Generalized Riemann Functions, Their Weights, and the Complete Graph

By a Riemann function we mean $f:\mathbb{Z}^n\to\mathbb{Z}$ such that $f(\mathbf{d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and $d_1+\cdots+d_n+C$ constant, $C$, large. adding $1$ to the Baker-Norine rank of graph, one gets an equivalent function, similarly related functions.
 To each associate $W: \mathbb{Z}^n\to \mathbb{Z}$ via Möbius inversion call weight function. We give evidence seems organize structure in simpler way: first, $f$ satisfies Riemann-Roch formula iff its symmetry condition. Second, will calculate certain graphs show quite simple describe; do this on two vertices complete graphs.
 For graphs, build work Cori Le Borgne who gave linear time method compute graph. The associated has extremely sparse (i.e., mostly zero). Our computation leads new algorithm rank, likely Borgne, but seemingly general $\mathbf{d}\in \mathbb{Z}^n$, namely$$r_{{\rm BN},K_n}(\mathbf{d}) =-1+\biggl| \biggl\{ i=0,\ldots,\deg(\mathbf{d}) \ \Bigm|\ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le \deg(\mathbf{d})-i\biggr\} \biggr|.$$However, which requires \mathbb{Z}^n$ be sorted parking easier evaluate $\mathbf{d}$.
 study functions natural generalization functions, with many same properties exhibited by functions.