Theorem selberg 26601

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 6756	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (Λ‘𝑛) = (Λ‘𝑑))
2		oveq2 7263	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑑 → (𝑥 / 𝑛) = (𝑥 / 𝑑))
3	2	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑑)))
4	1, 3	oveq12d 7273	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑑 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))))
5	4	cbvsumv 15336	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))
6		fzfid 13621	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
7		elfznn 13214	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
8	7	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
9		vmacl 26172	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
10	8, 9	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
11	10	recnd 10934	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
12		elfznn 13214	. . . . . . . . . . . . . . . . 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 26172	. . . . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℂ)
17	6, 11, 16	fsummulc2 15424	. . . . . . . . . . . . 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 12699	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ+)
19		rpdivcl 12684	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → (𝑥 / 𝑑) ∈ ℝ+)
20	18, 19	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ+)
21	20	rpred 12701	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
22		chpval 26176	. . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)))
25	13	nncnd 11919	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
26	7	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℕ)
27	26	nncnd 11919	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
28	26	nnne0d 11953	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
29	25, 27, 28	divcan3d 11686	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
30	29	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘((𝑑 · 𝑚) / 𝑑)) = (Λ‘𝑚))
31	30	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = ((Λ‘𝑑) · (Λ‘𝑚)))
32	31	sumeq2dv 15343	. . . . . . . . . . . . 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 15343	. . . . . . . . . . 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	syl5eq 2791	. . . . . . . . . 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 7278	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → (Λ‘(𝑛 / 𝑑)) = (Λ‘((𝑑 · 𝑚) / 𝑑)))
37	36	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
38		rpre 12667	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
39		ssrab2 4009	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
40		simprr 769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
41	39, 40	sselid 3915	. . . . . . . . . . . . . . . 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 13214	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
45	44	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
46		dvdsdivcl 15953	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
47	45, 46	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
48	39, 47	sselid 3915	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ ℕ)
49		vmacl 26172	. . . . . . . . . . . . . . 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 10936	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
52	51	recnd 10934	. . . . . . . . . . . 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 26242	. . . . . . . . . 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 7270	. . . . . . . 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 13621	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
58		vmacl 26172	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
59	45, 58	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
60	59	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	44	nnrpd 12699	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
62		rpdivcl 12684	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
63	61, 62	sylan2 592	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
64	63	rpred 12701	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
65		chpcl 26178	. . . . . . . . . . . 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 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
68	60, 67	mulcld 10926	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
69	45	nnrpd 12699	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
70		relogcl 25636	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
72	71	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
73	60, 72	mulcld 10926	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
74	57, 68, 73	fsumadd 15380	. . . . . . . 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 13621	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
76		dvdsssfz1 15955	. . . . . . . . . . . . 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 8971	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ∈ Fin)
79	78, 51	fsumrecl 15374	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
80	79	recnd 10934	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
81	57, 80, 73	fsumadd 15380	. . . . . . . 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 11107	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛)))
84	83	oveq2d 7271	. . . . . . . . 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 10930	. . . . . . . . 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 15343	. . . . . . 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 26596	. . . . . . . . 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 15343	. . . . . . 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 7278	. . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
93	92	oveq1d 7270	. . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2))
94	93	oveq2d 7271	. . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
95		mucl 26195	. . . . . . . . . 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 12356	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
98	61	ad2antrl 724	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ+)
99	41	nnrpd 12699	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ+)
100	98, 99	rpdivcld 12718	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ+)
101		relogcl 25636	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℝ)
102	101	recnd 10934	. . . . . . . . . 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 13790	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ)
105	97, 104	mulcld 10926	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ)
106	94, 38, 105	dvdsflsumcom 26242	. . . . . 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 6760	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
108	107	oveq1d 7270	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2))
109	108	oveq2d 7271	. . . . . . . 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 15343	. . . . . . 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 15343	. . . . . 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 7270	. . . 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 7270	. . 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 5173	. 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 26599	. 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 395   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  1c1 10803   + caddc 10805   · cmul 10807   − cmin 11135   / cdiv 11562  ℕcn 11903  2c2 11958  ℤcz 12249  ℝ⁺crp 12659  ...cfz 13168  ⌊cfl 13438  ↑cexp 13710  𝑂(1)co1 15123  Σcsu 15325   ∥ cdvds 15891  logclog 25615  Λcvma 26146  ψcchp 26147  μcmu 26149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-fac 13916  df-bc 13945  df-hash 13973  df-shft 14706  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-limsup 15108  df-clim 15125  df-rlim 15126  df-o1 15127  df-lo1 15128  df-sum 15326  df-ef 15705  df-e 15706  df-sin 15707  df-cos 15708  df-tan 15709  df-pi 15710  df-dvds 15892  df-gcd 16130  df-prm 16305  df-pc 16466  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-fbas 20507  df-fg 20508  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-lp 22195  df-perf 22196  df-cn 22286  df-cnp 22287  df-haus 22374  df-cmp 22446  df-tx 22621  df-hmeo 22814  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-xms 23381  df-ms 23382  df-tms 23383  df-cncf 23947  df-limc 24935  df-dv 24936  df-ulm 25441  df-log 25617  df-cxp 25618  df-atan 25922  df-em 26047  df-vma 26152  df-chp 26153  df-mu 26155

This theorem is referenced by:  selbergb  26602  selberg2  26604  selbergs  26627