Definition df-zeta 26974

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 26973	. 2 class ζ
2		c1 11128	. . . . . . 7 class 1
3		c2 12293	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1539	. . . . . . . . 9 class 𝑠
6		cmin 11464	. . . . . . . . 9 class −
7	2, 5, 6	co 7403	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26514	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7403	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7403	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1539	. . . . . . 7 class 𝑓
13	5, 12	cfv 6530	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11132	. . . . . 6 class ·
15	10, 13, 14	co 7403	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12499	. . . . . 6 class ℕ0
17		cc0 11127	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1539	. . . . . . . . 9 class 𝑛
20		cfz 13522	. . . . . . . . 9 class ...
21	17, 19, 20	co 7403	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11465	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1539	. . . . . . . . . . 11 class 𝑘
25		cexp 14077	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7403	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14318	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7403	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7403	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11130	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7403	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7403	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7403	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15700	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7403	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7403	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11892	. . . . . . 7 class /
38	34, 36, 37	co 7403	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15700	. . . . 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 11125	. . . . 5 class ℂ
42	2	csn 4601	. . . . 5 class {1}
43	41, 42	cdif 3923	. . . 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 24818	. . . 4 class –cn→
46	43, 41, 45	co 7403	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7359	. 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)