \mathscr {B}_{s,\delta }\left( bm_{\delta }\frac{h+z}{cm_{\chi }}+y\right) \\ & (a,b,c \in \mathbb {N} \,\,\text {and}\,\, x,y,z\in \mathbb {R}), \end{aligned} \end{aligned}\) where \(\psi , \delta , \chi \) are Dirichlet characters modulo \(m_{\psi },m_{\delta }, m_{\chi }\) , respectively, and \(\mathscr {B}_{r,\psi }(x), \mathscr {B}_{s,\delta }(x)\) are generalized Bernoulli functions. In this paper, we prove Petersson-Knopp type identities for the sum \(\varPhi _{r,s}\left( \begin{matrix} a & b & c \\ 0 & 0 & 0 \\ \psi & \delta & \chi \end{matrix} \right) \) which generalize the results of Petersson-Knopp, Can and others.

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Petersson-Knopp Type Identities for Generalized Dedekind-Rademacher Sums Attached to Three Dirichlet Characters

  • 会议论文

  • 发表日期：

    2025年

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  • Brad Isaacson

摘要

In a previous paper, we defined the generalized Dedekind-Rademacher sum attached to three Dirichlet characters \(\begin{aligned} \begin{aligned} & \varPhi _{r,s}\left( \begin{matrix} a & b & c \\ x & y & z \\ \psi & \delta & \chi \end{matrix} \right) = \sum _{h\,(\text {mod} \,cm_{\chi })} \chi (h) \mathscr {B}_{r,\psi }\left( am_{\psi }\frac{h+z}{cm_{\chi }}+x\right) \mathscr {B}_{s,\delta }\left( bm_{\delta }\frac{h+z}{cm_{\chi }}+y\right) \\ & (a,b,c \in \mathbb {N} \,\,\text {and}\,\, x,y,z\in \mathbb {R}), \end{aligned} \end{aligned}\) where \(\psi , \delta , \chi \) are Dirichlet characters modulo \(m_{\psi },m_{\delta }, m_{\chi }\) , respectively, and \(\mathscr {B}_{r,\psi }(x), \mathscr {B}_{s,\delta }(x)\) are generalized Bernoulli functions. In this paper, we prove Petersson-Knopp type identities for the sum \(\varPhi _{r,s}\left( \begin{matrix} a & b & c \\ 0 & 0 & 0 \\ \psi & \delta & \chi \end{matrix} \right) \) which generalize the results of Petersson-Knopp, Can and others.