TsumDivisorsAntidiagonal.lean`

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The Lambert series rearrangement, hypothesis-driven

This file proves the general Lambert series rearrangement ∑_{m≥1} c(m)rᵐ/(1-rᵐ) = ∑_{N≥1} (∑_{d ∣ N} c(d))rᴺ over a topological field, with the analytic inputs taken as hypotheses rather than derived from a norm. The rearrangement itself — regrouping the double series ∑_{m,j} c(m)r^{mj} along the fibres of (m, j) ↦ mj, which are exactly the divisor pairs of N — is pure algebra plus unconditional-summability bookkeeping. Only the two analytic facts (the geometric row expansions and the summability of the double series) genuinely use the topology, and they are what the hypotheses below isolate.

Two formulations

tsum_lambert_of_summable, over an arbitrary Hausdorff topological field, takes both analytic inputs as hypotheses:
- hgeo — each row has its geometric expansion, HasSum (fun j ↦ r^{m(j+1)}) (rᵐ/(1-rᵐ));
- hsum — the double series is (unconditionally) summable.
tsum_lambert_of_summable_of_isTopologicallyNilpotent, over a complete uniform field, replaces the per-row hgeo by the single, cleaner hypothesis that r is topologically nilpotent (IsTopologicallyNilpotent r, i.e. rⁿ → 0). This is a trade, not a free weakening. hgeo implies IsTopologicallyNilpotent r (a HasSum family is null along the cofinite filter — see the example motivating the companion), so the hypothesis is genuinely weaker; but reconstructing each row's HasSum from rⁿ → 0 still needs the row to be summable, and the only source of that is hsum, whose fibres are summable only when 𝕜 is complete (Summable.comp_injective). Completeness cannot be dropped: over a bare topological field rⁿ → 0 does not imply the geometric row is unconditionally summable. Topological nilpotency then supplies each row's value through the sequential geometric limit ∑_{j<N} (rᵐ)ʲ → (1-rᵐ)⁻¹ (isTopologicallyNilpotent_iff_tendsto_geomSum), which Summable.hasSum_iff_tendsto_nat upgrades to the required HasSum.

Both intended instantiations are complete, so the second formulation is the convenient one there; the first is kept for its full topological-field generality.

How this differs from the existing mathlib lemma

Mathlib's tsum_pow_div_one_sub_eq_tsum_sigma (in Mathlib.NumberTheory.TsumDivisorsAntidiagonal, whence the name of this file) states ∑' n : ℕ+, n ^ k * r ^ n / (1 - r ^ n) = ∑' n : ℕ+, σ k n * r ^ n for a complete nontrivially normed field and ‖r‖ < 1. Two restrictions are baked into it:

The analysis is normed. Summability of the double series and the geometric row expansions are established through ‖·‖-estimates, so the lemma is unusable over a nonarchimedean local field in the ValuativeRel framework, which carries no canonical NormedField instance (its valuation takes values in an abstract rank-one group with zero, not in ℝ).
The coefficients are the powers c(d) = dᵏ (so the divisor sums are σ k). The rearrangement works for arbitrary coefficients; the Tate curve needs c(d) = -((5d³ + 7d⁵)/12), which is not a power of d.

The two known Lambert lemmas become corollaries by discharging the hypotheses in their respective settings:

over a complete nontrivially normed field, by hasSum_geometric_of_norm_lt_one / tendsto_pow_atTop_nhds_zero_of_norm_lt_one and summable_prod_mul_pow — this rederivation of tsum_pow_div_one_sub_eq_tsum_sigma is carried out below (as an example; in a mathlib PR it would simply become the new proof of that lemma);
over a nonarchimedean local field, by TateCurve.hasSum_geometric_succ / TateCurve.tendsto_pow_nhds_zero and TateCurve.summable_of_valuation_le_pow — recovering TateCurve.tsum_lambert of FLT.KnownIn1980s.EllipticCurves.TateCurveBaseChange (rewiring that file through this one is left for the upstreaming pass).

