Definition df-zeta 27178

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 27177	. 2 class ζ
2		c1 11096	. . . . . . 7 class 1
3		c2 12290	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1569	. . . . . . . . 9 class 𝑠
6		cmin 11436	. . . . . . . . 9 class −
7	2, 5, 6	co 7410	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 26720	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7410	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7410	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1569	. . . . . . 7 class 𝑓
13	5, 12	cfv 6536	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11100	. . . . . 6 class ·
15	10, 13, 14	co 7410	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12499	. . . . . 6 class ℕ0
17		cc0 11095	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1569	. . . . . . . . 9 class 𝑛
20		cfz 13530	. . . . . . . . 9 class ...
21	17, 19, 20	co 7410	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11437	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1569	. . . . . . . . . . 11 class 𝑘
25		cexp 14093	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7410	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14334	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7410	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7410	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11098	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7410	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7410	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7410	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15733	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7410	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7410	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11866	. . . . . . 7 class /
38	34, 36, 37	co 7410	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15733	. . . . 5 class Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
40	15, 39	wceq 1570	. . . 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 11093	. . . . 5 class ℂ
42	2	csn 4589	. . . . 5 class {1}
43	41, 42	cdif 3902	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3079	. . 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 25035	. . . 4 class –cn→
46	43, 41, 45	co 7410	. . 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 1570	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)