Metamath Proof Explorer

Theorem dchrmusumlem

Description: The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by n , is bounded. Equation 9.4.16 of Shapiro, p. 379. (Contributed by Mario Carneiro, 12-May-2016)

Ref Expression
Hypotheses rpvmasum.z ⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
rpvmasum.l ⊢ 𝐿 = ( ℤRHom ‘ 𝑍 )
rpvmasum.a ⊢ ( 𝜑 → 𝑁 ∈ ℕ )
dchrmusum.g ⊢ 𝐺 = ( DChr ‘ 𝑁 )
dchrmusum.d ⊢ 𝐷 = ( Base ‘ 𝐺 )
dchrmusum.1 ⊢ 1 = ( 0g ‘ 𝐺 )
dchrmusum.b ⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
dchrmusum.n1 ⊢ ( 𝜑 → 𝑋 ≠ 1 )
dchrmusum.f ⊢ 𝐹 = ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) )
dchrmusum.c ⊢ ( 𝜑 → 𝐶 ∈ ( 0 [,) +∞ ) )
dchrmusum.t ⊢ ( 𝜑 → seq 1 ( + , 𝐹 ) ⇝ 𝑇 )
dchrmusum.2 ⊢ ( 𝜑 → ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , 𝐹 ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑇 ) ) ≤ ( 𝐶 / 𝑦 ) )
Assertion dchrmusumlem ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) )

Proof

Step Hyp Ref Expression
1 rpvmasum.z ⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
2 rpvmasum.l ⊢ 𝐿 = ( ℤRHom ‘ 𝑍 )
3 rpvmasum.a ⊢ ( 𝜑 → 𝑁 ∈ ℕ )
4 dchrmusum.g ⊢ 𝐺 = ( DChr ‘ 𝑁 )
5 dchrmusum.d ⊢ 𝐷 = ( Base ‘ 𝐺 )
6 dchrmusum.1 ⊢ 1 = ( 0g ‘ 𝐺 )
7 dchrmusum.b ⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
8 dchrmusum.n1 ⊢ ( 𝜑 → 𝑋 ≠ 1 )
9 dchrmusum.f ⊢ 𝐹 = ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) )
10 dchrmusum.c ⊢ ( 𝜑 → 𝐶 ∈ ( 0 [,) +∞ ) )
11 dchrmusum.t ⊢ ( 𝜑 → seq 1 ( + , 𝐹 ) ⇝ 𝑇 )
12 dchrmusum.2 ⊢ ( 𝜑 → ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , 𝐹 ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑇 ) ) ≤ ( 𝐶 / 𝑦 ) )
13 fzfid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∈ Fin )
14 7 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑋 ∈ 𝐷 )
15 elfzelz ⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑛 ∈ ℤ )
16 15 adantl ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℤ )
17 4 1 5 2 14 16 dchrzrhcl ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) ∈ ℂ )
18 elfznn ⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑛 ∈ ℕ )
19 18 adantl ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℕ )
20 mucl ⊢ ( 𝑛 ∈ ℕ → ( μ ‘ 𝑛 ) ∈ ℤ )
21 19 20 syl ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( μ ‘ 𝑛 ) ∈ ℤ )
22 21 zred ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( μ ‘ 𝑛 ) ∈ ℝ )
23 22 19 nndivred ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( μ ‘ 𝑛 ) / 𝑛 ) ∈ ℝ )
24 23 recnd ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( μ ‘ 𝑛 ) / 𝑛 ) ∈ ℂ )
25 17 24 mulcld ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) ∈ ℂ )
26 13 25 fsumcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) ∈ ℂ )
27 climcl ⊢ ( seq 1 ( + , 𝐹 ) ⇝ 𝑇 → 𝑇 ∈ ℂ )
28 11 27 syl ⊢ ( 𝜑 → 𝑇 ∈ ℂ )
29 28 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑇 ∈ ℂ )
30 26 29 mulcld ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) ∈ ℂ )
31 1 2 3 4 5 6 7 8 9 10 11 12 dchrisumn0 ⊢ ( 𝜑 → 𝑇 ≠ 0 )
32 31 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑇 ≠ 0 )
33 30 29 32 divrecd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) / 𝑇 ) = ( ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) · ( 1 / 𝑇 ) ) )
34 26 29 32 divcan4d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) / 𝑇 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) )
35 33 34 eqtr3d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) · ( 1 / 𝑇 ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) )
36 35 mpteq2dva ⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) · ( 1 / 𝑇 ) ) ) = ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) ) )
37 28 31 reccld ⊢ ( 𝜑 → ( 1 / 𝑇 ) ∈ ℂ )
38 37 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 / 𝑇 ) ∈ ℂ )
39 1 2 3 4 5 6 7 8 9 10 11 12 dchrmusum2 ⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) ) ∈ 𝑂(1) )
40 rpssre ⊢ ℝ+ ⊆ ℝ
41 o1const ⊢ ( ( ℝ+ ⊆ ℝ ∧ ( 1 / 𝑇 ) ∈ ℂ ) → ( 𝑥 ∈ ℝ+ ↦ ( 1 / 𝑇 ) ) ∈ 𝑂(1) )
42 40 37 41 sylancr ⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( 1 / 𝑇 ) ) ∈ 𝑂(1) )
43 30 38 39 42 o1mul2 ⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) · 𝑇 ) · ( 1 / 𝑇 ) ) ) ∈ 𝑂(1) )
44 36 43 eqeltrrd ⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( μ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) )