Imports (2)

Imported by 2 files

theorem tsum_lambert_of_summable Documented declaration: tsum_lambert_of_summable line 86 source
Source documentation
The Lambert series rearrangement ∑_{m≥1} c(m)rᵐ/(1-rᵐ) = ∑_{N≥1} (∑_{d ∣ N} c(d))rᴺ over a Hausdorff topological field, with the analytic inputs as hypotheses: hgeo says each row rᵐ/(1-rᵐ) has its geometric expansion ∑_{j≥1} r^{mj}, and hsum says the double series is (unconditionally) summable. Given these, the double series is regrouped along the fibres of (m, j) ↦ mj, which are exactly the divisor pairs of N.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, TopologicalSpace, IsTopologicalRing, T2Space, Nat, PNat, HasSum, HPow.hPow, HMul.hMul, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, HSub.hSub, SummationFilter.unconditional, Summable, Prod, Prod.fst, Prod.snd, Eq, tsum and 2 more
Proof uses: Eq.mpr, id, congrFun', SummationFilter, congrArg, optParam, mul_div_assoc, HasSum.mul_left, Subtype, Membership.mem, Finset, Nat.divisorsAntidiagonal, Function.comp, Sigma, Equiv, sigmaAntidiagonalEquivProd, Sigma.mk, of_eq_true, Eq.trans, True and 34 more
94 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem tsum_lambert_of_summable (r : 𝕜) (c : ℕ → 𝕜)
    (hgeo : ∀ m : ℕ+, HasSum (fun j : ℕ ↦ r ^ ((m : ℕ) * (j + 1)))
      (r ^ (m : ℕ) / (1 - r ^ (m : ℕ))))
    (hsum : Summable fun p : ℕ+ × ℕ+ ↦ c p.1 * r ^ ((p.1 : ℕ) * (p.2 : ℕ))) :
    ∑' m : ℕ+, c m * r ^ (m : ℕ) / (1 - r ^ (m : ℕ)) =
      ∑' N : ℕ+, (∑ d ∈ (N : ℕ).divisors, c d) * r ^ (N : ℕ) := by
…
```
example (example at line 114) line 114 source
Source comment
`hgeo → IsTopologicallyNilpotent r`: the row hypothesis is strictly stronger than topological nilpotency, which is why the completeness-based companion below can take the weaker `IsTopologicallyNilpotent r` in its place. Only the first row `hgeo 1` is needed — a `HasSum` family is null along the cofinite filter (`= atTop` on `ℕ`), i.e. `rⁿ → 0`.
```
example (r : 𝕜)
    (hgeo : ∀ m : ℕ+, HasSum (fun j : ℕ ↦ r ^ ((m : ℕ) * (j + 1)))
      (r ^ (m : ℕ) / (1 - r ^ (m : ℕ)))) :
    IsTopologicallyNilpotent r := by
…
```
theorem isTopologicallyNilpotent_iff_tendsto_geomSum Documented declaration: isTopologicallyNilpotent_iff_tendsto_geomSum line 127 source
Source documentation
The sequential geometric limit that identifies each row's value in the companion below: over a Hausdorff topological field, r is topologically nilpotent iff the geometric partial sums ∑_{n<N} rⁿ converge to (1-r)⁻¹. This sequential convergence is weaker than HasSum, but combined with row summability from hsum (through Summable.hasSum_iff_tendsto_nat) it recovers the row HasSum that tsum_lambert_of_summable takes as hgeo.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, TopologicalSpace, IsTopologicalRing, T2Space, Iff, IsTopologicallyNilpotent, Filter.Tendsto, Nat, Finset.sum, Finset.range, HPow.hPow, Filter.atTop, nhds, Inv.inv, HSub.hSub, OfNat.ofNat
Proof uses: Iff.intro, Ne, Eq, False, one_ne_zero, tendsto_nhds_unique, tendsto_const_nhds, Eq.mp, congrFun', Filter, congrArg, funext, one_pow, Eq.symm, Eq.mpr, HDiv.hDiv, id, geom_sum_eq, Filter.Tendsto.div_const, Filter.Tendsto.sub_const and 53 more
119 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
theorem isTopologicallyNilpotent_iff_tendsto_geomSum (r : 𝕜) :
    IsTopologicallyNilpotent r ↔
      Filter.Tendsto (fun N ↦ ∑ n ∈ Finset.range N, r ^ n) Filter.atTop (nhds (1 - r)⁻¹) := by
…
```
theorem tsum_lambert_of_summable_of_isTopologicallyNilpotent Documented declaration: tsum_lambert_of_summable_of_isTopologicallyNilpotent line 166 source
Source documentation
