Theorem selberg 21195

Description: Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that sum_ n <_ x , Λ ( n ) log n + sum_ m x. n <_ x , Λ ( m )Λ ( n ) = 2 x log x + O ( x ). Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)

Assertion

Ref	Expression
selberg	|- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O ( 1 )

Distinct variable group:   x, n

Proof of Theorem selberg

Dummy variables d m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5687	. . . . . . . . . . . . 13 |- ( n = d -> (Λ ` n ) = (Λ ` d ) )
2		oveq2 6048	. . . . . . . . . . . . . 14 |- ( n = d -> ( x / n ) = ( x / d ) )
3	2	fveq2d 5691	. . . . . . . . . . . . 13 |- ( n = d -> (ψ ` ( x / n ) ) = (ψ ` ( x / d ) ) )
4	1, 3	oveq12d 6058	. . . . . . . . . . . 12 |- ( n = d -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) )
5	4	cbvsumv 12445	. . . . . . . . . . 11 |- sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) ( (Λ ` d ) x. (ψ ` ( x / d ) ) )
6		fzfid 11267	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / d ) ) ) e. Fin )
7		elfznn 11036	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN )
8	7	adantl 453	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN )
9		vmacl 20854	. . . . . . . . . . . . . . . 16 |- ( d e. NN -> (Λ ` d ) e. RR )
10	8, 9	syl 16	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. RR )
11	10	recnd 9070	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. CC )
12		elfznn 11036	. . . . . . . . . . . . . . . . 17 |- ( m e. ( 1 ... ( |_ ` ( x / d ) ) ) -> m e. NN )
13	12	adantl 453	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. NN )
14		vmacl 20854	. . . . . . . . . . . . . . . 16 |- ( m e. NN -> (Λ ` m ) e. RR )
15	13, 14	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. RR )
16	15	recnd 9070	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. CC )
17	6, 11, 16	fsummulc2 12522	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` d ) x. sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` m ) ) )
18	7	nnrpd 10603	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
19		rpdivcl 10590	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ )
20	18, 19	sylan2 461	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ )
21	20	rpred 10604	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR )
22		chpval 20858	. . . . . . . . . . . . . . 15 |- ( ( x / d ) e. RR -> (ψ ` ( x / d ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) )
23	21, 22	syl 16	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / d ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) )
24	23	oveq2d 6056	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) = ( (Λ ` d ) x. sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) ) )
25	13	nncnd 9972	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. CC )
26	7	ad2antlr 708	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. NN )
27	26	nncnd 9972	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. CC )
28	26	nnne0d 10000	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d =/= 0 )
29	25, 27, 28	divcan3d 9751	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( d x. m ) / d ) = m )
30	29	fveq2d 5691	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` ( ( d x. m ) / d ) ) = (Λ ` m ) )
31	30	oveq2d 6056	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) = ( (Λ ` d ) x. (Λ ` m ) ) )
32	31	sumeq2dv 12452	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` m ) ) )
33	17, 24, 32	3eqtr4d 2446	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
34	33	sumeq2dv 12452	. . . . . . . . . . 11 |- ( x e. RR+ -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
35	5, 34	syl5eq 2448	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
36		oveq1 6047	. . . . . . . . . . . . 13 |- ( n = ( d x. m ) -> ( n / d ) = ( ( d x. m ) / d ) )
37	36	fveq2d 5691	. . . . . . . . . . . 12 |- ( n = ( d x. m ) -> (Λ ` ( n / d ) ) = (Λ ` ( ( d x. m ) / d ) ) )
38	37	oveq2d 6056	. . . . . . . . . . 11 |- ( n = ( d x. m ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
39		rpre 10574	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
40		ssrab2 3388	. . . . . . . . . . . . . . . . 17 |- { y e. NN | y || n } C_ NN
41		simprr 734	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. { y e. NN | y || n } )
42	40, 41	sseldi 3306	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. NN )
43	42	anassrs 630	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> d e. NN )
44	43, 9	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> (Λ ` d ) e. RR )
45		elfznn 11036	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
46	45	adantl 453	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
47		dvdsdivcl 20919	. . . . . . . . . . . . . . . . 17 |- ( ( n e. NN /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. { y e. NN | y || n } )
48	46, 47	sylan 458	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. { y e. NN | y || n } )
49	40, 48	sseldi 3306	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. NN )
50		vmacl 20854	. . . . . . . . . . . . . . 15 |- ( ( n / d ) e. NN -> (Λ ` ( n / d ) ) e. RR )
51	49, 50	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> (Λ ` ( n / d ) ) e. RR )
52	44, 51	remulcld 9072	. . . . . . . . . . . . 13 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. RR )
53	52	recnd 9070	. . . . . . . . . . . 12 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. CC )
54	53	anasss 629	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. CC )
55	38, 39, 54	dvdsflsumcom 20926	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
56	35, 55	eqtr4d 2439	. . . . . . . . 9 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) )
57	56	oveq1d 6055	. . . . . . . 8 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) )
58		fzfid 11267	. . . . . . . . 9 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
59		vmacl 20854	. . . . . . . . . . . 12 |- ( n e. NN -> (Λ ` n ) e. RR )
60	46, 59	syl 16	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. RR )
61	60	recnd 9070	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
62	45	nnrpd 10603	. . . . . . . . . . . . . 14 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. RR+ )
63		rpdivcl 10590	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. RR+ ) -> ( x / n ) e. RR+ )
64	62, 63	sylan2 461	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR+ )
