feat(NumberTheory/EulerProduct): general statements about Euler produ…

…cts (#8410)

This adds a folder `NumberTheory.EulerProduct` and within it the file `Basic.lean`, which contains some general results on Euler products (for functions from the naturals into a complete normed field, with suitable properties).

Mathlib.lean

@@ -2658,6 +2658,7 @@ import Mathlib.NumberTheory.Dioph
 import Mathlib.NumberTheory.DiophantineApproximation
 import Mathlib.NumberTheory.DirichletCharacter.Basic
 import Mathlib.NumberTheory.Divisors
+import Mathlib.NumberTheory.EulerProduct.Basic
 import Mathlib.NumberTheory.FLT.Basic
 import Mathlib.NumberTheory.FLT.Four
 import Mathlib.NumberTheory.FermatPsp

Mathlib/NumberTheory/EulerProduct/Basic.lean

@@ -0,0 +1,195 @@
+/-
+Copyright (c) 2023 Michael Stoll. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Michael Stoll
+-/
+import Mathlib.Analysis.Normed.Field.InfiniteSum
+import Mathlib.Analysis.SpecificLimits.Normed
+import Mathlib.NumberTheory.ArithmeticFunction
+import Mathlib.NumberTheory.SmoothNumbers
+
+/-!
+# Euler Products
+
+The main result in this file is `EulerProduct.eulerProduct`, which says that
+if `f : ℕ → R` is norm-summable, where `R` is a complete normed commutative ring and `f` is
+multiplicative on coprime arguments with `f 0 = 0`, then
+`∏ p in primesBelow N, ∑' e : ℕ, f (p^e)` tends to `∑' n, f n` as `N` tends to infinity.
+`Nat.ArithmeticFunction.IsMultiplicative.eulerProduct` is a version of
+`EulerProduct.eulerProduct` for multiplicative arithmetic functions in the sense of
+`Nat.ArithmeticFunction.IsMultiplicative`.
+
+There is also a version `EulerProduct.eulerProduct_completely_multiplicative`, which states that
+`∏ p in primesBelow N, (1 - f p)⁻¹` tends to `∑' n, f n` as `N` tends to infinity,
+when `f` is completely multiplicative with values in a complete normed field `F`
+(implemented as `f : ℕ →*₀ F`).
+
+An intermediate step is `EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum`
+(and its variant `EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum_geometric`),
+which relates the finite product over primes to the sum of `f n` over `N`-smooth `n`.
+
+## Tags
+
+Euler product
+-/
+
+/-- If `f` is multiplicative and summable, then its values at natural numbers `> 1`
+have norm strictly less than `1`. -/
+lemma Summable.norm_lt_one {F : Type*} [NormedField F] [CompleteSpace F] {f : ℕ →* F}
+    (hsum : Summable f) {p : ℕ} (hp : 1 < p) :
+    ‖f p‖ < 1 := by
+  refine summable_geometric_iff_norm_lt_1.mp ?_
+  simp_rw [← map_pow]
+  exact hsum.comp_injective <| Nat.pow_right_injective hp
+
+open scoped Topology
+
+open Nat BigOperators
+
+namespace EulerProduct
+
+section General
+
+/-!
+### General Euler Products
+
+In this section we consider multiplicative (on coprime arguments) functions `f : ℕ → R`,
+where `R` is a complete normed commutative ring. The main result is `EulerProduct.eulerProduct`.
+-/
+
+variable {R : Type*} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R}
+variable (hf₁ : f 1 = 1) (hmul : ∀ {m n}, Nat.Coprime m n → f (m * n) = f m * f n)
+
+/-- We relate a finite product over primes to an infinite sum over smooth numbers. -/
+lemma summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum
+    (hsum : ∀ {p : ℕ}, p.Prime → Summable (fun n : ℕ ↦ ‖f (p ^ n)‖)) (N : ℕ) :
+    Summable (fun m : N.smoothNumbers ↦ ‖f m‖) ∧
+      HasSum (fun m : N.smoothNumbers ↦ f m) (∏ p in N.primesBelow, ∑' (n : ℕ), f (p ^ n)) := by
+  induction' N with N ih
+  · rw [smoothNumbers_zero, primesBelow_zero, Finset.prod_empty]
+    exact ⟨(Set.finite_singleton 1).summable (‖f ·‖), hf₁ ▸ hasSum_singleton 1 f⟩
+  · rw [primesBelow_succ]
+    split_ifs with hN
+    · constructor
+      · simp_rw [← (equivProdNatSmoothNumbers hN).summable_iff, Function.comp_def,
+                 equivProdNatSmoothNumbers_apply', map_prime_pow_mul hmul hN]
+        refine Summable.of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun _ ↦ norm_mul_le ..) ?_
+        apply summable_mul_of_summable_norm (f := fun (x : ℕ) ↦ ‖f (N ^ x)‖)
+          (g := fun (x : N.smoothNumbers) ↦ ‖f x.val‖) <;>
+          simp_rw [norm_norm]
+        exacts [hsum hN, ih.1]
+      · simp_rw [Finset.prod_insert (not_mem_primesBelow N),
+                 ← (equivProdNatSmoothNumbers hN).hasSum_iff, Function.comp_def,
+                 equivProdNatSmoothNumbers_apply', map_prime_pow_mul hmul hN]
+        apply (hsum hN).of_norm.hasSum.mul ih.2
+        -- `exact summable_mul_of_summable_norm (hsum hN) ih.1` gives a time-out
+        convert summable_mul_of_summable_norm (hsum hN) ih.1
+    · rwa [smoothNumbers_succ hN]
+
+/-- A version of `EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum`
+in terms of the value of the series. -/
+lemma prod_primesBelow_tsum_eq_tsum_smoothNumbers (hsum : Summable (‖f ·‖)) (N : ℕ) :
+    ∏ p in N.primesBelow, ∑' (n : ℕ), f (p ^ n) = ∑' m : N.smoothNumbers, f m :=
+  (summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum hf₁ hmul
+    (fun hp ↦ hsum.comp_injective <| Nat.pow_right_injective hp.one_lt) _).2.tsum_eq.symm
+
+/-- The following statement says that summing over `N`-smooth numbers
+for large enough `N` gets us arbitrarily close to the sum over all natural numbers
+(assuming `f` is norm-summable and `f 0 = 0`; the latter since `0` is not smooth). -/
