Multiple Dedekind-Rademacher sums in function fields
The \textit{Dedekind sum} can be defined by NEWLINE\[NEWLINE s(h,k) := {1 \over {4k}} \sum_{ r=1 }^{ k-1 } \cot \left( {{\pi r} \over k} \right) \cot \left( {{\pi hr} \over k} \right) , NEWLINE\]NEWLINE where \(h\) and \(k\) are positive, relatively prime integers. Dedekind sums first appeared in Dedekind's work on the \(\eta\)-function and have been generalized in various ways, e.g., by \textit{D. Zagier} [Math. Ann. 202, 149--172 (1973; Zbl 0237.10025)] who considered an arbitrary number of cotangent factors, and by \textit{A. Bayad} and \textit{A. Raouj} [J. Number Theory 132, No. 2, 332--347 (2012; Zbl 1247.11058)] who replaced the cotangent by cotangent derivatives.NEWLINENEWLINEThe paper under review continues the development of Dedekind sums in function fields [\textit{Y. Hamahata}, Proc. Japan Acad., Ser. A 84, No. 7, 89--92 (2008; Zbl 1225.11055); \textit{S. Okada}, J. Number Theory 130, No. 8, 1750--1762 (2010; Zbl 1197.11155); \textit{A. Bayad} and \textit{Y. Hamahata}, Acta Arith. 152, No. 1, 71--80 (2012; Zbl 1301.11047)]. Let \(A:= {\mathbb F}_q[x]\) and \(\Lambda\) be an \(A\)-lattice, i.e., a finitely generated \(A\)-module satisfying certain discreteness conditions for a completion of the quotient field of \(A\), and let NEWLINE\[NEWLINE e_\Lambda (z) := z \prod_{ \lambda \in \Lambda } \!{}^\prime \left( 1 - {z \over \lambda} \right) NEWLINE\]NEWLINE where the \('\) indicates that the sum/product is only taken over nonzero and nonsingular values. Note that \(e_\Lambda (z)^{ -1 } \) is a natural analogue of the cotangent function in this setting, and in [Zbl 1301.11047] the authors studied NEWLINE\[NEWLINE s_\Lambda \left( a_n; a_1, a_2, \dots, a_{ n-1 } \right) = {{(-1)^{ n-1 }} \over { a_n }} \sum_{ \lambda \in \Lambda/a_n \Lambda } \!\!\!\!\!\!{}^\prime \;\;e_\Lambda \left( {{ a_1 \lambda } \over { a_n }} \right)^{ -1 } \cdots \;e_\Lambda \left( {{ a_{n-1} \lambda } \over { a_n }} \right)^{ -1 } , NEWLINE\]NEWLINE a function-field analogue of Zagier's higher-dimensional Dedekind sum mentioned above. In the present paper the authors replace the \((-1)\)-powers by arbitrary negative powers, giving rise to an analogue of cotangent derivatives. The main result is a \textit{reciprocity theorem}: when the arguments are coprime, a certain sum of multiple function-field Dedekind sums equals an expression involving analogues of Eisenstein series in this setting. The authors also prove a \textit{Petersson-Knopp} identity for the multiple function-field Dedekind sums, in analogy of such identities for Dedekind sums [\textit{M. I. Knopp}, J. Number Theory 12, 2--9 (1980; Zbl 0423.10015)] and their multivariate versions [\textit{M. Beck}, Acta Arith. 109, No. 2, 109--130 (2003; Zbl 1061.11043)]. They finish by exhibiting an appearance of the multiple function-field Dedekind sums in connection with the \textit{Goss \(L\)-function} [\textit{D. Goss}, Basic structures of function field arithmetic. Berlin: Springer (1998; Zbl 0892.11021)].
scientific article; zbMATH DE number 6332005