Uniqueness of a normalized formal point-count zeta series

Over the complex numbers, the differential equation

Z' = Z * A

together with the constant coefficient of Z determines a formal power series uniquely. This gives the formal bridge needed after proving Euler's coefficient identity for an effective-divisor zeta series: that series is the canonical exponential formalPointCountZeta.

theorem BGS.HasseWeil.powerSeries_eq_of_constantCoeff_eq_of_derivative_eq_mul (F G A : PowerSeries ℂ) (hconstant : PowerSeries.constantCoeff F = PowerSeries.constantCoeff G) (hF : PowerSeries.derivative F = F * A) (hG : PowerSeries.derivative G = G * A) : F = G

Two formal power series with the same constant coefficient and the same logarithmic-derivative equation are equal.

theorem BGS.HasseWeil.eq_formalPointCountZeta_of_normalized_pointCountDerivative (Z : PowerSeries ℂ) (pointCount : ℕ → ℕ) (hZ0 : PowerSeries.constantCoeff Z = 1) (htrace : HasFormalZetaPointCountDerivative Z pointCount) : Z = formalPointCountZeta pointCount

A normalized series satisfying Euler's point-count derivative identity is the canonical exponential point-count zeta series.