+lemma norm_tsum_smoothNumbers_sub_tsum_lt (hsum : Summable f) (hf₀ : f 0 = 0)
+    {ε : ℝ} (εpos : 0 < ε) :
+    ∃ N₀ : ℕ, ∀ N ≥ N₀, ‖(∑' m : ℕ, f m) - (∑' m : N.smoothNumbers, f m)‖ < ε := by
+  obtain ⟨N₀, hN₀⟩ :=
+    summable_iff_nat_tsum_vanishing.mp hsum (Metric.ball 0 ε) <| Metric.ball_mem_nhds 0 εpos
+  simp_rw [mem_ball_zero_iff] at hN₀
+  refine ⟨N₀, fun N hN₁ ↦ ?_⟩
+  convert hN₀ _ <| N.smoothNumbers_compl.trans fun _ ↦ hN₁.le.trans
+  simp_rw [← tsum_subtype_add_tsum_subtype_compl hsum N.smoothNumbers,
+    add_sub_cancel', tsum_eq_tsum_diff_singleton (N.smoothNumbers)ᶜ hf₀]
+
+open Filter
+
+/-- The *Euler Product* for multiplicative (on coprime arguments) functions.
+If `f : ℕ → R`, where `R` is a complete normed commutative ring, `f 0 = 0`,
+`f 1 = 1`, `f` is multiplicative on coprime arguments,
+and `‖f ·‖` is summable, then `∏' p : {p : ℕ | p.Prime}, ∑' e, f (p ^ e) = ∑' n, f n`.
+Since there are no infinite products yet in Mathlib, we state it in the form of
+convergence of finite partial products. -/
+-- TODO: Change to use `∏'` once infinite products are in Mathlib
+theorem eulerProduct (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) :
+    Tendsto (fun n : ℕ ↦ ∏ p in primesBelow n, ∑' e, f (p ^ e)) atTop (𝓝 (∑' n, f n)) := by
+  rw [Metric.tendsto_nhds]
+  intro ε εpos
+  simp only [Finset.mem_range, eventually_atTop, ge_iff_le]
+  obtain ⟨N₀, hN₀⟩ := norm_tsum_smoothNumbers_sub_tsum_lt hsum.of_norm hf₀ εpos
+  use N₀
+  convert hN₀ using 3 with m
+  rw [dist_eq_norm, norm_sub_rev, prod_primesBelow_tsum_eq_tsum_smoothNumbers hf₁ hmul hsum m]
+
+/-- The *Euler Product* for a multiplicative arithmetic function `f` with values in a
+complete normed commutative ring `R`: if `‖f ·‖` is summable, then
+`∏' p : {p : ℕ | p.Prime}, ∑' e, f (p ^ e) = ∑' n, f n`.
+Since there are no infinite products yet in Mathlib, we state it in the form of
+convergence of finite partial products. -/
+-- TODO: Change to use `∏'` once infinite products are in Mathlib
+nonrec theorem _root_.Nat.ArithmeticFunction.IsMultiplicative.eulerProduct
+    {f : ArithmeticFunction R} (hf : f.IsMultiplicative) (hsum : Summable (‖f ·‖)) :
+    Tendsto (fun n : ℕ ↦ ∏ p in primesBelow n, ∑' e, f (p ^ e)) atTop (𝓝 (∑' n, f n)) :=
+  eulerProduct hf.1 hf.2 hsum f.map_zero
+
+end General
+
+section CompletelyMultiplicative
+
+/-!
+### Euler Products for completely multiplicative functions
+
+We now assume that `f` is completely multiplicative and has values in a complete normed field `F`.
+Then we can use the formula for geometric series to simplify the statement. This leads to
+`EulerProduct.eulerProduct_completely_multiplicative`.
+-/
+
+variable {F : Type*} [NormedField F] [CompleteSpace F]
+
+/-- Given a (completely) multiplicative function `f : ℕ → F`, where `F` is a normed field,
+such that `‖f p‖ < 1` for all primes `p`, we can express the sum of `f n` over all `N`-smooth
+positive integers `n` as a product of `(1 - f p)⁻¹` over the primes `p < N`. At the same time,
+we show that the sum involved converges absolutely. -/
+lemma summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric {f : ℕ →* F}
+    (h : ∀ {p : ℕ}, p.Prime → ‖f p‖ < 1) (N : ℕ) :
+    Summable (fun m : N.smoothNumbers ↦ ‖f m‖) ∧
+      HasSum (fun m : N.smoothNumbers ↦ f m) (∏ p in N.primesBelow, (1 - f p)⁻¹) := by
+  have hmul {m n} (_ : Nat.Coprime m n) := f.map_mul m n
+  convert summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum f.map_one hmul ?_ N with M hM <;>
+    simp_rw [map_pow]
+  · exact (tsum_geometric_of_norm_lt_1 <| h <| prime_of_mem_primesBelow hM).symm
+  · intro p hp
+    refine Summable.of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun _ ↦ norm_pow_le ..) ?_
+    exact summable_geometric_iff_norm_lt_1.mpr <| (norm_norm (f p)).symm ▸ h hp
+
+/-- A version of `EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric`
+in terms of the value of the series. -/
+lemma prod_primesBelow_geometric_eq_tsum_smoothNumbers {f : ℕ →* F} (hsum : Summable f) (N : ℕ) :
+    ∏ p in N.primesBelow, (1 - f p)⁻¹ = ∑' m : N.smoothNumbers, f m := by
+  refine (summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric ?_ N).2.tsum_eq.symm
+  exact fun {_} hp ↦ hsum.norm_lt_one hp.one_lt
+
+open Filter in
+/-- The *Euler Product* for completely multiplicative functions.
+If `f : ℕ →*₀ F`, where `F` is a complete normed field
+and `‖f ·‖` is summable, then `∏' p : {p : ℕ | p.Prime}, (1 - f p)⁻¹ = ∑' n, f n`.
+Since there are no infinite products yet in Mathlib, we state it in the form of
+convergence of finite partial products. -/
+-- TODO: Change to use `∏'` once infinite products are in Mathlib
+theorem eulerProduct_completely_multiplicative {f : ℕ →*₀ F} (hsum : Summable fun x => ‖f x‖) :
+    Tendsto (fun n : ℕ ↦ ∏ p in primesBelow n, (1 - f p)⁻¹) atTop (𝓝 (∑' n, f n)) := by
+  convert eulerProduct f.map_one (fun {m n} _ ↦ f.map_mul m n) hsum f.map_zero with N p hN
+  simp_rw [map_pow]
+  refine (tsum_geometric_of_norm_lt_1 <| summable_geometric_iff_norm_lt_1.mp ?_).symm
+  refine Summable.of_norm ?_
+  convert hsum.comp_injective <| pow_right_injective (prime_of_mem_primesBelow hN).one_lt
+  simp only [norm_pow, Function.comp_apply, map_pow]
+
+end CompletelyMultiplicative
+
+end EulerProduct

