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[Merged by Bors] - feat(NumberTheory/LSeries): introduce notations #11253

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[Merged by Bors] - feat(NumberTheory/LSeries): introduce notations #11253

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23 changes: 22 additions & 1 deletion Mathlib/NumberTheory/LSeries/Basic.lean
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Expand Up @@ -36,6 +36,13 @@ Given a sequence `f: ℕ → ℂ`, we define the corresponding L-series.
* `LSeriesSummable.isBigO_rpow`: if the `LSeries` of `f` is summable at `s`,
then `f = O(n^(re s))`.

## Notation

We introduce `L` as notation for `LSeries` and `↗f` as notation for `fun n : ℕ ↦ (f n : ℂ)`,
both scoped to `LSeries.notation`. The latter makes it convenient to use arithmetic functions
or Dirichlet characters (or anything that coerces to a function `N → R`, where `ℕ` coerces
to `N` and `R` coerces to `ℂ`) as arguments to `LSeries` etc.

## Tags

L-series
Expand All @@ -48,7 +55,6 @@ L-series
* Move `LSeries_add` and friends to a new file on algebraic operations on L-series
-/


open scoped BigOperators

open Complex
Expand Down Expand Up @@ -186,6 +192,21 @@ theorem LSeriesSummable_iff_of_re_eq_re {f : ℕ → ℂ} {s s' : ℂ} (h : s.re
#align nat.arithmetic_function.l_series_summable_iff_of_re_eq_re LSeriesSummable_iff_of_re_eq_re


/-!
### Notation
-/

@[inherit_doc]
scoped[LSeries.notation] notation "L" => LSeries
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Will there be a follow-up PR that uses the L notation in the files in NumberTheory/LSeries/*? I think it would be good to have that.


/-- We introduce notation `↗f` for `f` interpreted as a function `ℕ → ℂ`.

Let `R` be a ring with a coercion to `ℂ`. Then we can write `↗χ` when `χ : DirichletCharacter R`
or `↗f` when `f : ArithmeticFunction R` or simply `f : N → R` with a coercion from `ℕ` to `N`
as an argument to `LSeries`, `LSeriesHasSum`, `LSeriesSummable` etc. -/
scoped[LSeries.notation] notation:max "↗" f:max => fun n : ℕ ↦ (f n : ℂ)


/-!
### Criteria for and consequences of summability of L-series

Expand Down