{
  "description": "Inventory residue/meromorphic API",
  "subagent_type": "Explore",
  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We need a precise capability inventory of the vendored complex-analysis API to plan a formalization of a rectangle-contour residue argument for \u03b6\u2032/\u03b6 with a higher-order pole.\n\nSurvey THOROUGHLY and report with exact declaration names + file paths + full signatures (hypotheses and conclusions) for:\n\n1. In D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\FromPNTPlus\\ResidueCalcOnRectangles.lean (1153 lines):\n   - Full signature of `RectangleIntegral'_eq_sumResiduesIn` (ALL hypotheses \u2014 especially whether HasSimplePolesOn is among them, and what RectangleIntegral' vs RectangleIntegral are)\n   - The `residue` placeholder def, `sumResiduesIn`, `HasSimplePolesOn`, and any lemmas for computing residues (residue_eq_of_tendsto, ResidueMult, logDerivResidue family \u2014 full signatures)\n   - `RectanglePullToNhdOfPole*` signatures\n   - Any higher-order-pole residue machinery (search for \"order\", \"principalPart\", \"MeromorphicNF\", \"iterated deriv\")\n2. In the same directory: Rectangle.lean and ZetaBounds.lean \u2014 list the main public theorems (one line each: name + what it proves). Especially anything about \u03b6\u2032/\u03b6 bounds, zeta residue at 1, logDeriv.\n3. In the vendored Mathlib at D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib:\n   - Does Mathlib have its own `residue` API? Search Analysis/Complex/ and Geometry/ for \"residue\" (case-insensitive, def/theorem names). Report what exists in THIS vendored version.\n   - `MeromorphicAt` / `meromorphicOrderAt` API: which file, and list key lemmas (especially: order of product/sum, meromorphicOrderAt_eq_int_iff, normal form MeromorphicNFAt, any Laurent-series or principal-part API, anything relating the order to derivatives).\n   - Digamma: file Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.lean \u2014 list main declarations.\n   - Search for \"Stieltjes\" and \"laurent\" (case-insensitive) related to riemannZeta \u2014 any Laurent expansion of \u03b6 at s=1?\n   - riemannZeta API: which files define it, the functional equation theorem names, trivial zeros theorem names (\u03b6(\u22122n)=0), pole at 1 API.\n   - Jensen's formula / Borel\u2013Carath\u00e9odory / Hadamard factorization / Weierstrass products: anything? Search Complex analysis dirs for \"Jensen\", \"Borel\", \"Carath\", \"Hadamard\", \"Weierstrass\", \"factorization\".\n4. Check what `Zeta23/FromPNTPlus/MediumPNT.lean` and `StrongPNTPrefix.lean` export at a high level (2-3 sentences each).\n\nReport as a structured list with exact names and signatures where I asked. Be thorough \u2014 this is the foundation of a multi-session design document. No need to read our project files (research/), only zeta-23-lean and the vendored Mathlib."
}

{
  "status": "completed",
  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We need a precise capability inventory of the vendored complex-analysis API to plan a formalization of a rectangle-contour residue argument for \u03b6\u2032/\u03b6 with a higher-order pole.\n\nSurvey THOROUGHLY and report with exact declaration names + file paths + full signatures (hypotheses and conclusions) for:\n\n1. In D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\FromPNTPlus\\ResidueCalcOnRectangles.lean (1153 lines):\n   - Full signature of `RectangleIntegral'_eq_sumResiduesIn` (ALL hypotheses \u2014 especially whether HasSimplePolesOn is among them, and what RectangleIntegral' vs RectangleIntegral are)\n   - The `residue` placeholder def, `sumResiduesIn`, `HasSimplePolesOn`, and any lemmas for computing residues (residue_eq_of_tendsto, ResidueMult, logDerivResidue family \u2014 full signatures)\n   - `RectanglePullToNhdOfPole*` signatures\n   - Any higher-order-pole residue machinery (search for \"order\", \"principalPart\", \"MeromorphicNF\", \"iterated deriv\")\n2. In the same directory: Rectangle.lean and ZetaBounds.lean \u2014 list the main public theorems (one line each: name + what it proves). Especially anything about \u03b6\u2032/\u03b6 bounds, zeta residue at 1, logDeriv.\n3. In the vendored Mathlib at D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib:\n   - Does Mathlib have its own `residue` API? Search Analysis/Complex/ and Geometry/ for \"residue\" (case-insensitive, def/theorem names). Report what exists in THIS vendored version.\n   - `MeromorphicAt` / `meromorphicOrderAt` API: which file, and list key lemmas (especially: order of product/sum, meromorphicOrderAt_eq_int_iff, normal form MeromorphicNFAt, any Laurent-series or principal-part API, anything relating the order to derivatives).\n   - Digamma: file Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.lean \u2014 list main declarations.\n   - Search for \"Stieltjes\" and \"laurent\" (case-insensitive) related to riemannZeta \u2014 any Laurent expansion of \u03b6 at s=1?\n   - riemannZeta API: which files define it, the functional equation theorem names, trivial zeros theorem names (\u03b6(\u22122n)=0), pole at 1 API.\n   - Jensen's formula / Borel\u2013Carath\u00e9odory / Hadamard factorization / Weierstrass products: anything? Search Complex analysis dirs for \"Jensen\", \"Borel\", \"Carath\", \"Hadamard\", \"Weierstrass\", \"factorization\".\n4. Check what `Zeta23/FromPNTPlus/MediumPNT.lean` and `StrongPNTPrefix.lean` export at a high level (2-3 sentences each).\n\nReport as a structured list with exact names and signatures where I asked. Be thorough \u2014 this is the foundation of a multi-session design document. No need to read our project files (research/), only zeta-23-lean and the vendored Mathlib.",
  "agentId": "a35f4584ec1204acf",
  "agentType": "Explore",
  "content": [
