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By a {\em Riemann function} we mean a function $f\colon{\mathbbZ}^n\to{\mathbb Z}$ such that $f({\bf d})$ is equals $0$ for $d_1+\cdots+d_n$sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for$d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rankfunction of a graph, one gets an equivalent Riemann function, and similarly forrelated rank functions.  To each Riemann function we associate a related function $W\colon{\mathbbZ}^n\to{\mathbb Z}$ via M\"obius inversion that we call the {\em weight} of theRiemann function. We give evidence that the weight seems to organize thestructure of a Riemann function in a simpler way: first, a Riemann function $f$satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetrycondition. Second, we will calculate the weight of the Baker-Norine rank forcertain graphs and show that the weight function is quite simple to describe;we do this for graphs on two vertices and for the complete graph.  For the complete graph, we build on the work of Cori and Le Borgne who gave alinear time method to compute the Baker-Norine rank of the complete graph. Theassociated weight function has a simple formula and is extremely sparse (i.e.,mostly zero). Our computation of the weight function leads to another lineartime algorithm to compute the Baker-Norine rank, via a formula likely relatedto one of Cori and Le Borgne, but seemingly simpler, namely $$ r_{{\rmBN},K_n}({\bf d}) = -1+\biggl| \biggl\{ i=0,\ldots,{\rm deg}({\bf d}) \ \Bigm|\ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le {\rm deg}({\bfd})-i \biggr\} \biggr|. $$  Our study of weight functions leads to a natural generalization of Riemannfunctions, with many of the same properties exhibited by Riemann functions.

Generalized Riemann Functions, Their Weights, and the Complete Graph