Definition df-zeta 26952

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 26951	. 2 class ζ
2		c1 11014	. . . . . . 7 class 1
3		c2 12187	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1540	. . . . . . . . 9 class 𝑠
6		cmin 11351	. . . . . . . . 9 class −
7	2, 5, 6	co 7352	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26492	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7352	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7352	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1540	. . . . . . 7 class 𝑓
13	5, 12	cfv 6486	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11018	. . . . . 6 class ·
15	10, 13, 14	co 7352	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12388	. . . . . 6 class ℕ0
17		cc0 11013	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1540	. . . . . . . . 9 class 𝑛
20		cfz 13409	. . . . . . . . 9 class ...
21	17, 19, 20	co 7352	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11352	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1540	. . . . . . . . . . 11 class 𝑘
25		cexp 13970	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7352	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14211	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7352	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7352	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11016	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7352	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7352	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7352	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15595	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7352	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7352	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11781	. . . . . . 7 class /
38	34, 36, 37	co 7352	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15595	. . . . 5 class Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
40	15, 39	wceq 1541	. . . 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 11011	. . . . 5 class ℂ
42	2	csn 4575	. . . . 5 class {1}
43	41, 42	cdif 3895	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3048	. . 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 24797	. . . 4 class –cn→
46	43, 41, 45	co 7352	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7308	. 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 1541	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)