Definition df-zeta 26940

Description: Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except 1, but going from the alternating zeta function to the regular zeta function requires dividing by 1 − 2↑(1 − 𝑠), which has zeroes other than 1. To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.)

Assertion

Ref	Expression
df-zeta	⊢ ζ = (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))

Distinct variable group:   𝑓,𝑘,𝑛,𝑠

Detailed syntax breakdown of Definition df-zeta

Step	Hyp	Ref	Expression
1		czeta 26939	. 2 class ζ
2		c1 11029	. . . . . . 7 class 1
3		c2 12201	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1539	. . . . . . . . 9 class 𝑠
6		cmin 11365	. . . . . . . . 9 class −
7	2, 5, 6	co 7353	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26480	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7353	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7353	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1539	. . . . . . 7 class 𝑓
13	5, 12	cfv 6486	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11033	. . . . . 6 class ·
15	10, 13, 14	co 7353	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12402	. . . . . 6 class ℕ0
17		cc0 11028	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1539	. . . . . . . . 9 class 𝑛
20		cfz 13428	. . . . . . . . 9 class ...
21	17, 19, 20	co 7353	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11366	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1539	. . . . . . . . . . 11 class 𝑘
25		cexp 13986	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7353	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14227	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7353	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7353	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11031	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7353	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7353	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7353	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15611	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7353	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7353	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11795	. . . . . . 7 class /
38	34, 36, 37	co 7353	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15611	. . . . 5 class Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
40	15, 39	wceq 1540	. . . 4 wff ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
41		cc 11026	. . . . 5 class ℂ
42	2	csn 4579	. . . . 5 class {1}
43	41, 42	cdif 3902	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3044	. . 3 wff ∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
45		ccncf 24785	. . . 4 class –cn→
46	43, 41, 45	co 7353	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7309	. 2 class (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))
48	1, 47	wceq 1540	1 wff ζ = (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))

This definition is referenced by: (None)