[Merged by Bors] - feat: sums over residue classes #11189
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bors d+ |
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@sgouezel I have now split the lemma |
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@sgouezel I'll merge this in the afternoon (CET), unless there is further feedback by then. |
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This adds a file `Analysis.SumOverResidueClass`, whose main result is
```lean
/-- A decreasing sequence of real numbers is summable on a residue class
if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f
```
We then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.
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kbuzzard
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This adds a file `Analysis.SumOverResidueClass`, whose main result is
```lean
/-- A decreasing sequence of real numbers is summable on a residue class
if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f
```
We then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.
dagurtomas
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Mar 22, 2024
This adds a file `Analysis.SumOverResidueClass`, whose main result is
```lean
/-- A decreasing sequence of real numbers is summable on a residue class
if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f
```
We then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.
utensil
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Mar 26, 2024
This adds a file `Analysis.SumOverResidueClass`, whose main result is
```lean
/-- A decreasing sequence of real numbers is summable on a residue class
if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f
```
We then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.
Louddy
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that referenced
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Apr 15, 2024
This adds a file `Analysis.SumOverResidueClass`, whose main result is
```lean
/-- A decreasing sequence of real numbers is summable on a residue class
if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f
```
We then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.
callesonne
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that referenced
this pull request
Apr 22, 2024
This adds a file `Analysis.SumOverResidueClass`, whose main result is
```lean
/-- A decreasing sequence of real numbers is summable on a residue class
if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f
```
We then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.
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This adds a file
Analysis.SumOverResidueClass, whose main result isWe then use this to show that the harmonic series still diverges when restricted to a residue class.
This is needed for the proof that the abscissa of absolute convergence of a Dirichlet L-series is 1.