Definition df-zeta 26980

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 26979	. 2 class ζ
2		c1 11027	. . . . . . 7 class 1
3		c2 12200	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1540	. . . . . . . . 9 class 𝑠
6		cmin 11364	. . . . . . . . 9 class −
7	2, 5, 6	co 7358	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26520	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7358	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7358	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1540	. . . . . . 7 class 𝑓
13	5, 12	cfv 6492	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11031	. . . . . 6 class ·
15	10, 13, 14	co 7358	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12401	. . . . . 6 class ℕ0
17		cc0 11026	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1540	. . . . . . . . 9 class 𝑛
20		cfz 13423	. . . . . . . . 9 class ...
21	17, 19, 20	co 7358	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11365	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1540	. . . . . . . . . . 11 class 𝑘
25		cexp 13984	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7358	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14225	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7358	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7358	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11029	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7358	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7358	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7358	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15609	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7358	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7358	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11794	. . . . . . 7 class /
38	34, 36, 37	co 7358	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15609	. . . . 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 11024	. . . . 5 class ℂ
42	2	csn 4580	. . . . 5 class {1}
43	41, 42	cdif 3898	. . . 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 24825	. . . 4 class –cn→
46	43, 41, 45	co 7358	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7314	. 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)