Definition df-zeta 26994

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 26993	. 2 class ζ
2		c1 11033	. . . . . . 7 class 1
3		c2 12230	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1541	. . . . . . . . 9 class 𝑠
6		cmin 11371	. . . . . . . . 9 class −
7	2, 5, 6	co 7361	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26535	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7361	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7361	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1541	. . . . . . 7 class 𝑓
13	5, 12	cfv 6493	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11037	. . . . . 6 class ·
15	10, 13, 14	co 7361	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12431	. . . . . 6 class ℕ0
17		cc0 11032	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1541	. . . . . . . . 9 class 𝑛
20		cfz 13455	. . . . . . . . 9 class ...
21	17, 19, 20	co 7361	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11372	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1541	. . . . . . . . . . 11 class 𝑘
25		cexp 14017	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7361	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14258	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7361	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7361	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11035	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7361	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7361	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7361	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15642	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7361	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7361	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11801	. . . . . . 7 class /
38	34, 36, 37	co 7361	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15642	. . . . 5 class Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
40	15, 39	wceq 1542	. . . 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 11030	. . . . 5 class ℂ
42	2	csn 4568	. . . . 5 class {1}
43	41, 42	cdif 3887	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3052	. . 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 24856	. . . 4 class –cn→
46	43, 41, 45	co 7361	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7317	. 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 1542	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)