    "type='text' text=\"Below is a read-only capability inventory of the vendored complex-analysis API, organized by your four questions. Exact declaration names, file paths, and full signatures are given where requested.\\n\\n---\\n\\n## 1. `Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean`\\n\\nFile: `D:/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean`.\\n\\n### Rectangle integrals\\n\\n```lean\\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E\\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E\\n\\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E\\n-- = HIntegral f z.re w.re z.im - HIntegral f z.re w.re w.im\\n--   + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\\n\\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E\\n-- = (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\\n```\\n\\nSo `RectangleIntegral` is the raw boundary integral; `RectangleIntegral'` is normalized by `2\u03c0i`.\\n\\n### Pull-to-pole lemmas\\n\\n```lean\\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\\n    (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227\\n      z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\\n    (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p)\\n    (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\\n    (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\\\ {p})) :\\n    RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082\\n\\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102}\\n    (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im)\\n    (hp : Rectangle z w \u2208 \ud835\udcdd p)\\n    (fHolo : HolomorphicOn f (Rectangle z w \\\\ {p})) :\\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\\n      RectangleIntegral f z w =\\n      RectangleIntegral f (-c - I * c + p) (c + I * c + p)\\n\\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\\n    (fHolo : HolomorphicOn f (Rectangle z w \\\\ {p})) :\\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\\n      RectangleIntegral' f z w =\\n      RectangleIntegral' f (-c - I * c + p) (c + I * c + p)\\n```\\n\\n### Simple-pole residue scaffold\\n\\n```lean\\ndef HasSimplePolesOn (f : \u2102 \u2192 \u2102) (s : Set \u2102) : Prop :=\\n  \u2200 z \u2208 s, (-1 : \u2124) \u2264 meromorphicOrderAt f z\\n```\\n\\n```lean\\nnoncomputable def residue (f : \u2102 \u2192 \u2102) (z\u2080 : \u2102) : \u2102 :=\\n  Filter.limUnder (nhdsWithin z\u2080 {z\u2080}\u1d9c) (fun z \u21a6 (z - z\u2080) * f z)\\n\\nnoncomputable def sumResiduesIn (f : \u2102 \u2192 \u2102) (S : Set \u2102) : \u2102 :=\\n  \u2211' z : S, residue f z\\n```\\n\\n```lean\\nlemma residue_eq_of_tendsto {f : \u2102 \u2192 \u2102} {p c : \u2102}\\n    (h : Filter.Tendsto (fun z \u21a6 (z - p) * f z)\\n       (nhdsWithin p {p}\u1d9c) (nhds c)) :\\n    residue f p = c\\n\\nlemma residue_analyticAt_eq_zero {f : \u2102 \u2192 \u2102} {p : \u2102}\\n    (hf : AnalyticAt \u2102 f p) :\\n    residue f p = 0\\n\\nlemma simplePole_sub_residue_isBigO_one {f : \u2102 \u2192 \u2102} {p : \u2102}\\n    (hf : MeromorphicAt f p)\\n    (hord : meromorphicOrderAt f p = (-1 : \u2124)) :\\n    (f - (fun z \u21a6 residue f p / (z - p))) =O[nhdsWithin p {p}\u1d9c] (1 : \u2102 \u2192 \u2102)\\n\\nlemma residue_eq_zero_of_not_pole_of_meromorphicAt {F : \u2102 \u2192 \u2102} {s : \u2102}\\n    (hs_mero : MeromorphicAt F s)\\n    (hs_not_pole : 0 \u2264 meromorphicOrderAt F s) :\\n    residue F s = 0\\n\\nlemma sumResiduesIn_inter_eq_of_set_eq {F : \u2102 \u2192 \u2102} {Rn S2 P : Set \u2102}\\n    (h_set_eq : Rn \u2229 P = S2 \u2229 P)\\n    (h_residue_zero : \u2200 s \u2208 S2, s \u2209 P \u2192 residue F s = 0) :\\n    sumResiduesIn F (Rn \u2229 P) = sumResiduesIn F S2\\n```\\n\\n### Rectangle residue theorem for simple poles\\n\\n```lean\\nlemma RectangleIntegral'_eq_sumResiduesIn {f : \u2102 \u2192 \u2102} {z w : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (f_mero : MeromorphicOn f (Rectangle z w))\\n    (f_no_poles_boundary : Disjoint (RectangleBorder z w)\\n       {z | meromorphicOrderAt f z < 0})\\n    (f_poles_finite : (Rectangle z w \u2229 {z | meromorphicOrderAt f z < 0}).Finite)\\n    (f_simple_poles : HasSimplePolesOn f (Rectangle z w)) :\\n    RectangleIntegral' f z w =\\n    sumResiduesIn f (Rectangle z w \u2229 {z | meromorphicOrderAt f z < 0})\\n```\\n\\nSo **`HasSimplePolesOn is indeed a hypothesis`**, and the conclusion is a `tsum` over the set of points where `meromorphicOrderAt f z < 0`.\\n\\n### Other simple-pole residue lemmas in the same file\\n\\n```lean\\nlemma ResidueTheoremAtOrigin' {z w c : \u2102}\\n    (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) :\\n    RectangleIntegral (fun s => c / s) z w = 2 * I * \u03c0 * c\\n\\ntheorem ResidueTheoremInRectangle\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p) :\\n    RectangleIntegral' (fun s => c / (s - p)) z w = c\\n\\nlemma ResidueTheoremOnRectangleWithSimplePole {f g : \u2102 \u2192 \u2102} {z w p A : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\\n    (gHolo : HolomorphicOn g (Rectangle z w))\\n    (principalPart : Set.EqOn\\n       (f - fun s \u21a6 A / (s - p)) g (Rectangle z w \\\\ {p})) :\\n    RectangleIntegral' f z w = A\\n\\ntheorem ResidueTheoremOnRectangleWithSimplePole' {f : \u2102 \u2192 \u2102} {z w p A : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\\n    (fHolo : HolomorphicOn f (Rectangle z w \\\\ {p}))\\n    (near_p : (f - (fun s \u21a6 A / (s - p))) =O[\ud835\udcdd[\u2260] p] (1 : \u2102 \u2192 \u2102)) :\\n    RectangleIntegral' f z w = A\\n```\\n\\n### Higher-order-pole / normal-form machinery used by the proof\\n\\nThere is **no** general higher-order residue formula or `iteratedDeriv`-based residue computation. The file works around this by assuming `HasSimplePolesOn`. The main normal-form/order lemmas used are:\\n\\n| Name | Kind | Statement summary |\\n|------|------|-------------------|\\n| `meromorphicOrderAt_eq_neg_one_of_simplePole` | private | under `HasSimplePolesOn`, any negative order is exactly `-1` |\\n| `residue_toMeromorphicNFOn_eq_residue` | private | residue is unchanged by `toMeromorphicNFOn` |\\n| `principalPart_meromorphicOn` | private | finite sum `\u2211 c p / (s-p)` is meromorphic |\\n| `sub_principalPart_analyticAt_of_not_mem_poles` | private | subtracting the principal part is analytic away from the chosen poles |\\n| `meromorphicOrderAt_sub_principalPart_nonneg` | private | after subtracting the principal part, order at each pole becomes `\u2265 0` |\\n| `holoPart_holomorphicOn` | private | the \u201cholomorphic part\u201d after subtracting principal part is holomorphic on the rectangle |\\n| `principalPart_borderIntegrable` | private | the principal part is integrable on the rectangle boundary |\\n| `rectangleIntegral'_toMeromorphicNFOn_eq` | private | boundary integral unchanged by `toMeromorphicNFOn` |\\n| `toMeromorphicNFOn_add_integral` | private | splits `RectangleIntegral'` of `fNF` into holomorphic part + principal part |\\n| `rectangleIntegral'_sum_div_sub` | private | integral of a finite sum of simple-pole terms equals sum of coefficients |\\n| `sumResiduesIn_inter_eq_of_set_eq` | public | re-indexes `sumResiduesIn` using a set equality |\\n| `residue_eq_zero_of_not_pole_of_meromorphicAt` | public | residue is `0` at non-negative-order points |\\n\\nNo `iteratedDeriv` residue formula appears in this file.