65	64	rpred 10604	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
66		chpcl 20860	. . . . . . . . . . . 12 |- ( ( x / n ) e. RR -> (ψ ` ( x / n ) ) e. RR )
67	65, 66	syl 16	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. RR )
68	67	recnd 9070	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. CC )
69	61, 68	mulcld 9064	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
70	46	nnrpd 10603	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
71		relogcl 20426	. . . . . . . . . . . 12 |- ( n e. RR+ -> ( log ` n ) e. RR )
72	70, 71	syl 16	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. RR )
73	72	recnd 9070	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. CC )
74	61, 73	mulcld 9064	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( log ` n ) ) e. CC )
75	58, 69, 74	fsumadd 12487	. . . . . . . 8 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) )
76		fzfid 11267	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... n ) e. Fin )
77		sgmss 20842	. . . . . . . . . . . . 13 |- ( n e. NN -> { y e. NN | y || n } C_ ( 1 ... n ) )
78	46, 77	syl 16	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } C_ ( 1 ... n ) )
79		ssfi 7288	. . . . . . . . . . . 12 |- ( ( ( 1 ... n ) e. Fin /\ { y e. NN | y || n } C_ ( 1 ... n ) ) -> { y e. NN | y || n } e. Fin )
80	76, 78, 79	syl2anc 643	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } e. Fin )
81	80, 52	fsumrecl 12483	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. RR )
82	81	recnd 9070	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. CC )
83	58, 82, 74	fsumadd 12487	. . . . . . . 8 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) )
84	57, 75, 83	3eqtr4d 2446	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
85	73, 68	addcomd 9224	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` n ) + (ψ ` ( x / n ) ) ) = ( (ψ ` ( x / n ) ) + ( log ` n ) ) )
86	85	oveq2d 6056	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = ( (Λ ` n ) x. ( (ψ ` ( x / n ) ) + ( log ` n ) ) ) )
87	61, 68, 73	adddid 9068	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( (ψ ` ( x / n ) ) + ( log ` n ) ) ) = ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
88	86, 87	eqtrd 2436	. . . . . . . 8 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
89	88	sumeq2dv 12452	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
90		logsqvma2 21190	. . . . . . . . 9 |- ( n e. NN -> sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
91	46, 90	syl 16	. . . . . . . 8 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
92	91	sumeq2dv 12452	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
93	84, 89, 92	3eqtr4d 2446	. . . . . 6 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) )
94	36	fveq2d 5691	. . . . . . . . 9 |- ( n = ( d x. m ) -> ( log ` ( n / d ) ) = ( log ` ( ( d x. m ) / d ) ) )
95	94	oveq1d 6055	. . . . . . . 8 |- ( n = ( d x. m ) -> ( ( log ` ( n / d ) ) ^ 2 ) = ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) )
96	95	oveq2d 6056	. . . . . . 7 |- ( n = ( d x. m ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
97		mucl 20877	. . . . . . . . . 10 |- ( d e. NN -> ( mmu ` d ) e. ZZ )
98	42, 97	syl 16	. . . . . . . . 9 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. ZZ )
99	98	zcnd 10332	. . . . . . . 8 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. CC )
100	62	ad2antrl 709	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> n e. RR+ )
101	42	nnrpd 10603	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. RR+ )
102	100, 101	rpdivcld 10621	. . . . . . . . . 10 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( n / d ) e. RR+ )
103		relogcl 20426	. . . . . . . . . . 11 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. RR )
104	103	recnd 9070	. . . . . . . . . 10 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. CC )
105	102, 104	syl 16	. . . . . . . . 9 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( log ` ( n / d ) ) e. CC )
106	105	sqcld 11476	. . . . . . . 8 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( ( log ` ( n / d ) ) ^ 2 ) e. CC )
107	99, 106	mulcld 9064	. . . . . . 7 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) e. CC )
108	96, 39, 107	dvdsflsumcom 20926	. . . . . 6 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
109	29	fveq2d 5691	. . . . . . . . . 10 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( log ` ( ( d x. m ) / d ) ) = ( log ` m ) )
110	109	oveq1d 6055	. . . . . . . . 9 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) = ( ( log ` m ) ^ 2 ) )
111	110	oveq2d 6056	. . . . . . . 8 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
112	111	sumeq2dv 12452	. . . . . . 7 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
113	112	sumeq2dv 12452	. . . . . 6 |- ( x e. RR+ -> sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
114	93, 108, 113	3eqtrd 2440	. . . . 5 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
115	114	oveq1d 6055	. . . 4 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) )
116	115	oveq1d 6055	. . 3 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) = ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
117	116	mpteq2ia 4251	. 2 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) = ( x e. RR+ |-> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
118		eqid 2404	. . 3 |- ( ( ( ( log ` ( x / d ) ) ^ 2 ) + ( 2 - ( 2 x. ( log ` ( x / d ) ) ) ) ) / d ) = ( ( ( ( log ` ( x / d ) ) ^ 2 ) + ( 2 - ( 2 x. ( log ` ( x / d ) ) ) ) ) / d )
119	118	selberglem2 21193	. 2 |- ( x e. RR+ |-> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O ( 1 )
120	117, 119	eqeltri 2474	1 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O ( 1 )

Syntax hints:   /\ wa 359   = wceq 1649   e. wcel 1721   {crab 2670   C_ wss 3280   class class class wbr 4172   e. cmpt 4226   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   1c1 8947   + caddc 8949   x. cmul 8951   - cmin 9247   / cdiv 9633   NNcn 9956   2c2 10005   ZZcz 10238   RR+crp 10568   ...cfz 10999   |_cfl 11156   ^cexp 11337   O ( 1 )co1 12235   sum_csu 12434   || cdivides 12807   logclog 20405  Λcvma 20827  ψcchp 20828   mmucmu 20830

This theorem is referenced by:  selbergb  21196  selberg2  21198  selbergs  21221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-em 20784  df-vma 20833  df-chp 20834  df-mu 20836