Theorem selberg 27592

Description: Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Σ𝑛 ≤ 𝑥, Λ(𝑛)log𝑛 + Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛) = 2𝑥log𝑥 + 𝑂(𝑥). Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)

Assertion

Ref	Expression
selberg	⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Distinct variable group:   𝑥,𝑛

Proof of Theorem selberg

Dummy variables 𝑑 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (Λ‘𝑛) = (Λ‘𝑑))
2		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑑 → (𝑥 / 𝑛) = (𝑥 / 𝑑))
3	2	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑑)))
4	1, 3	oveq12d 7449	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑑 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))))
5	4	cbvsumv 15732	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))
6		fzfid 14014	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
7		elfznn 13593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
8	7	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
9		vmacl 27161	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
10	8, 9	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
11	10	recnd 11289	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
12		elfznn 13593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ)
13	12	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ)
14		vmacl 27161	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℝ)
16	15	recnd 11289	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℂ)
17	6, 11, 16	fsummulc2 15820	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘𝑚)))
18	7	nnrpd 13075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ+)
19		rpdivcl 13060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → (𝑥 / 𝑑) ∈ ℝ+)
20	18, 19	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ+)
21	20	rpred 13077	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
22		chpval 27165	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑑) ∈ ℝ → (ψ‘(𝑥 / 𝑑)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚))
23	21, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚))
24	23	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)))
25	13	nncnd 12282	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
26	7	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℕ)
27	26	nncnd 12282	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
28	26	nnne0d 12316	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
29	25, 27, 28	divcan3d 12048	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
30	29	fveq2d 6910	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘((𝑑 · 𝑚) / 𝑑)) = (Λ‘𝑚))
31	30	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = ((Λ‘𝑑) · (Λ‘𝑚)))
32	31	sumeq2dv 15738	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘𝑚)))
33	17, 24, 32	3eqtr4d 2787	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
34	33	sumeq2dv 15738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
35	5, 34	eqtrid 2789	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
36		fvoveq1 7454	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → (Λ‘(𝑛 / 𝑑)) = (Λ‘((𝑑 · 𝑚) / 𝑑)))
37	36	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
38		rpre 13043	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
39		ssrab2 4080	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
40		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
41	39, 40	sselid 3981	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ)
42	41	anassrs 467	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 𝑑 ∈ ℕ)
43	42, 9	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (Λ‘𝑑) ∈ ℝ)
44		elfznn 13593	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
45	44	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
46		dvdsdivcl 16353	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
47	45, 46	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
48	39, 47	sselid 3981	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ ℕ)
49		vmacl 27161	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 / 𝑑) ∈ ℕ → (Λ‘(𝑛 / 𝑑)) ∈ ℝ)
50	48, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (Λ‘(𝑛 / 𝑑)) ∈ ℝ)
51	43, 50	remulcld 11291	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
52	51	recnd 11289	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
53	52	anasss 466	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
54	37, 38, 53	dvdsflsumcom 27231	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
55	35, 54	eqtr4d 2780	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))))
56	55	oveq1d 7446	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))))
57		fzfid 14014	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
58		vmacl 27161	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
59	45, 58	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
60	59	recnd 11289	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	44	nnrpd 13075	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
62		rpdivcl 13060	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
63	61, 62	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
64	63	rpred 13077	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
65		chpcl 27167	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
66	64, 65	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
67	66	recnd 11289	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
68	60, 67	mulcld 11281	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
69	45	nnrpd 13075	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
70		relogcl 26617	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
72	71	recnd 11289	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
73	60, 72	mulcld 11281	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
74	57, 68, 73	fsumadd 15776	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))))
75		fzfid 14014	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
76		dvdsssfz1 16355	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ (1...𝑛))
77	45, 76	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ (1...𝑛))
78	75, 77	ssfid 9301	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ∈ Fin)
79	78, 51	fsumrecl 15770	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
80	79	recnd 11289	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
81	57, 80, 73	fsumadd 15776	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))))
82	56, 74, 81	3eqtr4d 2787	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
83	72, 67	addcomd 11463	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛)))
84	83	oveq2d 7447	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛))))
85	60, 67, 72	adddid 11285	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛))) = (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))))
86	84, 85	eqtrd 2777	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))))
87	86	sumeq2dv 15738	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))))
88		logsqvma2 27587	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
89	45, 88	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
90	89	sumeq2dv 15738	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
91	82, 87, 90	3eqtr4d 2787	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)))
92		fvoveq1 7454	. . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
93	92	oveq1d 7446	. . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2))
94	93	oveq2d 7447	. . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
95		mucl 27184	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
96	41, 95	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ)
97	96	zcnd 12723	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
98	61	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ+)
99	41	nnrpd 13075	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ+)
100	98, 99	rpdivcld 13094	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ+)
101		relogcl 26617	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℝ)
102	101	recnd 11289	. . . . . . . . . 10 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℂ)
103	100, 102	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (log‘(𝑛 / 𝑑)) ∈ ℂ)
104	103	sqcld 14184	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ)
105	97, 104	mulcld 11281	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ)
106	94, 38, 105	dvdsflsumcom 27231	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
107	29	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
108	107	oveq1d 7446	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2))
109	108	oveq2d 7447	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘𝑚)↑2)))
110	109	sumeq2dv 15738	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)))
111	110	sumeq2dv 15738	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)))
112	91, 106, 111	3eqtrd 2781	. . . . 5 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)))
113	112	oveq1d 7446	. . . 4 ⊢ (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) = (Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥))
114	113	oveq1d 7446	. . 3 ⊢ (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) = ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥))))
115	114	mpteq2ia 5245	. 2 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥))))
116		eqid 2737	. . 3 ⊢ ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) = ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑)
117	116	selberglem2 27590	. 2 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
118	115, 117	eqeltri 2837	1 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3436   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154  1c1 11156   + caddc 11158   · cmul 11160   − cmin 11492   / cdiv 11920  ℕcn 12266  2c2 12321  ℤcz 12613  ℝ⁺crp 13034  ...cfz 13547  ⌊cfl 13830  ↑cexp 14102  𝑂(1)co1 15522  Σcsu 15722   ∥ cdvds 16290  logclog 26596  Λcvma 27135  ψcchp 27136  μcmu 27138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-o1 15526  df-lo1 15527  df-sum 15723  df-ef 16103  df-e 16104  df-sin 16105  df-cos 16106  df-tan 16107  df-pi 16108  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-ulm 26420  df-log 26598  df-cxp 26599  df-atan 26910  df-em 27036  df-vma 27141  df-chp 27142  df-mu 27144

This theorem is referenced by:  selbergb  27593  selberg2  27595  selbergs  27618