\\n\\n---\\n\\n## 2. `Rectangle.lean` and `ZetaBounds.lean` (main public theorems)\\n\\n### `Zeta23/FromPNTPlus/Rectangle.lean`\\n\\nKey public declarations:\\n\\n- `RectangleBorder z w` \u2014 the boundary of the axis-parallel rectangle with corners `z,w`.\\n- `Square p c` \u2014 axis-parallel square centered at `p` with half-side `c`.\\n- `segment_reProdIm_segment_eq_convexHull` \u2014 rectangle = convex hull of its four corners.\\n- `rectangle_in_convex` \u2014 if the four corners lie in a convex set, the whole rectangle lies in it.\\n- `mem_Rect` \u2014 membership in a rectangle in terms of real/imag inequalities.\\n- `rect_subset_iff` \u2014 `Rectangle z' w' \u2286 Rectangle z w` iff both corners are inside.\\n- `RectSubRect` / `RectSubRect'` \u2014 nested rectangles under coordinate inequalities.\\n- `rectangle_disjoint_singleton` / `rectangleBorder_disjoint_singleton` \u2014 conditions for a point to be disjoint from a rectangle/border.\\n- `rectangle_subset_punctured_rect` / `rectangleBorder_subset_punctured_rect` \u2014 shrink a rectangle around a pole.\\n- `rectangle_mem_nhds_iff` \u2014 a rectangle is a neighborhood of `p` iff `p` is in the open product interval.\\n- `Complex.nhds_hasBasis_square` \u2014 neighborhoods of `p` have a basis of squares `Square p c`.\\n- `square_mem_nhds` / `square_subset_square` / `SmallSquareInRectangle` \u2014 square neighborhoods can be chosen inside a given rectangle.\\n- `not_mem_rectangleBorder_of_rectangle_mem_nhds` \u2014 interior points are not on the border.\\n- Many `mapsTo_rectangle*_NoP` lemmas \u2014 boundary parametrizations map into punctured rectangles.\\n\\n### `Zeta23/FromPNTPlus/ZetaBounds.lean`\\n\\nMain public theorems (one line each):\\n\\n| Theorem | What it proves |\\n|---------|----------------|\\n| `ResidueOfTendsTo` | If `(s-p)f(s) \u2192 A` and `f` is holomorphic on a punctured neighborhood of `p`, then `f - A/(s-p)` is bounded near `p`. |\\n| `analyticAt_riemannZeta` | `\u03b6` is analytic at every `s \u2260 1`. |\\n| `differentiableAt_deriv_riemannZeta` | `\u03b6'` is differentiable at every `s \u2260 1`. |\\n| `riemannZetaResidue` | `\u03b6` has simple-pole residue `1` at `s = 1` (boundedness form). |\\n| `logDerivResidue'` / `logDerivResidue` / `logDerivResidue''` | If `f` has a simple pole with residue `A \u2260 0` and `f` is nonzero on a punctured neighborhood, then `f'/f + 1/(s-p) =O[\ud835\udcdd[\u2260]p] 1`, and a boundedness corollary. |\\n| `ResidueMult` | If `f - A/(s-p) =O 1` near `p` and `g` is holomorphic at `p`, then `f*g - A*g(p)/(s-p) =O 1`. |\\n| `riemannZetaLogDerivResidue` | `-\u03b6'/\u03b6 - 1/(s-1)` is bounded near `s = 1`. |\\n| `riemannZetaLogDerivResidueBigO` | `-\u03b6'/\u03b6 - 1/(s-1) =O[\ud835\udcdd[\u2260]1] 1`. |\\n| `ZetaUpperBnd` | `\u2016\u03b6(\u03c3+it)\u2016 \u2264 C log |t|` for `3 < |t|` and `1 - A/log|t| \u2264 \u03c3 \u2264 2`. |\\n| `ZetaDerivUpperBnd` | `\u2016\u03b6'(\u03c3+it)\u2016 \u2264 C (log |t|)\u00b2` in the same region. |\\n| `ZetaLowerBound3` | Lower bound `c(\u03c3-1)^{3/4}/(log|t|)^{1/4} \u2264 \u2016\u03b6(\u03c3+it)\u2016` for `1 < \u03c3 \u2264 2`. |\\n| `ZetaInvBnd` | `1/\u2016\u03b6(\u03c3+it)\u2016 \u2264 C (log|t|)^7` in a narrow strip around `Re s = 1`. |\\n| `ZetaLowerBnd` | `c/(log|t|)^7 \u2264 \u2016\u03b6(\u03c3+it)\u2016` for `\u03c3 < 1` in a narrow strip. |\\n| `ZetaZeroFree` | `\u03b6(\u03c3+it) \u2260 0` for `3 < |t|` and `\u03c3` in a narrow strip to the left of `1`. |\\n| `LogDerivZetaBnd` | `\u2016\u03b6'/\u03b6(\u03c3+it)\u2016 \u2264 C (log|t|)^9` in a narrow strip around `1`. |\\n| `LogDerivZetaBndUnif` | Uniform version of the above for `\u03c3 \u2265 1 - A/(log|t|)^9`. |\\n| `ZetaNoZerosOn1Line` | `\u03b6(1+it) \u2260 0` for all real `t`. |\\n| `ZetaNoZerosInBox` | For every `T`, there is `\u03c3\u2082 < 1` such that `\u03b6` has no zeros in `[[\u03c3\u2082,2]] \u00d7\u2102 [[-T,T]]`. |\\n| `LogDerivZetaHoloOn` | `\u03b6'/\u03b6` is holomorphic on any set not containing `1` and where `\u03b6` has no zeros. |\\n| `LogDerivZetaHolcSmallT` | `\u03b6'/\u03b6` is holomorphic on a small box around `1` minus `{1}`. |\\n| `LogDerivZetaHolcLargeT` | `\u03b6'/\u03b6` is holomorphic on a large box `[[1-A/log^9 T,2]] \u00d7\u2102 [[-T,T]] \\\\ {1}`. |\\n| `dlog_riemannZeta_bdd_on_vertical_lines_generalized` | `\u2016-\u03b6'(\u03c3\u2081+it)/\u03b6(\u03c3\u2081+it)\u2016 \u2264 \u2016\u03b6'(\u03c3\u2080)/\u03b6(\u03c3\u2080)\u2016` for `1 < \u03c3\u2080 \u2264 \u03c3\u2081`. |\\n| `triv_bound_zeta` | `\u2016-\u03b6'(\u03c3\u2080+it)/\u03b6(\u03c3\u2080+it)\u2016 \u2264 (\u03c3\u2080-1)\u207b\u00b9 + C` for `\u03c3\u2080 > 1`. |\\n\\n---\\n\\n## 3. Vendored Mathlib inventory\\n\\n### Complex / contour `residue` API\\n\\n**There is no native complex-contour `residue` API in this vendored Mathlib.** Searching `Mathlib/Analysis/Complex` and `Mathlib/Geometry` for \u201cresidue\u201d yields only:\\n\\n- `Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean` \u2014 algebraic-geometry residue fields (`X.residueField x`, `X.residue x`), unrelated to contour integration.\\n\\nNo file `Mathlib/Analysis/Complex/Residue.lean` or similar exists.\\n\\n### Meromorphic order / normal-form API\\n\\nThe core API is in:\\n\\n- `Mathlib/Analysis/Meromorphic/Basic.lean` \u2014 `MeromorphicAt`.\\n- `Mathlib/Analysis/Meromorphic/Order.lean` \u2014 `meromorphicOrderAt`.\\n- `Mathlib/Analysis/Meromorphic/NormalForm.lean` \u2014 `MeromorphicNFAt`, `toMeromorphicNFAt`, `toMeromorphicNFOn`.\\n- `Mathlib/Analysis/Meromorphic/FactorizedRational.lean` \u2014 factorized rational functions, zeros/poles extraction.\\n- `Mathlib/Analysis/Meromorphic/TrailingCoefficient.lean` \u2014 `meromorphicTrailingCoeffAt`.