dchrvmasumlem

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 = ( 0_g ‘ 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	dchrvmasumlem	⊢ ( 𝜑 → ( 𝑥 ∈ ℝ⁺ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 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 = ( 0_g ‘ 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	1 2 3 4 5 6 7 8 9 10 11 12	dchrisumn0	⊢ ( 𝜑 → 𝑇 ≠ 0 )
14	13	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → 𝑇 ≠ 0 )
15		ifnefalse	⊢ ( 𝑇 ≠ 0 → if ( 𝑇 = 0 , ( log ‘ 𝑥 ) , 0 ) = 0 )
16	14 15	syl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → if ( 𝑇 = 0 , ( log ‘ 𝑥 ) , 0 ) = 0 )
17	16	oveq2d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) + if ( 𝑇 = 0 , ( log ‘ 𝑥 ) , 0 ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) + 0 ) )
18		fzfid	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∈ Fin )
19	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑋 ∈ 𝐷 )
20		elfzelz	⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑛 ∈ ℤ )
21	20	adantl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℤ )
22	4 1 5 2 19 21	dchrzrhcl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) ∈ ℂ )
23		elfznn	⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑛 ∈ ℕ )
24	23	adantl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℕ )
25		vmacl	⊢ ( 𝑛 ∈ ℕ → ( Λ ‘ 𝑛 ) ∈ ℝ )
26		nndivre	⊢ ( ( ( Λ ‘ 𝑛 ) ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( Λ ‘ 𝑛 ) / 𝑛 ) ∈ ℝ )
27	25 26	mpancom	⊢ ( 𝑛 ∈ ℕ → ( ( Λ ‘ 𝑛 ) / 𝑛 ) ∈ ℝ )
28	24 27	syl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( Λ ‘ 𝑛 ) / 𝑛 ) ∈ ℝ )
29	28	recnd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( Λ ‘ 𝑛 ) / 𝑛 ) ∈ ℂ )
30	22 29	mulcld	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ∈ ℂ )
31	18 30	fsumcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ∈ ℂ )
32	31	addridd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) + 0 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) )
33	17 32	eqtrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ⁺ ) → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) + if ( 𝑇 = 0 , ( log ‘ 𝑥 ) , 0 ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) )
34	33	mpteq2dva	⊢ ( 𝜑 → ( 𝑥 ∈ ℝ⁺ ↦ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) + if ( 𝑇 = 0 , ( log ‘ 𝑥 ) , 0 ) ) ) = ( 𝑥 ∈ ℝ⁺ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ) )
35	1 2 3 4 5 6 7 8 9 10 11 12	dchrvmasumif	⊢ ( 𝜑 → ( 𝑥 ∈ ℝ⁺ ↦ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) + if ( 𝑇 = 0 , ( log ‘ 𝑥 ) , 0 ) ) ) ∈ 𝑂(1) )
36	34 35	eqeltrrd	⊢ ( 𝜑 → ( 𝑥 ∈ ℝ⁺ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) )

Metamath Proof Explorer

Theorem dchrvmasumlem

Proof