Theorem selberg 26696

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 6774	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (Λ‘𝑛) = (Λ‘𝑑))
2		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑑 → (𝑥 / 𝑛) = (𝑥 / 𝑑))
3	2	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑑)))
4	1, 3	oveq12d 7293	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑑 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))))
5	4	cbvsumv 15408	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))
6		fzfid 13693	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
7		elfznn 13285	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
8	7	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
9		vmacl 26267	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
10	8, 9	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
11	10	recnd 11003	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
12		elfznn 13285	. . . . . . . . . . . . . . . . 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 26267	. . . . . . . . . . . . . . . 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 11003	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℂ)
17	6, 11, 16	fsummulc2 15496	. . . . . . . . . . . . 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 12770	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ+)
19		rpdivcl 12755	. . . . . . . . . . . . . . . . 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 12772	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
22		chpval 26271	. . . . . . . . . . . . . . 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 7291	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)))
25	13	nncnd 11989	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
26	7	ad2antlr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℕ)
27	26	nncnd 11989	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
28	26	nnne0d 12023	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
29	25, 27, 28	divcan3d 11756	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
30	29	fveq2d 6778	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘((𝑑 · 𝑚) / 𝑑)) = (Λ‘𝑚))
31	30	oveq2d 7291	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = ((Λ‘𝑑) · (Λ‘𝑚)))
32	31	sumeq2dv 15415	. . . . . . . . . . . . 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 2788	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
34	33	sumeq2dv 15415	. . . . . . . . . . 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 2790	. . . . . . . . . 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 7298	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → (Λ‘(𝑛 / 𝑑)) = (Λ‘((𝑑 · 𝑚) / 𝑑)))
37	36	oveq2d 7291	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
38		rpre 12738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
39		ssrab2 4013	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
40		simprr 770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
41	39, 40	sselid 3919	. . . . . . . . . . . . . . . 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 13285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
45	44	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
46		dvdsdivcl 16025	. . . . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ ℕ)
49		vmacl 26267	. . . . . . . . . . . . . . 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 11005	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
52	51	recnd 11003	. . . . . . . . . . . 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 26337	. . . . . . . . . 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 2781	. . . . . . . . 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 7290	. . . . . . . 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 13693	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
58		vmacl 26267	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
59	45, 58	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
60	59	recnd 11003	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	44	nnrpd 12770	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
62		rpdivcl 12755	. . . . . . . . . . . . . 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 12772	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
65		chpcl 26273	. . . . . . . . . . . 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 11003	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
68	60, 67	mulcld 10995	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
69	45	nnrpd 12770	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
70		relogcl 25731	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
72	71	recnd 11003	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
73	60, 72	mulcld 10995	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
74	57, 68, 73	fsumadd 15452	. . . . . . . 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 13693	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
76		dvdsssfz1 16027	. . . . . . . . . . . . 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 9042	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ∈ Fin)
79	78, 51	fsumrecl 15446	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
80	79	recnd 11003	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
81	57, 80, 73	fsumadd 15452	. . . . . . . 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 2788	. . . . . . 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 11177	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛)))
84	83	oveq2d 7291	. . . . . . . . 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 10999	. . . . . . . . 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 2778	. . . . . . . 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 15415	. . . . . . 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 26691	. . . . . . . . 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 15415	. . . . . . 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 2788	. . . . . 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 7298	. . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
93	92	oveq1d 7290	. . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2))
94	93	oveq2d 7291	. . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
95		mucl 26290	. . . . . . . . . 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 12427	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
98	61	ad2antrl 725	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ+)
99	41	nnrpd 12770	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ+)
100	98, 99	rpdivcld 12789	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ+)
101		relogcl 25731	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℝ)
102	101	recnd 11003	. . . . . . . . . 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 13862	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ)
105	97, 104	mulcld 10995	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ)
106	94, 38, 105	dvdsflsumcom 26337	. . . . . 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 6778	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
108	107	oveq1d 7290	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2))
109	108	oveq2d 7291	. . . . . . . 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 15415	. . . . . . 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 15415	. . . . . 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 2782	. . . . 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 7290	. . . 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 7290	. . 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 5177	. 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 2738	. . 3 ⊢ ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) = ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑)
117	116	selberglem2 26694	. 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 2835	1 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   ⊆ wss 3887   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  ℝcr 10870  1c1 10872   + caddc 10874   · cmul 10876   − cmin 11205   / cdiv 11632  ℕcn 11973  2c2 12028  ℤcz 12319  ℝ⁺crp 12730  ...cfz 13239  ⌊cfl 13510  ↑cexp 13782  𝑂(1)co1 15195  Σcsu 15397   ∥ cdvds 15963  logclog 25710  Λcvma 26241  ψcchp 26242  μcmu 26244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-o1 15199  df-lo1 15200  df-sum 15398  df-ef 15777  df-e 15778  df-sin 15779  df-cos 15780  df-tan 15781  df-pi 15782  df-dvds 15964  df-gcd 16202  df-prm 16377  df-pc 16538  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031  df-ulm 25536  df-log 25712  df-cxp 25713  df-atan 26017  df-em 26142  df-vma 26247  df-chp 26248  df-mu 26250

This theorem is referenced by:  selbergb  26697  selberg2  26699  selbergs  26722