`B_a = P_{1/q}(G_a)`: the Euler transform with parameter `q` is inverted by the plethystic power `1/q`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Formula_StateCombinatorics
import Definitions.Def_TypeDensity_Reciprocity_Defs
import Definitions.Def_TypeDensity_Reciprocity_Quantum
import Theorems.Thm_TypeDensity_constantCoeff_logSer
import Theorems.Thm_TypeDensity_expSer_logSer
import Theorems.Thm_TypeDensity_pPow_pPow
import Theorems.Thm_TypeDensity_constantCoeff_clusterSer
import Theorems.Thm_TypeDensity_scalar_one
import Theorems.Thm_TypeDensity_plethSum_moebiusSum
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheory.ArithmeticFunction.Moebius
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.RamificationInertia.Basic
import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.PowerSeries.Expand
import Mathlib.RingTheory.PowerSeries.Log
set_option autoImplicit false

open TypeDensity

theorem TypeDensity.clusterSer_eq_pPow (a : ℕ) :
    clusterSer a = pPow (RatFunc.X : RatFunc ℚ)⁻¹ (eulerSer 0 a) := by sorry