Definition df-zeta 26525

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 26524	. 2 class ζ
2		c1 11113	. . . . . . 7 class 1
3		c2 12269	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1540	. . . . . . . . 9 class 𝑠
6		cmin 11446	. . . . . . . . 9 class −
7	2, 5, 6	co 7411	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26071	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7411	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7411	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1540	. . . . . . 7 class 𝑓
13	5, 12	cfv 6543	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11117	. . . . . 6 class ·
15	10, 13, 14	co 7411	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12474	. . . . . 6 class ℕ0
17		cc0 11112	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1540	. . . . . . . . 9 class 𝑛
20		cfz 13486	. . . . . . . . 9 class ...
21	17, 19, 20	co 7411	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11447	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1540	. . . . . . . . . . 11 class 𝑘
25		cexp 14029	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7411	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14264	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7411	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7411	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11115	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7411	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7411	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7411	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15634	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7411	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7411	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11873	. . . . . . 7 class /
38	34, 36, 37	co 7411	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15634	. . . . 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 11110	. . . . 5 class ℂ
42	2	csn 4628	. . . . 5 class {1}
43	41, 42	cdif 3945	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3061	. . 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 24399	. . . 4 class –cn→
46	43, 41, 45	co 7411	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7366	. 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)