The Lambert series rearrangement from topological nilpotency, over a complete uniform field: the companion to tsum_lambert_of_summable that trades the per-row hypothesis hgeo for the single assumption IsTopologicallyNilpotent r, at the cost of assuming 𝕜 complete.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, UniformSpace, IsUniformAddGroup, IsTopologicalRing, CompleteSpace, T2Space, Nat, IsTopologicallyNilpotent, Summable, Prod, PNat, HMul.hMul, Prod.fst, HPow.hPow, Prod.snd, SummationFilter.unconditional, Eq, tsum, HDiv.hDiv, HSub.hSub and 3 more
Proof uses: HasSum, HAdd.hAdd, Function.Injective, Prod.mk, Nat.succPNat, Nat.succPNat_injective, And.right, Iff.mp, And, Prod.ext_iff, Summable.comp_injective, id, Eq.mp, Filter.Tendsto, Function.comp, Filter.atTop, nhds, congrFun', Filter, congrArg and 83 more
116 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
theorem tsum_lambert_of_summable_of_isTopologicallyNilpotent (r : 𝕜) (c : ℕ → 𝕜)
    (htn : IsTopologicallyNilpotent r)
    (hsum : Summable fun p : ℕ+ × ℕ+ ↦ c p.1 * r ^ ((p.1 : ℕ) * (p.2 : ℕ))) :
    ∑' m : ℕ+, c m * r ^ (m : ℕ) / (1 - r ^ (m : ℕ)) =
      ∑' N : ℕ+, (∑ d ∈ (N : ℕ).divisors, c d) * r ^ (N : ℕ) := by
…
```
theorem hasSum_geometric_pow_mul_succ The geometric series of rᵐ, indexed from the first term: for ‖r‖ < 1 and m ≠ 0, ∑_{j≥1} r^{mj} = rᵐ/(1-rᵐ) line 220 source
Source documentation
The geometric series of rᵐ, indexed from the first term: for ‖r‖ < 1 and m ≠ 0, ∑_{j≥1} r^{mj} = rᵐ/(1-rᵐ). This discharges the hgeo hypothesis of tsum_lambert_of_summable over a normed field (no completeness is needed: the limit is explicit).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NontriviallyNormedField, LT.lt, Real, Norm.norm, OfNat.ofNat, Nat, Ne, HasSum, HPow.hPow, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HSub.hSub, SummationFilter.unconditional
Proof uses: Eq.mpr, id, Eq, congrFun', congrArg, norm_pow, pow_lt_one₀, norm_nonneg, funext, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat and 26 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem hasSum_geometric_pow_mul_succ (hr : ‖r‖ < 1) {m : ℕ} (hm : m ≠ 0) :
    HasSum (fun j : ℕ ↦ r ^ (m * (j + 1))) (r ^ m / (1 - r ^ m)) := by
…
```
theorem summable_prod_mul_pow' The variant of summable_prod_mul_pow with the weight on the first coordinate line 232 source
Source documentation
The variant of summable_prod_mul_pow with the weight on the first coordinate. This discharges the hsum hypothesis of tsum_lambert_of_summable for the weights dᵏ over a complete normed field.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NontriviallyNormedField, CompleteSpace, NormSMulClass, Int, Nat, LT.lt, Real, Norm.norm, OfNat.ofNat, Summable, Prod, PNat, HMul.hMul, HPow.hPow, Prod.fst, Prod.snd, SummationFilter.unconditional
Proof uses: Summable.congr, Prod.swap, Summable.prod_symm, summable_prod_mul_pow, of_eq_true, Eq, Eq.trans, True, congrFun', congrArg, mul_comm, eq_self
81 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem summable_prod_mul_pow' [NormSMulClass ℤ 𝕜] (k : ℕ) (hr : ‖r‖ < 1) :
    Summable fun p : ℕ+ × ℕ+ ↦ ((p.1 : ℕ) : 𝕜) ^ k * r ^ ((p.1 : ℕ) * (p.2 : ℕ)) := by
…
```
example (example at line 237) line 237 source
example (example at line 251) line 251 source
Source comment
Confirmation that the completeness-based companion is genuinely usable in this setting: a `NontriviallyNormedField` with `[CompleteSpace]` supplies the uniform-field instances the variant needs, and `‖r‖ < 1` discharges `IsTopologicallyNilpotent r`.
```
example (r : 𝕜) (c : ℕ → 𝕜) (hr : ‖r‖ < 1)
    (hsum : Summable fun p : ℕ+ × ℕ+ ↦ c p.1 * r ^ ((p.1 : ℕ) * (p.2 : ℕ))) :
    ∑' m : ℕ+, c m * r ^ (m : ℕ) / (1 - r ^ (m : ℕ)) =
      ∑' N : ℕ+, (∑ d ∈ (N : ℕ).divisors, c d) * r ^ (N : ℕ) :=
…
```