Definition df-zeta 26445

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 26444	. 2 class ζ
2		c1 11093	. . . . . . 7 class 1
3		c2 12249	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar 𝑠
5	4	cv 1540	. . . . . . . . 9 class 𝑠
6		cmin 11426	. . . . . . . . 9 class −
7	2, 5, 6	co 7393	. . . . . . . 8 class (1 − 𝑠)
8		ccxp 25993	. . . . . . . 8 class ↑𝑐
9	3, 7, 8	co 7393	. . . . . . 7 class (2↑𝑐(1 − 𝑠))
10	2, 9, 6	co 7393	. . . . . 6 class (1 − (2↑𝑐(1 − 𝑠)))
11		vf	. . . . . . . 8 setvar 𝑓
12	11	cv 1540	. . . . . . 7 class 𝑓
13	5, 12	cfv 6532	. . . . . 6 class (𝑓‘𝑠)
14		cmul 11097	. . . . . 6 class ·
15	10, 13, 14	co 7393	. . . . 5 class ((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠))
16		cn0 12454	. . . . . 6 class ℕ0
17		cc0 11092	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar 𝑛
19	18	cv 1540	. . . . . . . . 9 class 𝑛
20		cfz 13466	. . . . . . . . 9 class ...
21	17, 19, 20	co 7393	. . . . . . . 8 class (0...𝑛)
22	2	cneg 11427	. . . . . . . . . . 11 class -1
23		vk	. . . . . . . . . . . 12 setvar 𝑘
24	23	cv 1540	. . . . . . . . . . 11 class 𝑘
25		cexp 14009	. . . . . . . . . . 11 class ↑
26	22, 24, 25	co 7393	. . . . . . . . . 10 class (-1↑𝑘)
27		cbc 14244	. . . . . . . . . . 11 class C
28	19, 24, 27	co 7393	. . . . . . . . . 10 class (𝑛C𝑘)
29	26, 28, 14	co 7393	. . . . . . . . 9 class ((-1↑𝑘) · (𝑛C𝑘))
30		caddc 11095	. . . . . . . . . . 11 class +
31	24, 2, 30	co 7393	. . . . . . . . . 10 class (𝑘 + 1)
32	31, 5, 8	co 7393	. . . . . . . . 9 class ((𝑘 + 1)↑𝑐𝑠)
33	29, 32, 14	co 7393	. . . . . . . 8 class (((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
34	21, 33, 23	csu 15614	. . . . . . 7 class Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠))
35	19, 2, 30	co 7393	. . . . . . . 8 class (𝑛 + 1)
36	3, 35, 25	co 7393	. . . . . . 7 class (2↑(𝑛 + 1))
37		cdiv 11853	. . . . . . 7 class /
38	34, 36, 37	co 7393	. . . . . 6 class (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))
39	16, 38, 18	csu 15614	. . . . 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 11090	. . . . 5 class ℂ
42	2	csn 4622	. . . . 5 class {1}
43	41, 42	cdif 3941	. . . 4 class (ℂ ∖ {1})
44	40, 4, 43	wral 3060	. . 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 24321	. . . 4 class –cn→
46	43, 41, 45	co 7393	. . 3 class ((ℂ ∖ {1})–cn→ℂ)
47	44, 11, 46	crio 7348	. 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)