Definition df-zeta 26977

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 26976	. 2 class ζ
2		c1 11039	. . . . . . 7 class 1
3		c2 12236	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1541	. . . . . . . . 9 class 𝑠
6		cmin 11377	. . . . . . . . 9 class −
7	2, 5, 6	co 7367	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26519	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7367	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7367	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1541	. . . . . . 7 class 𝑓
13	5, 12	cfv 6498	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11043	. . . . . 6 class ·
15	10, 13, 14	co 7367	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12437	. . . . . 6 class ℕ0
17		cc0 11038	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1541	. . . . . . . . 9 class 𝑛
20		cfz 13461	. . . . . . . . 9 class ...
21	17, 19, 20	co 7367	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11378	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1541	. . . . . . . . . . 11 class 𝑘
25		cexp 14023	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7367	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14264	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7367	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7367	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11041	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7367	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7367	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7367	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15648	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7367	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7367	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11807	. . . . . . 7 class /
38	34, 36, 37	co 7367	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15648	. . . . 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 11036	. . . . 5 class ℂ
42	2	csn 4567	. . . . 5 class {1}
43	41, 42	cdif 3886	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3051	. . 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 24843	. . . 4 class –cn→
46	43, 41, 45	co 7367	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7323	. 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)