Definition df-zeta 26068

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 26067	. 2 class ζ
2		c1 10803	. . . . . . 7 class 1
3		c2 11958	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1538	. . . . . . . . 9 class 𝑠
6		cmin 11135	. . . . . . . . 9 class −
7	2, 5, 6	co 7255	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 25616	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7255	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7255	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1538	. . . . . . 7 class 𝑓
13	5, 12	cfv 6418	. . . . . 6 class (𝑓‘𝑠)
14		cmul 10807	. . . . . 6 class ·
15	10, 13, 14	co 7255	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12163	. . . . . 6 class ℕ0
17		cc0 10802	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1538	. . . . . . . . 9 class 𝑛
20		cfz 13168	. . . . . . . . 9 class ...
21	17, 19, 20	co 7255	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11136	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1538	. . . . . . . . . . 11 class 𝑘
25		cexp 13710	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7255	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 13944	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7255	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7255	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 10805	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7255	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7255	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7255	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15325	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7255	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7255	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11562	. . . . . . 7 class /
38	34, 36, 37	co 7255	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15325	. . . . 5 class Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
40	15, 39	wceq 1539	. . . 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 10800	. . . . 5 class ℂ
42	2	csn 4558	. . . . 5 class {1}
43	41, 42	cdif 3880	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3063	. . 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 23945	. . . 4 class –cn→
46	43, 41, 45	co 7255	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7211	. 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 1539	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)