Mathlib/NumberTheory/SmoothNumbers.lean

@@ -111,6 +111,14 @@ lemma Prime.smoothNumbers_coprime {p n : ℕ} (hp : p.Prime) (hn : n ∈ smoothN
   rw [hp.coprime_iff_not_dvd, ← mem_factors_iff_dvd hn.1 hp]
   exact fun H ↦ (hn.2 p H).false
 
+/-- If `f : ℕ → F` is multiplicative on coprime arguments, `p` is a prime and `m` is `p`-smooth,
+then `f (p^e * m) = f (p^e) * f m`. -/
+lemma map_prime_pow_mul {F : Type*} [CommSemiring F] {f : ℕ → F}
+    (hmul : ∀ {m n}, Nat.Coprime m n → f (m * n) = f m * f n) {p : ℕ} (hp : p.Prime) (e : ℕ)
+    {m : p.smoothNumbers} :
+    f (p ^ e * m) = f (p ^ e) * f m :=
+  hmul <| Coprime.pow_left _ <| hp.smoothNumbers_coprime <| Subtype.mem m
+
 open List Perm in
 /-- We establish the bijection from `ℕ × smoothNumbers p` to `smoothNumbers (p+1)`
 given by `(e, n) ↦ p^e * n` when `p` is a prime. See `Nat.smoothNumbers_succ` for