`[xⁿ] ψ_k f = ψ_k([x^{n/k}] f)` if `k ∣ n`, and `0` otherwise (`k ≠ 0`).

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Reciprocity_Defs
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.RatFunc.AsPolynomial
import Mathlib.RingTheory.PowerSeries.Expand
import Mathlib.RingTheory.PowerSeries.Log
set_option autoImplicit false

open TypeDensity

theorem TypeDensity.coeff_adamsSer (k : ℕ) (hk : k ≠ 0) (f : Ser) (n : ℕ) :
    PowerSeries.coeff n (adamsSer k f) =
      if k ∣ n then adamsCoef k (PowerSeries.coeff (n / k) f) else 0 := by sorry