\\n\\nKey definitions and lemmas:\\n\\n```lean\\n-- Basic.lean\\ndef MeromorphicAt (f : \ud835\udd5c \u2192 E) (x : \ud835\udd5c) :=\\n  \u2203 (n : \u2115), AnalyticAt \ud835\udd5c (fun z \u21a6 (z - x) ^ n \u2022 f z) x\\n```\\n\\n```lean\\n-- Order.lean\\nnoncomputable def meromorphicOrderAt (f : \ud835\udd5c \u2192 E) (x : \ud835\udd5c) : WithTop \u2124\\n\\nlemma meromorphicOrderAt_eq_top_iff :\\n    meromorphicOrderAt f x = \u22a4 \u2194 \u2200\u1da0 z in \ud835\udcdd[\u2260] x, f z = 0\\n\\nlemma meromorphicOrderAt_eq_int_iff {n : \u2124} (hf : MeromorphicAt f x) :\\n    meromorphicOrderAt f x = n \u2194\\n    \u2203 g : \ud835\udd5c \u2192 E, AnalyticAt \ud835\udd5c g x \u2227 g x \u2260 0 \u2227\\n      \u2200\u1da0 z in \ud835\udcdd[\u2260] x, f z = (z - x) ^ n \u2022 g z\\n\\nlemma meromorphicOrderAt_ne_top_iff {f : \ud835\udd5c \u2192 E} {z\u2080 : \ud835\udd5c} (hf : MeromorphicAt f z\u2080) :\\n    meromorphicOrderAt f z\u2080 \u2260 \u22a4 \u2194\\n    \u2203 (g : \ud835\udd5c \u2192 E), AnalyticAt \ud835\udd5c g z\u2080 \u2227 g z\u2080 \u2260 0 \u2227\\n      f =\u1da0[\ud835\udcdd[\u2260] z\u2080] fun z \u21a6 (z - z\u2080) ^ ((meromorphicOrderAt f z\u2080).untop\u2080) \u2022 g z\\n```\\n\\nOrder under arithmetic:\\n\\n```lean\\ntheorem meromorphicOrderAt_mul {f g : \ud835\udd5c \u2192 \ud835\udd5c'} (hf : MeromorphicAt f x)\\n    (hg : MeromorphicAt g x) :\\n    meromorphicOrderAt (f * g) x =\\n    meromorphicOrderAt f x + meromorphicOrderAt g x\\n\\ntheorem meromorphicOrderAt_prod {x : \ud835\udd5c} {\u03b9 : Type*} {s : Finset \u03b9} {f : \u03b9 \u2192 \ud835\udd5c \u2192 \ud835\udd5c'}\\n    (hf : \u2200 i \u2208 s, MeromorphicAt (f i) x) :\\n    meromorphicOrderAt (\u220f i \u2208 s, f i) x =\\n    \u2211 i \u2208 s, meromorphicOrderAt (f i) x\\n\\ntheorem meromorphicOrderAt_add (hf\u2081 : MeromorphicAt f\u2081 x) (hf\u2082 : MeromorphicAt f\u2082 x) :\\n    min (meromorphicOrderAt f\u2081 x) (meromorphicOrderAt f\u2082 x) \u2264\\n    meromorphicOrderAt (f\u2081 + f\u2082) x\\n\\ntheorem meromorphicOrderAt_add_of_ne\\n    (hf\u2081 : MeromorphicAt f\u2081 x) (hf\u2082 : MeromorphicAt f\u2082 x)\\n    (h : meromorphicOrderAt f\u2081 x \u2260 meromorphicOrderAt f\u2082 x) :\\n    meromorphicOrderAt (f\u2081 + f\u2082) x =\\n    min (meromorphicOrderAt f\u2081 x) (meromorphicOrderAt f\u2082 x)\\n```\\n\\nOrder and derivatives:\\n\\n```lean\\nlemma meromorphicOrderAt_deriv_eq_sub_one [CompleteSpace E] {f : \ud835\udd5c \u2192 E} {x : \ud835\udd5c} {n : \u2124}\\n    (hn : (n : \ud835\udd5c) \u2260 0) (hf : meromorphicOrderAt f x = \u2191n) :\\n    meromorphicOrderAt (deriv f) x = \u2191(n - 1)\\n\\nlemma meromorphicOrderAt_deriv [CompleteSpace E] {f : \ud835\udd5c \u2192 E} {x : \ud835\udd5c} {n : \u2124}\\n    (hn : (\u2191(n + 1) : \ud835\udd5c) \u2260 0) (hf : meromorphicOrderAt f x = \u2191(n + 1)) :\\n    meromorphicOrderAt (deriv f) x = \u2191n\\n```\\n\\nNormal form:\\n\\n```lean\\n-- NormalForm.lean\\ndef MeromorphicNFAt :=\\n  f =\u1da0[\ud835\udcdd x] 0 \u2228\\n    \u2203 (n : \u2124) (g : \ud835\udd5c \u2192 E), AnalyticAt \ud835\udd5c g x \u2227 g x \u2260 0 \u2227\\n      f =\u1da0[\ud835\udcdd x] (\u00b7 - x) ^ n \u2022 g\\n\\nnoncomputable def toMeromorphicNFAt (f : \ud835\udd5c \u2192 E) (x : \ud835\udd5c) : \ud835\udd5c \u2192 E\\nnoncomputable def toMeromorphicNFOn (f : \ud835\udd5c \u2192 E) (U : Set \ud835\udd5c) : \ud835\udd5c \u2192 E\\n\\ntheorem meromorphicNFAt_toMeromorphicNFAt :\\n    MeromorphicNFAt (toMeromorphicNFAt f x) x\\n\\ntheorem meromorphicNFOn_toMeromorphicNFOn :\\n    MeromorphicNFOn (toMeromorphicNFOn f U) U\\n\\ntheorem toMeromorphicNFOn_eqOn_codiscrete (hf : MeromorphicOn f U) :\\n    f =\u1da0[Filter.codiscreteWithin U] toMeromorphicNFOn f U\\n\\ntheorem MeromorphicOn.toMeromorphicNFOn_eq_self_on_nhdsNE\\n    (hf : MeromorphicOn f U) (hx : x \u2208 U) :\\n    toMeromorphicNFOn f U =\u1da0[\ud835\udcdd[\u2260] x] f\\n\\ntheorem meromorphicOrderAt_toMeromorphicNFOn (hf : MeromorphicOn f U) (hx : x \u2208 U) :\\n    meromorphicOrderAt (toMeromorphicNFOn f U) x = meromorphicOrderAt f x\\n\\ntheorem toMeromorphicNFOn_eq_toMeromorphicNFAt_on_nhds\\n    (hf : MeromorphicOn f U) (hx : x \u2208 U) :\\n    toMeromorphicNFOn f U =\u1da0[\ud835\udcdd x] toMeromorphicNFAt f x\\n```\\n\\nPrincipal-part-like factorization:\\n\\n```lean\\n-- FactorizedRational.lean\\ntheorem MeromorphicOn.extract_zeros_poles {f : \ud835\udd5c \u2192 E} (h\u2081f : MeromorphicOn f U)\\n    (h\u2082f : \u2200 u : U, meromorphicOrderAt f u \u2260 \u22a4)\\n    (h\u2083f : (divisor f U).support.Finite) :\\n    \u2203 g : \ud835\udd5c \u2192 E, AnalyticOnNhd \ud835\udd5c g U \u2227 (\u2200 u : U, g u \u2260 0) \u2227\\n      f =\u1da0[codiscreteWithin U] (\u220f\u1da0 u, (\u00b7 - u) ^ divisor f U u) \u2022 g\\n\\ntheorem MeromorphicOn.extract_zeros_poles_log {f g : \ud835\udd5c \u2192 E} ...\\n    (h : f =\u1da0[codiscreteWithin U] (\u220f\u1da0 u, (\u00b7 - u) ^ D u) \u2022 g) :\\n    (log \u2016f \u00b7\u2016) =\u1da0[codiscreteWithin U]\\n      \u2211\u1da0 u, (D u * log \u2016\u00b7 - u\u2016) + (log \u2016g \u00b7\u2016)\\n```\\n\\nThere is **no** dedicated Laurent-series expansion or principal-part coefficient API beyond this normal-form / factorized-rational machinery.\\n\\n### Digamma\\n\\nFile: `Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.lean`.\\n\\n```lean\\nnoncomputable def Complex.digamma : \u2102 \u2192 \u2102 := logDeriv Gamma\\n\\ntheorem digamma_def : digamma = logDeriv Gamma\\n\\n@[simp] theorem digamma_zero : digamma 0 = 0\\n\\ntheorem digamma_one : digamma 1 = - Real.eulerMascheroniConstant\\n\\ntheorem digamma_one_half : digamma (1 / 2) = - 2 * log 2 - Real.eulerMascheroniConstant\\n\\ntheorem digamma_apply_add_one (s : \u2102) (hs : \u2200 m : \u2115, s \u2260 - m) :\\n    digamma (s + 1) = digamma s + s\u207b\u00b9\\n\\ntheorem meromorphic_digamma : Meromorphic digamma\\n```\\n\\n### Stieltjes / Laurent related to `riemannZeta`\\n\\n- **Stieltjes**: `Mathlib` has a general theory of Stieltjes functions/measures (`MeasureTheory/Measure/Stieltjes.lean`, `Probability/Kernel/Disintegration/MeasurableStieltjes.lean`, etc.), but **nothing named \u201cStieltjes\u201d is tied to `riemannZeta`** in this vendored version.\\n- **Laurent**: generic Laurent polynomial/series files exist (`Algebra/Polynomial/Laurent.lean`, `Algebra/Vertex/...`), and `Analysis/Complex/ValueDistribution/Cartan.lean` mentions a \u201cJensen-type identity.\u201d There is **no Laurent expansion of `\u03b6` at `s = 1`**. The closest result is the asymptotic expansion in `NumberTheory/Harmonic/ZetaAsymp.lean`:\\n\\n```lean\\ntheorem tendsto_riemannZeta_sub_one_div :\\n    Tendsto (fun s : \u2102 \u21a6 riemannZeta s - 1 / (s - 1))\\n      (\ud835\udcdd[\u2260] 1) (\ud835\udcdd Real.eulerMascheroniConstant)\\n```\\n\\ni.e. the principal part is `1/(s-1)` and the constant term is `\u03b3`.\\n\\n### `riemannZeta` API\\n\\nDefinitions and main results are in `Mathlib/NumberTheory/LSeries/RiemannZeta.lean`.