Nonvanishing of cyclotomic character L-functions at s = 1

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import Definitions.Def_TauCeti_NumberTheory_ArithmeticDirichletSeries_Basic
import Definitions.Def_TauCeti_NumberTheory_ArithmeticDirichletSeries_NormCoeff
import Definitions.Def_TauCeti_NumberTheory_ArithmeticDirichletSeries_Weight
import Definitions.Def_TauCeti_NumberTheory_Chebotarev_GaloisCharacter_Cyclotomic_Series
import Definitions.Def_TauCeti_NumberTheory_Chebotarev_GaloisCharacter_Weight
import Definitions.Def_TauCeti_NumberTheory_Chebotarev_RamifiedPrimes
import Definitions.Def_TauCeti_NumberTheory_NumberField_ArtinSymbol
import Definitions.Def_TauCeti_NumberTheory_NumberField_AutomorphismAction
import Definitions.Def_TauCeti_RingTheory_DedekindDomain_RamificationLocus
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharZero.Infinite
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Group.Pi.Units
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.GroupWithZero.Units.Fintype
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Algebra.Module.ZLattice.Covolume
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Algebra.Order.Ring.IsNonarchimedean
import Mathlib.Algebra.Order.Ring.Units
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.Algebra.Ring.Subgroup
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Analysis.AbsoluteValue.Equivalence
import Mathlib.Analysis.Analytic.Uniqueness
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.Complex.Order
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.MellinTransform
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.Group.Tannery
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Real.Pi.Bounds
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Summable
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Int.WithZero
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Card.Arithmetic
import Mathlib.Data.ZMod.Basic
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.Data.ZMod.Units
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Galois.Abelian
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.Galois.Infinite
import Mathlib.FieldTheory.KrullTopology
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Minpoly.IsConjRoot
import Mathlib.FieldTheory.Normal.Closure
import Mathlib.FieldTheory.Normal.Defs
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.GroupTheory.Abelianization.Defs
import Mathlib.GroupTheory.FiniteAbelian.Duality
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.IndexNormal
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.QuotientGroup.Basic
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic
import Mathlib.GroupTheory.Subgroup.Center
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.LinearAlgebra.Trace
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.NumberTheory.AbelSummation
import Mathlib.NumberTheory.ArithmeticFunction.Defs
import Mathlib.NumberTheory.ArithmeticFunction.LFunction
import Mathlib.NumberTheory.Cyclotomic.Gal
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.EulerProduct.ExpLog
import Mathlib.NumberTheory.Harmonic.ZetaAsymp
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.Convergence
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberTheory.LSeries.Dirichlet
import Mathlib.NumberTheory.LSeries.Linearity
import Mathlib.NumberTheory.LSeries.SumCoeff
import Mathlib.NumberTheory.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne
import Mathlib.NumberTheory.NumberField.ClassNumber
import Mathlib.NumberTheory.NumberField.Completion.FinitePlace
import Mathlib.NumberTheory.NumberField.Completion.InfinitePlace
import Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
import Mathlib.NumberTheory.NumberField.Cyclotomic.Galois
import Mathlib.NumberTheory.NumberField.DedekindZeta
import Mathlib.NumberTheory.NumberField.DirichletDensity
import Mathlib.NumberTheory.NumberField.Discriminant.Basic
import Mathlib.NumberTheory.NumberField.Discriminant.Different
import Mathlib.NumberTheory.NumberField.FractionalIdeal
import Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
import Mathlib.NumberTheory.NumberField.Ideal.Basic
import Mathlib.NumberTheory.NumberField.InfinitePlace.Basic
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
import Mathlib.NumberTheory.Padics.HeightOneSpectrum
import Mathlib.NumberTheory.RamificationInertia.Galois
import Mathlib.NumberTheory.RamificationInertia.Inertia
import Mathlib.NumberTheory.RamificationInertia.Unramified
import Mathlib.NumberTheory.ZetaValues
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Order.Northcott
import Mathlib.RingTheory.ClassGroup.Basic
import Mathlib.RingTheory.Complex
import Mathlib.RingTheory.DedekindDomain.AdicValuation
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.DedekindDomain.Different
import Mathlib.RingTheory.DedekindDomain.Factorization
import Mathlib.RingTheory.DedekindDomain.Ideal.Basic
import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
import Mathlib.RingTheory.DedekindDomain.SelmerGroup
import Mathlib.RingTheory.Frobenius
import Mathlib.RingTheory.Ideal.GoingUp
import Mathlib.RingTheory.Ideal.Int
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients
import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
import Mathlib.RingTheory.Ideal.Span
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Norm.Defs
import Mathlib.RingTheory.RamificationInertia.Basic
import Mathlib.RingTheory.RamificationInertia.Inertia
import Mathlib.RingTheory.RamificationInertia.Ramification
import Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.RingTheory.Trace.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain.Finite
import Mathlib.RingTheory.Unramified.Locus
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing
import Mathlib.Tactic.Group
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Topology.Algebra.InfiniteSum.Real
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.Order.Floor
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.MetricSpace.HausdorffDimension
import Mathlib.Topology.MetricSpace.Pseudo.Real
import Mathlib.Topology.UniformSpace.Real

section
set_option autoImplicit true
namespace TauCeti
end TauCeti
/-
Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: The Tau Ceti contributors
-/
/-!
# The continued L-series of a cyclotomic Galois character

For a finite Galois extension `F / K` of number fields and a character `χ` of `Gal(F/K)`,
`cyclotomicCharacterSeriesC K F χ` is a holomorphic continuation of the `L`-series of the ideal
weight `galoisCharacterWeight χ` to the half-plane `Re s > 1 - 1 / [K : ℚ]` when one exists, and
the `L`-series itself otherwise. For every `χ` it agrees with the `L`-series on `Re s > 1`.

