Analogies of Dedekind sums in function fields
Let \({\mathbb F}_q\) be a finite field, \(k={\mathbb F}_q(t)\), \(A={\mathbb F}_q[t]\), \(\Omega\) a completion of an algebraic closure of a completion of \(k\) with respect to the degree-valuation. Let \[ e(z):=\sum_{k=0}^\infty\frac{z^{q^k}}{D_k},\quad z\in\Omega, \] where \(D_0=1, D_k=(t^{q^k}-t)D_{k-1}^q, k\geq 1.\) Let \(a,b,c\in A\) with \((a,c)=(b,c)=1\) and \(c\) monic. For positive integers \(m\) with \(2m<q-1\) the author defines the Dedekind sum \[ D_m(a,b;c)=\frac{1}{c^m}\sum_{k\bmod c,k\not\equiv 0}e\left(\frac{\omega k}{c}\right)^{-q+1+2m}e\left(\frac{\omega a k}{c}\right)^{-m}e\left(\frac{\omega b k}{c}\right)^{-m} \] and proves several identities. For example, a reciprocity law: \[ D_1(a,b;c)+D_1(b,c;a)+D_1(c,a;b)=\frac{a^{q-1}+b^{q-1}+c^{q-1}-3}{abc(t^q-t)}. \]
scientific article; zbMATH DE number 5731633