\\n\\n```lean\\ndef completedRiemannZeta\u2080 (s : \u2102) : \u2102\\ndef completedRiemannZeta  (s : \u2102) : \u2102\\ndef riemannZeta            : \u2102 \u2192 \u2102 := hurwitzZetaEven 0\\n```\\n\\nKey theorems:\\n\\n```lean\\ntheorem differentiable_completedZeta\u2080 : Differentiable \u2102 completedRiemannZeta\u2080\\n\\ntheorem differentiableAt_completedZeta {s : \u2102} (hs : s \u2260 0) (hs' : s \u2260 1) :\\n    DifferentiableAt \u2102 completedRiemannZeta s\\n\\ntheorem differentiableAt_riemannZeta {s : \u2102} (hs' : s \u2260 1) :\\n    DifferentiableAt \u2102 riemannZeta s\\n\\ntheorem completedRiemannZeta\u2080_one_sub (s : \u2102) :\\n    completedRiemannZeta\u2080 (1 - s) = completedRiemannZeta\u2080 s\\n\\ntheorem completedRiemannZeta_one_sub (s : \u2102) :\\n    completedRiemannZeta (1 - s) = completedRiemannZeta s\\n\\ntheorem riemannZeta_one_sub {s : \u2102} (hs : \u2200 n : \u2115, s \u2260 -n) (hs' : s \u2260 1) :\\n    riemannZeta (1 - s) =\\n    2 * (2 * \u03c0) ^ (-s) * Gamma s * cos (\u03c0 * s / 2) * riemannZeta s\\n\\ntheorem riemannZeta_neg_two_mul_nat_add_one (n : \u2115) :\\n    riemannZeta (-2 * (n + 1)) = 0\\n\\ntheorem riemannZeta_residue_one :\\n    Tendsto (fun s \u21a6 (s - 1) * riemannZeta s) (\ud835\udcdd[\u2260] 1) (\ud835\udcdd 1)\\n\\ntheorem zeta_eq_tsum_one_div_nat_cpow {s : \u2102} (hs : 1 < re s) :\\n    riemannZeta s = \u2211' n : \u2115, 1 / (n : \u2102) ^ s\\n```\\n\\nSpecial values and pole data are further in:\\n\\n- `Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean`:\\n  - `tendsto_riemannZeta_sub_one_div` (principal part + \u03b3),\\n  - `riemannZeta_one` (`\u03b6 1 = (\u03b3 - log(4\u03c0))/2`),\\n  - `riemannZeta_one_ne_zero`,\\n  - `deriv_riemannZeta_zero`.\\n\\n- `Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean`:\\n  - `riemannZeta_two_mul_nat`,\\n  - `riemannZeta_neg_nat_eq_bernoulli`,\\n  - `riemannZeta_two`, `riemannZeta_four`.\\n\\n- `Mathlib/NumberTheory/LSeries/Nonvanishing.lean`:\\n  - `riemannZeta_ne_zero_of_one_le_re`,\\n  - `riemannZeta_ne_zero_of_one_lt_re`.\\n\\n- `Mathlib/NumberTheory/LSeries/ZetaZeros.lean`:\\n  - `riemannZetaZeros : Set \u2102`,\\n  - `isClosed_riemannZetaZeros`,\\n  - `isDiscrete_riemannZetaZeros`,\\n  - `IsCompact.inter_riemannZetaZeros_finite`.\\n\\n### Jensen / Borel\u2013Carath\u00e9odory / Hadamard / Weierstrass\\n\\n- **Borel\u2013Carath\u00e9odory**: `Mathlib/Analysis/Complex/BorelCaratheodory.lean`\\n  - `Complex.borelCaratheodory_zero`\\n  - `Complex.borelCaratheodory`\\n\\n- **Jensen\u2019s formula**: `Mathlib/Analysis/Complex/JensenFormula.lean`\\n  - `MeromorphicOn.circleAverage_log_norm`\\n  - `AnalyticOnNhd.circleAverage_log_norm`\\n  - `AnalyticOnNhd.sum_divisor_le`\\n\\n- **Hadamard three-lines theorem**: `Mathlib/Analysis/Complex/Hadamard.lean`\\n  - Several variants of `HadamardThreeLines` interpolation (not Hadamard factorization).\\n\\n- **Weierstrass theorem on uniform limits of holomorphic functions**: `Mathlib/Analysis/Complex/LocallyUniformLimit.lean`\\n  - `TendstoLocallyUniformlyOn.differentiableOn`\\n  - `TendstoLocallyUniformlyOn.deriv`\\n  - `differentiableOn_tsum_of_summable_norm`\\n  - `hasSum_deriv_of_summable_norm`\\n\\nThere is **no** Weierstrass factorization theorem or Hadamard factorization theorem in this vendored Mathlib. The project\u2019s `StrongPNTPrefix.lean` builds a finite Blaschke-factor argument by hand.\\n\\n---\\n\\n## 4. `MediumPNT.lean` and `StrongPNTPrefix.lean` exports\\n\\n### `Zeta23/FromPNTPlus/MediumPNT.lean`\\n\\nThis file implements the **smoothed explicit formula** step of the medium PNT proof. It defines the smoothed Chebyshev integral `SmoothedChebyshev` as a vertical Mellin inversion of `-\u03b6'/\u03b6` times a smooth cutoff, proves the Dirichlet-series identity `SmoothedChebyshevDirichlet` (the smoothed integral equals `\u2211 \u039b(n) \u00b7 Smooth1(n/X)`), and gives the approximation estimate `SmoothedChebyshevClose` showing that this smoothed sum is within `O(\u03b5 X log X)` of the Chebyshev psi function `\u03c8 X`.\\n\\n### `Zeta23/FromPNTPlus/StrongPNTPrefix.lean`\\n\\nThis is a sorry-free prefix of a Strong Prime Number Theorem development. It contains a self-contained complex-analysis toolkit: a local Borel\u2013Carath\u00e9odory derivative bound (`BorelCaratheodoryDeriv`), construction of a logarithm `J_B` of a nonvanishing analytic function (`LogOfAnalyticFunction`), a finite zero-set / Blaschke-factor setup (`ZeroFactor`, `Cf`, `BlaschkeB`), and culminates in `ZerosBound` \u2014 a bound on the weighted number of zeros of an analytic function in a smaller disk in terms of its sup norm on a larger disk. The remainder of the original `StrongPNT.lean` (after `ZerosBound`) is not ported.\""
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  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We are planning to formalize an explicit contour bound for the Li coefficients of \u03b6. I need a precise inventory of zero-counting infrastructure and the current Li-coefficient definitions.\n\nSurvey and report with exact declaration names + file paths + signatures:\n\n1. Zeta23 zero infrastructure, in D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\:\n   - Look in Defs/ and Defs.lean: how are nontrivial zeros of riemannZeta defined? Is there a zero-counting function (N(T) or similar)? Any API for: zeros are symmetric (\u03c1\u0304, 1\u2212\u03c1), zeros in critical strip, zero-free at Re \u2265 1?\n   - Search all of Zeta23/ for theorems about riemannZeta zeros (grep \"riemannZeta.*= 0\" or \"zeros\" in theorem/def names) \u2014 report the useful ones with signatures.\n   - Anything formalizing an explicit formula or zero-sum over \u03c1 (WeilEF/, ExplicitFormula/) \u2014 brief summary of what exists (we may reuse zero-enumeration techniques).\n2. In our package D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\:\n   - Find the definition of `liCoeff` (probably in LiSkeleton.lean or a Defs file): full definition, what zero set it sums over, how the tsum is set up, any summability lemmas already proved. Also `SInfty`, `sInftyLower`, `C0hat`, `Hhat`, `IIMaj`, `FAlt`, `Bn`, `marginTilde` \u2014 report each def's exact Lean code (they're short) and the file/line.\n   - Report the exact statements of `sInfty_lower_bound`, `marginTilde_convex`, `pos_of_convex_of_two_points` (signatures only).\n3. In the vendored Mathlib (D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib):\n   - Does Mathlib have von Mangoldt \u039b (ArithmeticFunction vonMangoldt)? The Chebyshev \u03c8? Brief note.\n   - Search for theorems about `riemannZeta` being nonzero (e.g., Re s \u2265 1) \u2014 report names.\n   - Any API for enumerating roots/zeros of analytic functions with counting (e.g., \"AnalyticOn... zeros finite on compacts\" \u2014 the identity theorem / discreteness of zeros)? Report names/files.\n\nReport as a structured list. Be thorough; exact names matter. Read-only."