For a cyclotomic extension `F = K(μ_m)` and a nontrivial character `χ`, the continuation exists,
so the series is analytic at `s = 1`, and its value at `s = 1` is nonzero. For the trivial
character the series is the Dedekind zeta function of `K` with the Euler factors at the ramified
primes deleted, which continues across `Re s = 1` apart from a single simple pole at `s = 1`.

## Main definitions

* `NumberField.Chebotarev.cyclotomicCharacterSeriesC`: the continued `L`-series of a Galois
  character.

## Main results

* `NumberField.Chebotarev.cyclotomicCharacterSeriesC_eq_LSeries`: on `Re s > 1` it is the
  `L`-series of `galoisCharacterWeight χ`.
* `NumberField.Chebotarev.logDeriv_cyclotomicCharacterSeriesC`: the logarithmic derivative
  agrees with that of the character `L`-series on `Re s > 1`.
* `NumberField.Chebotarev.differentiableOn_cyclotomicCharacterSeriesC`: for `F = K(μ_m)` and
  `χ` nontrivial it is holomorphic on `Re s > 1 - 1 / [K : ℚ]`.
* `NumberField.Chebotarev.analyticAt_cyclotomicCharacterSeriesC_one`: for `F = K(μ_m)` and `χ`
  nontrivial it is analytic at `s = 1`.
* `NumberField.Chebotarev.cyclotomicCharacterSeriesC_ne_zero_at_one`: for `F = K(μ_m)` and `χ`
  nontrivial it is nonzero at `s = 1`.

## References

* The nonvanishing argument at `s = 1`, in which the logarithm of the product of the `L`-series
  over all characters is a series with nonnegative coefficients that is unbounded as `s → 1⁺`, is
  analogous to `Chebotarev.artinLSeries_one_ne_zero` in AINTLIB (`github.com/CBirkbeck/aintlib`
  at commit `8102fa09bbf570f3e991adfdb2d6d70b48cb5b5e`, Apache-2.0),
  `projects/Chebotarev/CebotarevDensity/ZetaProduct.lean`, which proves the nonvanishing at
  `s = 1` of the Artin `L`-series of a nontrivial character of `Gal(K(μ_m)/K)`.
-/

 section

open _root_.Filter _root_.IsDedekindDomain _root_.NumberField _root_.TauCeti
open scoped _root_.Topology

namespace NumberField.Chebotarev
end NumberField.Chebotarev
section NumberField.Chebotarev
open NumberField NumberField.Chebotarev

variable (K F : Type*) [Field K] [NumberField K] [Field F] [NumberField F] [Algebra K F]
  [IsGalois K F]



variable {K F}





-- The continuation exists for a nontrivial ray class character: for `F = K(μ_m)` and a character
-- `χ` of `Gal(F/K)` with `χ ∘ cyclotomicArtin K F m` nontrivial, the `L`-series of the weight of
-- `χ` has a holomorphic continuation to `Re s > 1 - 1 / [K : ℚ]`.






-- As `s → 1⁺`, the prime sum over the primes of `K` that split completely in `F` tends to infinity.


-- At a completely split prime the Frobenius class is trivial, so its chosen representative is `1`.


-- The character sum of a power of the Galois weights at a prime is a nonnegative real number, and
-- at a completely split prime the sum of the weights themselves is `#Gal(F/K)`.


-- For `σ > 1`, summing over the characters the Euler-product logarithm series of their `L`-series
-- gives the real series `∑_{P, e} r(P, e) / (N(P) ^ σ) ^ (e + 1) / (e + 1)`, where `r(P, e)` is
-- the character sum of the `(e + 1)`-th powers of the weights at `P`.


-- For `F / K` abelian, the product over all characters `ψ` of `Gal(F/K)` of the `L`-series of
-- `galoisCharacterWeight ψ` tends to infinity in norm as `s → 1⁺` along the reals.




variable (K F) 

theorem NumberField.Chebotarev.cyclotomicCharacterSeriesC_ne_zero_at_one (m : ℕ) [_root_.NeZero m] [_root_.IsCyclotomicExtension {m} K F]
    (χ : (F ≃ₐ[K] F) →* ℂˣ) (hχ : χ ≠ 1) : _root_.NumberField.Chebotarev.cyclotomicCharacterSeriesC K F χ 1 ≠ 0 := by sorry