Definition df-zeta 26924

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 26923	. 2 class ζ
2		c1 11069	. . . . . . 7 class 1
3		c2 12241	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1539	. . . . . . . . 9 class 𝑠
6		cmin 11405	. . . . . . . . 9 class −
7	2, 5, 6	co 7387	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26464	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7387	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7387	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1539	. . . . . . 7 class 𝑓
13	5, 12	cfv 6511	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11073	. . . . . 6 class ·
15	10, 13, 14	co 7387	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12442	. . . . . 6 class ℕ0
17		cc0 11068	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1539	. . . . . . . . 9 class 𝑛
20		cfz 13468	. . . . . . . . 9 class ...
21	17, 19, 20	co 7387	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11406	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1539	. . . . . . . . . . 11 class 𝑘
25		cexp 14026	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7387	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14267	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7387	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7387	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11071	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7387	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7387	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7387	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15652	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7387	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7387	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11835	. . . . . . 7 class /
38	34, 36, 37	co 7387	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15652	. . . . 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 11066	. . . . 5 class ℂ
42	2	csn 4589	. . . . 5 class {1}
43	41, 42	cdif 3911	. . . 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 24769	. . . 4 class –cn→
46	43, 41, 45	co 7387	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7343	. 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)