Definition df-zeta 26515

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 26514	. 2 class ζ
2		c1 11110	. . . . . . 7 class 1
3		c2 12266	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1540	. . . . . . . . 9 class 𝑠
6		cmin 11443	. . . . . . . . 9 class −
7	2, 5, 6	co 7408	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26063	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7408	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7408	. . . . . 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 11114	. . . . . 6 class ·
15	10, 13, 14	co 7408	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12471	. . . . . 6 class ℕ0
17		cc0 11109	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1540	. . . . . . . . 9 class 𝑛
20		cfz 13483	. . . . . . . . 9 class ...
21	17, 19, 20	co 7408	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11444	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1540	. . . . . . . . . . 11 class 𝑘
25		cexp 14026	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7408	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14261	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7408	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7408	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11112	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7408	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7408	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7408	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15631	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7408	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7408	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11870	. . . . . . 7 class /
38	34, 36, 37	co 7408	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15631	. . . . 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 11107	. . . . 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 24391	. . . 4 class –cn→
46	43, 41, 45	co 7408	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7363	. 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)