Definition df-zeta 24740

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	|- zeta = ( iota_ f e. ( ( CC \ { 1 } ) -cn-> CC ) A. s e. ( CC \ { 1 } ) ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) ) = sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) ) )

Distinct variable group:   f, k, n, s

Detailed syntax breakdown of Definition df-zeta

Step	Hyp	Ref	Expression
1		czeta 24739	. 2 class zeta
2		c1 9937	. . . . . . 7 class 1
3		c2 11070	. . . . . . . 8 class 2
4		vs	. . . . . . . . . 10 setvar s
5	4	cv 1482	. . . . . . . . 9 class s
6		cmin 10266	. . . . . . . . 9 class -
7	2, 5, 6	co 6650	. . . . . . . 8 class ( 1 - s )
8		ccxp 24302	. . . . . . . 8 class ^c
9	3, 7, 8	co 6650	. . . . . . 7 class ( 2 ^c ( 1 - s ) )
10	2, 9, 6	co 6650	. . . . . 6 class ( 1 - ( 2 ^c ( 1 - s ) ) )
11		vf	. . . . . . . 8 setvar f
12	11	cv 1482	. . . . . . 7 class f
13	5, 12	cfv 5888	. . . . . 6 class ( f ` s )
14		cmul 9941	. . . . . 6 class x.
15	10, 13, 14	co 6650	. . . . 5 class ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) )
16		cn0 11292	. . . . . 6 class NN0
17		cc0 9936	. . . . . . . . 9 class 0
18		vn	. . . . . . . . . 10 setvar n
19	18	cv 1482	. . . . . . . . 9 class n
20		cfz 12326	. . . . . . . . 9 class ...
21	17, 19, 20	co 6650	. . . . . . . 8 class ( 0 ... n )
22	2	cneg 10267	. . . . . . . . . . 11 class -u 1
23		vk	. . . . . . . . . . . 12 setvar k
24	23	cv 1482	. . . . . . . . . . 11 class k
25		cexp 12860	. . . . . . . . . . 11 class ^
26	22, 24, 25	co 6650	. . . . . . . . . 10 class ( -u 1 ^ k )
27		cbc 13089	. . . . . . . . . . 11 class _C
28	19, 24, 27	co 6650	. . . . . . . . . 10 class ( n _C k )
29	26, 28, 14	co 6650	. . . . . . . . 9 class ( ( -u 1 ^ k ) x. ( n _C k ) )
30		caddc 9939	. . . . . . . . . . 11 class +
31	24, 2, 30	co 6650	. . . . . . . . . 10 class ( k + 1 )
32	31, 5, 8	co 6650	. . . . . . . . 9 class ( ( k + 1 ) ^c s )
33	29, 32, 14	co 6650	. . . . . . . 8 class ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) )
34	21, 33, 23	csu 14416	. . . . . . 7 class sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) )
35	19, 2, 30	co 6650	. . . . . . . 8 class ( n + 1 )
36	3, 35, 25	co 6650	. . . . . . 7 class ( 2 ^ ( n + 1 ) )
37		cdiv 10684	. . . . . . 7 class /
38	34, 36, 37	co 6650	. . . . . 6 class ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) )
39	16, 38, 18	csu 14416	. . . . 5 class sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) )
40	15, 39	wceq 1483	. . . 4 wff ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) ) = sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) )
41		cc 9934	. . . . 5 class CC
42	2	csn 4177	. . . . 5 class { 1 }
43	41, 42	cdif 3571	. . . 4 class ( CC \ { 1 } )
44	40, 4, 43	wral 2912	. . 3 wff A. s e. ( CC \ { 1 } ) ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) ) = sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) )
45		ccncf 22679	. . . 4 class -cn->
46	43, 41, 45	co 6650	. . 3 class ( ( CC \ { 1 } ) -cn-> CC )
47	44, 11, 46	crio 6610	. 2 class ( iota_ f e. ( ( CC \ { 1 } ) -cn-> CC ) A. s e. ( CC \ { 1 } ) ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) ) = sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) ) )
48	1, 47	wceq 1483	1 wff zeta = ( iota_ f e. ( ( CC \ { 1 } ) -cn-> CC ) A. s e. ( CC \ { 1 } ) ( ( 1 - ( 2 ^c ( 1 - s ) ) ) x. ( f ` s ) ) = sum_ n e. NN0 ( sum_ k e. ( 0 ... n ) ( ( ( -u 1 ^ k ) x. ( n _C k ) ) x. ( ( k + 1 ) ^c s ) ) / ( 2 ^ ( n + 1 ) ) ) )

This definition is referenced by: (None)