The rank of the multiplicativity defect: a coefficient-level law for the Davenport–Heilbronn function and its relatives

An Euler product of degree one is a Dirichlet series whose coefficients satisfy a(mn) = a(m)a(n). We measure the failure of this law by the matrix M(m,n) = a(mn) − a(m)a(n) and its rank. For the Davenport–Heilbronn function the failure is rank one and explicit (proved); in general a k-term combination of degree-one Euler products has a defect of rank at most k − 1 (proved). Both laws are checked to rounding on a finite block. A rank table predicted before computing scored 7 of 10 rows; the three misses are kicked functions, explained after the fact. The rank bounds the number of Euler-product terms from below but is not a classification: ζ + ε m−s has rank 2 and is no finite combination (proved). Nothing here touches a zero. Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, and the scripts, registered predictions and result files it cites.