Theorem selberg 27040

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 6888	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (Λ‘𝑛) = (Λ‘𝑑))
2		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑑 → (𝑥 / 𝑛) = (𝑥 / 𝑑))
3	2	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑑)))
4	1, 3	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑑 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))))
5	4	cbvsumv 15638	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))
6		fzfid 13934	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
7		elfznn 13526	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
8	7	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
9		vmacl 26611	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
10	8, 9	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
11	10	recnd 11238	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
12		elfznn 13526	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ)
13	12	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ)
14		vmacl 26611	. . . . . . . . . . . . . . . 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 11238	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℂ)
17	6, 11, 16	fsummulc2 15726	. . . . . . . . . . . . 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 13010	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ+)
19		rpdivcl 12995	. . . . . . . . . . . . . . . . 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 13012	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
22		chpval 26615	. . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)))
25	13	nncnd 12224	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
26	7	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℕ)
27	26	nncnd 12224	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
28	26	nnne0d 12258	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
29	25, 27, 28	divcan3d 11991	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
30	29	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘((𝑑 · 𝑚) / 𝑑)) = (Λ‘𝑚))
31	30	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = ((Λ‘𝑑) · (Λ‘𝑚)))
32	31	sumeq2dv 15645	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
34	33	sumeq2dv 15645	. . . . . . . . . . 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 2784	. . . . . . . . . 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 7428	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → (Λ‘(𝑛 / 𝑑)) = (Λ‘((𝑑 · 𝑚) / 𝑑)))
37	36	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
38		rpre 12978	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
39		ssrab2 4076	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
40		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
41	39, 40	sselid 3979	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ)
42	41	anassrs 468	. . . . . . . . . . . . . . 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 13526	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
45	44	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
46		dvdsdivcl 16255	. . . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ ℕ)
49		vmacl 26611	. . . . . . . . . . . . . . 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 11240	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
52	51	recnd 11238	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
53	52	anasss 467	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
54	37, 38, 53	dvdsflsumcom 26681	. . . . . . . . . 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 2775	. . . . . . . . 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 7420	. . . . . . . 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 13934	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
58		vmacl 26611	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
59	45, 58	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
60	59	recnd 11238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	44	nnrpd 13010	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
62		rpdivcl 12995	. . . . . . . . . . . . . 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 13012	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
65		chpcl 26617	. . . . . . . . . . . 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 11238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
68	60, 67	mulcld 11230	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
69	45	nnrpd 13010	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
70		relogcl 26075	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
72	71	recnd 11238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
73	60, 72	mulcld 11230	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
74	57, 68, 73	fsumadd 15682	. . . . . . . 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 13934	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
76		dvdsssfz1 16257	. . . . . . . . . . . . 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 9263	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ∈ Fin)
79	78, 51	fsumrecl 15676	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
80	79	recnd 11238	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
81	57, 80, 73	fsumadd 15682	. . . . . . . 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 2782	. . . . . . 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 11412	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛)))
84	83	oveq2d 7421	. . . . . . . . 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 11234	. . . . . . . . 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 2772	. . . . . . . 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 15645	. . . . . . 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 27035	. . . . . . . . 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 15645	. . . . . . 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 2782	. . . . . 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 7428	. . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
93	92	oveq1d 7420	. . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2))
94	93	oveq2d 7421	. . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
95		mucl 26634	. . . . . . . . . 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 12663	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
98	61	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ+)
99	41	nnrpd 13010	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ+)
100	98, 99	rpdivcld 13029	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ+)
101		relogcl 26075	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℝ)
102	101	recnd 11238	. . . . . . . . . 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 14105	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ)
105	97, 104	mulcld 11230	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ)
106	94, 38, 105	dvdsflsumcom 26681	. . . . . 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 6892	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
108	107	oveq1d 7420	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2))
109	108	oveq2d 7421	. . . . . . . 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 15645	. . . . . . 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 15645	. . . . . 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 2776	. . . . 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 7420	. . . 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 7420	. . 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 5250	. 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 2732	. . 3 ⊢ ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) = ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑)
117	116	selberglem2 27038	. 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 2829	1 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Syntax hints:   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   ⊆ wss 3947   class class class wbr 5147   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7405  ℂcc 11104  ℝcr 11105  1c1 11107   + caddc 11109   · cmul 11111   − cmin 11440   / cdiv 11867  ℕcn 12208  2c2 12263  ℤcz 12554  ℝ⁺crp 12970  ...cfz 13480  ⌊cfl 13751  ↑cexp 14023  𝑂(1)co1 15426  Σcsu 15628   ∥ cdvds 16193  logclog 26054  Λcvma 26585  ψcchp 26586  μcmu 26588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-xnn0 12541  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ioc 13325  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-fac 14230  df-bc 14259  df-hash 14287  df-shft 15010  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-limsup 15411  df-clim 15428  df-rlim 15429  df-o1 15430  df-lo1 15431  df-sum 15629  df-ef 16007  df-e 16008  df-sin 16009  df-cos 16010  df-tan 16011  df-pi 16012  df-dvds 16194  df-gcd 16432  df-prm 16605  df-pc 16766  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-hom 17217  df-cco 17218  df-rest 17364  df-topn 17365  df-0g 17383  df-gsum 17384  df-topgen 17385  df-pt 17386  df-prds 17389  df-xrs 17444  df-qtop 17449  df-imas 17450  df-xps 17452  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-mulg 18945  df-cntz 19175  df-cmn 19644  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-fbas 20933  df-fg 20934  df-cnfld 20937  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-lp 22631  df-perf 22632  df-cn 22722  df-cnp 22723  df-haus 22810  df-cmp 22882  df-tx 23057  df-hmeo 23250  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-xms 23817  df-ms 23818  df-tms 23819  df-cncf 24385  df-limc 25374  df-dv 25375  df-ulm 25880  df-log 26056  df-cxp 26057  df-atan 26361  df-em 26486  df-vma 26591  df-chp 26592  df-mu 26594

This theorem is referenced by:  selbergb  27041  selberg2  27043  selbergs  27066