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Abstract

Let d_i be strictly increasing positive integer degrees and let M_i be positive integers with log M_i/d_i→δ∈(0,∞). For the Euler product with M_i independent polynomial generators of degree d_i and one degree-one generator, we prove that the upper logarithmic coefficient rate is δ and the lower rate is max{δ-θ,0}, where θ=lim sup_i log(d_(i+1))/d_i. This gives an exact convergence criterion with no regularity assumption on the support gaps. We contrast it with normal connected locally finite graded domains of Krull dimension two, obtained by adapting an infinite-pole construction of Maclean. Their lower rate is δ/R, where R=lim sup_i d_(i+1)/d_i, and their upper rate is δ. In particular, normality, finite dimension and a degree-one nonzerodivisor do not jointly force convergence of the exponential Hilbert rate.