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  "status": "completed",
  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We are planning to formalize an explicit contour bound for the Li coefficients of \u03b6. I need a precise inventory of zero-counting infrastructure and the current Li-coefficient definitions.\n\nSurvey and report with exact declaration names + file paths + signatures:\n\n1. Zeta23 zero infrastructure, in D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\:\n   - Look in Defs/ and Defs.lean: how are nontrivial zeros of riemannZeta defined? Is there a zero-counting function (N(T) or similar)? Any API for: zeros are symmetric (\u03c1\u0304, 1\u2212\u03c1), zeros in critical strip, zero-free at Re \u2265 1?\n   - Search all of Zeta23/ for theorems about riemannZeta zeros (grep \"riemannZeta.*= 0\" or \"zeros\" in theorem/def names) \u2014 report the useful ones with signatures.\n   - Anything formalizing an explicit formula or zero-sum over \u03c1 (WeilEF/, ExplicitFormula/) \u2014 brief summary of what exists (we may reuse zero-enumeration techniques).\n2. In our package D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\:\n   - Find the definition of `liCoeff` (probably in LiSkeleton.lean or a Defs file): full definition, what zero set it sums over, how the tsum is set up, any summability lemmas already proved. Also `SInfty`, `sInftyLower`, `C0hat`, `Hhat`, `IIMaj`, `FAlt`, `Bn`, `marginTilde` \u2014 report each def's exact Lean code (they're short) and the file/line.\n   - Report the exact statements of `sInfty_lower_bound`, `marginTilde_convex`, `pos_of_convex_of_two_points` (signatures only).\n3. In the vendored Mathlib (D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib):\n   - Does Mathlib have von Mangoldt \u039b (ArithmeticFunction vonMangoldt)? The Chebyshev \u03c8? Brief note.\n   - Search for theorems about `riemannZeta` being nonzero (e.g., Re s \u2265 1) \u2014 report names.\n   - Any API for enumerating roots/zeros of analytic functions with counting (e.g., \"AnalyticOn... zeros finite on compacts\" \u2014 the identity theorem / discreteness of zeros)? Report names/files.\n\nReport as a structured list. Be thorough; exact names matter. Read-only.",
  "agentId": "a6939341f0a284148",
  "agentType": "Explore",
  "content": [
    "type='text' text=\"Below is a read-only inventory of the requested zero-counting and Li-coefficient infrastructure, with exact declaration names, file paths, and signatures.\\n\\n---\\n\\n## 1. Zeta23 zero infrastructure (`D:/Python/Quantum/zeta-23-lean/Zeta23/`)\\n\\n### 1.1 Definitions of nontrivial zeros and counting functions\\n\\nAll of these are in `Zeta23/Statement.lean`.\\n\\n- **`IsNontrivialZero`** (line 38)\\n  ```lean\\n  def IsNontrivialZero (\u03c1 : \u2102) : Prop := riemannZeta \u03c1 = 0 \u2227 0 < \u03c1.re \u2227 \u03c1.re < 1\\n  ```\\n- **`zeroMult`** (line 43) \u2014 multiplicity as the analytic order:\\n  ```lean\\n  def zeroMult (\u03c1 : \u2102) : \u2115 := (analyticOrderAt riemannZeta \u03c1).toNat\\n  ```\\n- **`zerosIn`** (line 46) \u2014 zeros with ordinate in a window:\\n  ```lean\\n  def zerosIn (T\u2081 T\u2082 : \u211d) : Set \u2102 :=\\n    {\u03c1 | IsNontrivialZero \u03c1 \u2227 T\u2081 < \u03c1.im \u2227 \u03c1.im \u2264 T\u2082}\\n  ```\\n- **`Ncount`** (line 49) \u2014 counted with multiplicity:\\n  ```lean\\n  def Ncount (T\u2081 T\u2082 : \u211d) : \u2115 := \u2211\u1da0 \u03c1 \u2208 zerosIn T\u2081 T\u2082, zeroMult \u03c1\\n  ```\\n- **`Ndist`** (line 52) \u2014 distinct zeros:\\n  ```lean\\n  def Ndist (T\u2081 T\u2082 : \u211d) : \u2115 := (zerosIn T\u2081 T\u2082).ncard\\n  ```\\n- **`N0`** (line 56) \u2014 critical-line zeros with multiplicity:\\n  ```lean\\n  def N0 (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    \u2211\u1da0 \u03c1 \u2208 zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | \u03c1.re = 1 / 2}, zeroMult \u03c1\\n  ```\\n- **`N0star`** (line 59) \u2014 critical-line zeros without multiplicity:\\n  ```lean\\n  def N0star (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    (zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | \u03c1.re = 1 / 2}).ncard\\n  ```\\n- **`N0simple`** (line 62) \u2014 simple critical-line zeros:\\n  ```lean\\n  def N0simple (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    (zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | \u03c1.re = 1 / 2} \u2229 {\u03c1 | zeroMult \u03c1 = 1}).ncard\\n  ```\\n- **`Nsimple`** (line 65) \u2014 simple zeros anywhere:\\n  ```lean\\n  def Nsimple (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    (zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | zeroMult \u03c1 = 1}).ncard\\n  ```\\n\\nThere is **no single-argument `N(T)`**; the formalization uses the dyadic/window counts above and an abstract version in `ZeroConfig`.\\n\\n### 1.2 Abstract zero configuration and counting (`Zeta23/Defs.lean`)\\n\\n- **`ZeroConfig`** (lines 120\u2013130) \u2014 an abstract locally finite multiset of strip zeros, closed under `reflect`:\\n  ```lean\\n  structure ZeroConfig where\\n    carrier : Set \u2102\\n    mult : \u2102 \u2192 \u2115\\n    one_le_mult : \u2200 \u03c1 \u2208 carrier, 1 \u2264 mult \u03c1\\n    strip : \u2200 \u03c1 \u2208 carrier, 0 \u2264 \u03c1.re \u2227 \u03c1.re \u2264 1\\n    reflect_mem : \u2200 \u03c1 \u2208 carrier, reflect \u03c1 \u2208 carrier\\n    mult_reflect : \u2200 \u03c1 \u2208 carrier, mult (reflect \u03c1) = mult \u03c1\\n    finite_window : \u2200 T\u2081 T\u2082 : \u211d, (carrier \u2229 {\u03c1 | T\u2081 < \u03c1.im \u2227 \u03c1.im \u2264 T\u2082}).Finite\\n  ```\\n- **`reflect`** (line 108):\\n  ```lean\\n  def reflect (\u03c1 : \u2102) : \u2102 := 1 - (starRingEnd \u2102) \u03c1\\n  ```\\n- Counting functions on `ZeroConfig` (lines 145\u2013161): `ZeroConfig.N`, `ZeroConfig.Nd`, `ZeroConfig.N0`, `ZeroConfig.N0star`, `ZeroConfig.N0s`, `ZeroConfig.Ns`.\\n- The zero-side matrix / Weil sum (lines 164\u2013170):\\n  ```lean\\n  def Wsummand (f g : \u211d \u2192 \u2102) (\u03c1 : \u2102) : \u2102 := ...\\n  def W (f g : \u211d \u2192 \u2102) : \u2102 := \u2211' \u03c1 : Z.carrier, Z.Wsummand f g \u03c1\\n  ```\\n\\n### 1.3 \u03b6 zeros packaged as a `ZeroConfig`\\n\\nIn `Zeta23/Statement/SeamClosed.lean`:\\n\\n- **`zetaSeam`** (line 22): discharges all four seam obligations.\\n- **`zetaZeroConfig`** (line 26):\\n  ```lean\\n  def zetaZeroConfig : ZeroConfig := zetaZeros zetaSeam\\n  ```\\n  so `carrier = {\u03c1 | IsNontrivialZero \u03c1}` and `mult = zeroMult`.\\n- Simp lemmas at lines 28\u201338 relate `zetaZeroConfig.N` to `Ncount`, etc.\\n- **`zerosIn_finite`** (line 47): every ordinate window is finite.