Definition df-zeta 26518

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 26517	. 2 class ζ
2		c1 11111	. . . . . . 7 class 1
3		c2 12267	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1541	. . . . . . . . 9 class 𝑠
6		cmin 11444	. . . . . . . . 9 class −
7	2, 5, 6	co 7409	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26064	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7409	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7409	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1541	. . . . . . 7 class 𝑓
13	5, 12	cfv 6544	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11115	. . . . . 6 class ·
15	10, 13, 14	co 7409	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12472	. . . . . 6 class ℕ0
17		cc0 11110	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1541	. . . . . . . . 9 class 𝑛
20		cfz 13484	. . . . . . . . 9 class ...
21	17, 19, 20	co 7409	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11445	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1541	. . . . . . . . . . 11 class 𝑘
25		cexp 14027	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7409	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14262	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7409	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7409	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11113	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7409	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7409	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7409	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15632	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7409	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7409	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11871	. . . . . . 7 class /
38	34, 36, 37	co 7409	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15632	. . . . 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 11108	. . . . 5 class ℂ
42	2	csn 4629	. . . . 5 class {1}
43	41, 42	cdif 3946	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3062	. . 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 24392	. . . 4 class –cn→
46	43, 41, 45	co 7409	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7364	. 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)