Definition df-zeta 27304

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 27303	. 2 class ζ
2		c1 11172	. . . . . . 7 class 1
3		c2 12366	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1569	. . . . . . . . 9 class 𝑠
6		cmin 11512	. . . . . . . . 9 class −
7	2, 5, 6	co 7408	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26846	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7408	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7408	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1569	. . . . . . 7 class 𝑓
13	5, 12	cfv 6527	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11176	. . . . . 6 class ·
15	10, 13, 14	co 7408	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12575	. . . . . 6 class ℕ0
17		cc0 11171	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1569	. . . . . . . . 9 class 𝑛
20		cfz 13608	. . . . . . . . 9 class ...
21	17, 19, 20	co 7408	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11513	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1569	. . . . . . . . . . 11 class 𝑘
25		cexp 14172	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7408	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14413	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7408	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7408	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11174	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7408	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7408	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7408	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15820	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7408	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7408	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11942	. . . . . . 7 class /
38	34, 36, 37	co 7408	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15820	. . . . 5 class Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
40	15, 39	wceq 1570	. . . 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 11169	. . . . 5 class ℂ
42	2	csn 4583	. . . . 5 class {1}
43	41, 42	cdif 3895	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3076	. . 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 25158	. . . 4 class –cn→
46	43, 41, 45	co 7408	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7364	. 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 1570	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 used by: (None)