\\n\\n### 1.4 Symmetry (\u03c1 \u21a6 1 \u2212 \u03c1\u0304)\\n\\nIn `Zeta23/ZetaReflect.lean`:\\n\\n- **`zeta_reflect_zero`** (line 278)\\n  ```lean\\n  theorem zeta_reflect_zero : \u2200 \u03c1, IsNontrivialZero \u03c1 \u2192 IsNontrivialZero (reflect \u03c1)\\n  ```\\n- **`zeta_mult_reflect`** (line 314)\\n  ```lean\\n  theorem zeta_mult_reflect : \u2200 \u03c1, IsNontrivialZero \u03c1 \u2192 zeroMult (reflect \u03c1) = zeroMult \u03c1\\n  ```\\n\\nUseful reflection geometry lemmas appear in `Zeta23/ZeroSide.lean` (lines 593\u2013611):\\n```lean\\nlemma reflect_re (\u03c1 : \u2102) : (reflect \u03c1).re = 1 - \u03c1.re\\nlemma reflect_im (\u03c1 : \u2102) : (reflect \u03c1).im = \u03c1.im\\nlemma reflect_eq_self_iff (\u03c1 : \u2102) : reflect \u03c1 = \u03c1 \u2194 \u03c1.re = 1 / 2\\n```\\n\\n### 1.5 Finiteness / discreteness / zero-free facts\\n\\nIn `Zeta23/Statement/Seam.lean`:\\n\\n- **`riemannZeta_analyticOnNhd_compl_one`** (line 30)\\n  ```lean\\n  lemma riemannZeta_analyticOnNhd_compl_one :\\n      AnalyticOnNhd \u2102 riemannZeta ({1}\u1d9c : Set \u2102)\\n  ```\\n- **`analyticOrderAt_riemannZeta_ne_top`** (line 40)\\n  ```lean\\n  lemma analyticOrderAt_riemannZeta_ne_top {s : \u2102} (hs : s \u2260 1) :\\n      analyticOrderAt riemannZeta s \u2260 \u22a4\\n  ```\\n- **`ZetaSeam.one_le_mult_holds`** (line 50)\\n  ```lean\\n  theorem ZetaSeam.one_le_mult_holds :\\n      \u2200 \u03c1, IsNontrivialZero \u03c1 \u2192 1 \u2264 zeroMult \u03c1\\n  ```\\n- **`riemannZeta_zeros_locallyFinite`** (line 63)\\n  ```lean\\n  lemma riemannZeta_zeros_locallyFinite (z : \u2102) :\\n      \u2203 t \u2208 \ud835\udcdd z, (t \u2229 {\u03c1 : \u2102 | \u03c1 \u2260 1 \u2227 riemannZeta \u03c1 = 0}).Finite\\n  ```\\n- **`ZetaSeam.finite_window_holds`** (line 91)\\n  ```lean\\n  theorem ZetaSeam.finite_window_holds (T\u2081 T\u2082 : \u211d) :\\n      ({\u03c1 | IsNontrivialZero \u03c1} \u2229 {\u03c1 | T\u2081 < \u03c1.im \u2227 \u03c1.im \u2264 T\u2082}).Finite\\n  ```\\n\\nZero-free on `Re s \u2265 1` is imported from Mathlib (see Part 3) and used heavily; in Zeta23 it is invoked as `riemannZeta_ne_zero_of_one_le_re`.\\n\\n### 1.6 Riemann\u2013von Mangoldt / zero counting\\n\\nIn `Zeta23/Hypotheses.lean` (lines 67\u201370) the abstract hypothesis is:\\n```lean\\nstructure RiemannVonMangoldt (Z : ZeroConfig) : Prop where\\n  main : \u2203 C T\u2080 : \u211d, \u2200 T : \u211d, T\u2080 \u2264 T \u2192\\n      |(Z.N T (2 * T) : \u211d) - T / (2 * Real.pi) * ell1 T| \u2264 C * Real.log T\\n  local_count : \u2203 A\u2080 : \u211d, 1 \u2264 A\u2080 \u2227 \u2200 t : \u211d,\\n      (Z.N t (t + 1) : \u211d) \u2264 A\u2080 * Real.log (|t| + 3)\\n```\\n\\nFor \u03b6 itself, in `Zeta23/RvM/LocalCount.lean`:\\n\\n- **`zeta_local_zero_count`** (line 266)\\n  ```lean\\n  theorem zeta_local_zero_count : \u2203 A\u2080 : \u211d, 1 \u2264 A\u2080 \u2227 \u2200 t : \u211d,\\n      (Ncount t (t + 1) : \u211d) \u2264 A\u2080 * Real.log (|t| + 3)\\n  ```\\n- **`zetaZeroConfig_local_count`** (line 294)\\n  ```lean\\n  theorem zetaZeroConfig_local_count : \u2203 A\u2080 : \u211d, 1 \u2264 A\u2080 \u2227 \u2200 t : \u211d,\\n      (zetaZeroConfig.N t (t + 1) : \u211d) \u2264 A\u2080 * Real.log (|t| + 3)\\n  ```\\n\\nAnd the full Riemann\u2013von Mangoldt theorem is in `Zeta23/RvM/Statement.lean` (line 20):\\n```lean\\ntheorem riemannVonMangoldt (h\u0393 : GammaFacts) : RiemannVonMangoldt zetaZeroConfig\\n```\\n\\n### 1.7 Explicit formula / zero-sum infrastructure\\n\\nIn `Zeta23/ExplicitFormula.lean`:\\n\\n- **`EF.literatureRHS`** (line 70) \u2014 the right-hand side of the literature explicit formula.\\n- **`EF.EF_lit`** (line 81) \u2014 literature-form hypothesis:\\n  ```lean\\n  def EF_lit (Z : ZeroConfig) : Prop :=\\n    \u2200 k : \u211d \u2192 \u2102, ContDiff \u211d 2 k \u2192 HasCompactSupport k \u2192\\n      Summable (fun \u03c1 : Z.carrier => (Z.mult \u03c1 : \u2102) * paperFT k (gammaOf \u03c1)) \u2227\\n      \u2211' \u03c1 : Z.carrier, (Z.mult \u03c1 : \u2102) * paperFT k (gammaOf \u03c1) = literatureRHS k\\n  ```\\n- **`EF.EF_paper`** (line 88) \u2014 paper spectral form `W(f,g) = \u222b h_f conj(h_g) \u03bd_X`.\\n- **`EF.prop_EF_of_lit`** (line 703) \u2014 proves paper form from literature form.\\n\\nThe Weil-EF directory (`Zeta23/WeilEF/`) builds `EF_lit` for \u03b6 unconditionally:\\n\\n- `Zeta23/WeilEF/ZeroSummability.lean`:\\n  - `WeilEF.zero_sum_inv_sq_gen`\\n  - `WeilEF.EF_zero_sum_summable_gen`\\n  - `WeilEF.zero_sum_inv_sq`\\n  - `WeilEF.EF_zero_sum_summable`\\n\\n- `Zeta23/WeilEF/ZeroSumLimit.lean` (line 26):\\n  ```lean\\n  theorem zero_sum_limit (hs : ZetaSeam) {k : \u211d \u2192 \u2102} (hk : ContDiff \u211d 2 k)\\n      (hkc : HasCompactSupport k) {R : \u2115 \u2192 \u211d} ... :\\n      Tendsto (fun j : \u2115 => \u2211 \u03c1 \u2208 Z j, (zeroMult \u03c1 : \u2102) * Hfn k \u03c1) atTop\\n        (\ud835\udcdd (\u2211' \u03c1 : (zetaZeros hs).carrier, ((zetaZeros hs).mult \u03c1 : \u2102) * Hfn k \u03c1))\\n  ```\\n\\n- `Zeta23/WeilEF/Contour.lean` (line 31): rectangle identity / weighted argument principle.\\n- `Zeta23/WeilEF/FullLineAssembly.lean` (line 39): `full_line_identity`.\\n- `Zeta23/WeilEF/Main.lean` (lines 51, 270):\\n  ```lean\\n  theorem EF_lit_zeta (hs : ZetaSeam) : Zeta23.EF.EF_lit (zetaZeros hs)\\n  theorem EF_lit_zetaZeroConfig : Zeta23.EF.EF_lit zetaZeroConfig\\n  ```\\n\\nThese give absolutely convergent zero-summation over `{\u03c1 : \u2102 // IsNontrivialZero \u03c1}` weighted by `zeroMult`, exactly the enumeration technique reusable for a contour-bound on Li coefficients.\\n\\n---\\n\\n## 2. NguyenRH1 Li-coefficient definitions (`D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiSkeleton.lean`)\\n\\n### 2.1 The Keiper\u2013Li coefficient `liCoeff`\\n\\n- **`liCoeff`** (line 39)\\n  ```lean\\n  def liCoeff (n : \u2115) : \u2102 :=\\n    tsum fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\\n      (zeroMult \u03c1 : \u2102) * (1 - (1 - 1 / (\u03c1 : \u2102)) ^ n)\\n  ```\\n  It sums over the **subtype of nontrivial zeros** `{\u03c1 : \u2102 // IsNontrivialZero \u03c1}` with weight `zeroMult \u03c1`.  \\n  **Summability note:** the file contains an \u201chonesty note\u201d that `tsum` of a non-summable family is `0` in Lean; no `liCoeff`-specific summability lemma is proved in this file. The summability is intended to follow from `\u03a3 |\u03c1|\u207b\u00b2 < \u221e`.\\n\\n### 2.2 Supporting definitions (exact code)\\n\\n- **`lagariasA1`** (line 68)\\n  ```lean\\n  def lagariasA1 : \u211d := (Real.log (4 * Real.pi) + Real.eulerMascheroniConstant) / 2\\n  ```\\n\\n- **`SInfty`** (line 73)\\n  ```lean\\n  def SInfty (n : \u2115) : \u211d :=\\n    (1 - lagariasA1) * n - 1 +\\n      tsum fun m : {m : \u2115 // Odd m \u2227 3 \u2264 m} =>\\n        ((1 - 1 / (m : \u211d)) ^ n - 1 + (n : \u211d) / (m : \u211d))\\n  ```\\n\\n- **`lagariasC1`** (line 79)\\n  ```lean\\n  def lagariasC1 : \u211d :=\\n    (Real.eulerMascheroniConstant - 1 - Real.log (2 * Real.pi)) / 2\\n  ```\\n\\n- **`sInftyLower`** (line 84)\\n  ```lean\\n  def sInftyLower (n : \u211d) : \u211d :=\\n    (1 / 2) * n * Real.log n + (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4\\n  ```\\n\\n- **`C0hat`** (line 429)\\n  ```lean\\n  def C0hat (t : \u211d) : \u211d :=\\n    0.433 * Real.log (t + 2) + 0.886 * Real.log (Real.log (t + 2)) + 2.9\\n  ```\\n\\n- **`Hhat`** (line 433)\\n  ```lean\\n  def Hhat (T : \u211d) : \u211d :=\\n    3 * C0hat T ^ 2 + 10.5 * Real.log (T + 2) + 48\\n  ```\\n\\n- **`IIMaj`** (line 438)\\n  ```lean\\n  def IIMaj (n : \u211d) : \u211d :=\\n    2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 *\\n      2 ^ (-(2 * Real.sqrt n - 2))\\n  ```\\n\\n- **`marginTilde`** (line 447)\\n  ```lean\\n  def marginTilde (F : \u211d \u2192 \u211d) (n : \u211d) : \u211d :=\\n    let T := Real.sqrt n + 1\\n    ((1 / 2) * n * Real.log n +\\n        (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4) -\\n      (2 * F T + 4 * T * ((2 / 3) * C0hat T + 10.5 * Real.log (T + 2) + 48) +\\n        40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) +\\n        IIMaj n + 1) + 1\\n  ```\\n\\n- **`FAlt`** (line 2097)\\n  ```lean\\n  def FAlt (T : \u211d) : \u211d :=\\n    (1 / (2 * Real.pi)) * (T * Real.log T) + 0.137 * Real.log T +\\n      0.443 * Real.log (Real.log T) + (7 / 8 + 1.588)\\n  ```\\n\\n- **`Bn`** (line 2106)\\n  ```lean\\n  def Bn (F : \u211d \u2192 \u211d) (n : \u211d) : \u211d :=\\n    let T := Real.sqrt n + 1\\n    2 * F T + 4 * T * ((2 / 3) * C0hat T + 10.5 * Real.log (T + 2) + 48) +\\n      40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) + IIMaj n + 1\\n  ```\\n\\n### 2.2 Requested theorem signatures\\n\\n- **`sInfty_lower_bound`** (line 320)\\n  ```lean\\n  theorem sInfty_lower_bound (n : \u2115) (hn : 20 \u2264 n) :\\n      sInftyLower n \u2264 SInfty n\\n  ```\\n\\n- **`marginTilde_convex`** (line 1849)\\n  ```lean\\n  theorem marginTilde_convex (F : \u211d \u2192 \u211d) (hF : ContDiffOn \u211d 2 F (Set.Ioi 1))\\n      (h\u03c8 : \u2200 T : \u211d, Real.sqrt 20 + 1 \u2264 T \u2192\\n        deriv (deriv F) T - deriv F T / (T - 1) < 0) :\\n      ConvexOn \u211d (Set.Ici 20) (marginTilde F)\\n  ```\\n\\n- **`pos_of_convex_of_two_points`** (line 2409)\\n  ```lean\\n  theorem pos_of_convex_of_two_points {f : \u211d \u2192 \u211d} {a N\u2080 N\u2081 x : \u211d}\\n      (hconv : ConvexOn \u211d (Set.Ici a) f)\\n      (hN : a < N\u2080) (hN01 : N\u2080 < N\u2081) (hx : N\u2081 \u2264 x)\\n      (h0 : 0 < f N\u2080) (h01 : f N\u2080 < f N\u2081) :\\n      0 < f x\\n  ```\\n\\n---\\n\\n## 3. Mathlib vendored API (`D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/`)\\n\\n### 3.1 von Mangoldt and Chebyshev \u03c8\\n\\n- **`ArithmeticFunction.vonMangoldt`**  \\n  File: `Mathlib/NumberTheory/ArithmeticFunction/VonMangoldt.lean` (line 65)\\n  ```lean\\n  noncomputable def vonMangoldt : ArithmeticFunction \u211d :=\\n    \u27e8fun n => if IsPrimePow n then Real.log (minFac n) else 0, if_neg not_isPrimePow_zero\u27e9\\n  ```\\n  Notation `\u039b` is available in the `ArithmeticFunction` locale.\\n\\n- **`Chebyshev.psi`**  \\n  File: `Mathlib/NumberTheory/Chebyshev.lean` (line 74)\\n  ```lean\\n  noncomputable def psi (x : \u211d) : \u211d :=\\n    \u2211 n \u2208 Ioc 0 \u230ax\u230b\u208a, \u039b n\\n  ```\\n  Notation `\u03c8` is available in the `Chebyshev` scope. The same file also defines `Chebyshev.theta` (\u03b8) and many explicit bounds.\\n\\n### 3.2 \u03b6 non-vanishing on / to the right of the 1-line\\n\\n- **`riemannZeta_ne_zero_of_one_le_re`**  \\n  File: `Mathlib/NumberTheory/LSeries/Nonvanishing.lean` (line 411)\\n  ```lean\\n  lemma _root_.riemannZeta_ne_zero_of_one_le_re \u2983s : \u2102\u2984 (hs : 1 \u2264 s.re) :\\n      riemannZeta s \u2260 0\\n  ```\\n\\n- **`riemannZeta_ne_zero_of_one_lt_re`**  \\n  File: `Mathlib/NumberTheory/LSeries/Dirichlet.lean` (line 328)\\n  ```lean\\n  lemma riemannZeta_ne_zero_of_one_lt_re {s : \u2102} (hs : 1 < s.re) : riemannZeta s \u2260 0\\n  ```\\n\\n### 3.3 Identity theorem / discreteness of zeros / finite zero sets\\n\\nRelevant general theorems are in `Mathlib/Analysis/Analytic/IsolatedZeros.lean` and `Mathlib/Analysis/Analytic/Order.lean`.\\n\\n- **`AnalyticAt.eventually_eq_zero_or_eventually_ne_zero`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 125)\\n  ```lean\\n  theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt \ud835\udd5c f z\u2080) :\\n      (\u2200\u1da0 z in \ud835\udcdd z\u2080, f z = 0) \u2228 \u2200\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z \u2260 0\\n  ```\\n\\n- **`AnalyticOnNhd.eqOn_zero_of_preconnected_of_frequently_eq_zero`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 214)\\n  ```lean\\n  theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hU : IsPreconnected U) (h\u2080 : z\u2080 \u2208 U) (hfw : \u2203\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z = 0) :\\n      EqOn f 0 U\\n  ```\\n\\n- **`AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 238)\\n  ```lean\\n  theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hg : AnalyticOnNhd \ud835\udd5c g U) (hU : IsPreconnected U) (h\u2080 : z\u2080 \u2208 U)\\n      (hfg : \u2203\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z = g z) :\\n      EqOn f g U\\n  ```\\n\\n- **`AnalyticOnNhd.eqOn_zero_or_eventually_ne_zero_of_preconnected`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 219)\\n  ```lean\\n  theorem eqOn_zero_or_eventually_ne_zero_of_preconnected (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hU : IsPreconnected U) :\\n      EqOn f 0 U \u2228 \u2200\u1da0 x in codiscreteWithin U, f x \u2260 0\\n  ```\\n\\n- **`AnalyticOnNhd.preimage_zero_mem_codiscreteWithin`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 346)\\n  ```lean\\n  theorem AnalyticOnNhd.preimage_mem_codiscreteWithin {U : Set \ud835\udd5c} {s : Set E} {f : \ud835\udd5c \u2192 E}\\n      (hfU : AnalyticOnNhd \ud835\udd5c f U) (h\u2082f : \u2200\u1da0 x in codiscreteWithin U, f x \u2260 f x)\\n      (hs : s \u2208 codiscreteWithin (f '' U)) :\\n      f \u207b\u00b9' s \u2208 codiscreteWithin U\\n  ```\\n\\n- **`Filter.codiscreteWithin_iff_locallyFiniteComplementWithin`**  \\n  `Mathlib/Topology/DiscreteSubset.lean` (line 317)\\n  ```lean\\n  theorem codiscreteWithin_iff_locallyFiniteComplementWithin [T1Space X] {s U : Set X} :\\n      s \u2208 codiscreteWithin U \u2194 \u2200 z \u2208 U, \u2203 t \u2208 \ud835\udcdd z, Set.Finite (t \u2229 (U \\\\ s))\\n  ```\\n  This is the bridge from \u201ccodiscrete\u201d to \u201clocally finite complement,\u201d i.e. zeros are isolated in the domain.\\n\\n- **`analyticOrderAt_ne_top_of_isPreconnected`**  \\n  `Mathlib/Analysis/Analytic/Order.lean` (line 627)\\n  ```lean\\n  theorem analyticOrderAt_ne_top_of_isPreconnected {x y : \ud835\udd5c} (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hU : IsPreconnected U) (h\u2081x : x \u2208 U) (hy : y \u2208 U) (h\u2082x : analyticOrderAt f x \u2260 \u22a4) :\\n      analyticOrderAt f y \u2260 \u22a4\\n  ```\\n\\n- **Concrete finiteness on compacts for \u03b6 zeros**  \\n  `Mathlib/NumberTheory/LSeries/ZetaZeros.lean` defines:\\n  ```lean\\n  noncomputable def riemannZetaZeros : Set \u2102 := riemannZeta \u207b\u00b9' {0}\\n  lemma isClosed_riemannZetaZeros\\n  lemma isDiscrete_riemannZetaZeros\\n  lemma IsCompact.inter_riemannZetaZeros_finite {S : Set \u2102} (hS : IsCompact S) :\\n      (S \u2229 riemannZetaZeros).Finite\\n  ```\\n\\nThese are the main tools available for proving that an analytic function\u2019s zero set is discrete and hence finite on compact subsets, and for controlling zero-counting via `codiscreteWithin`.\\n\\n---\""
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