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Theorem selberg 22682
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.
StepHypRef Expression
1 fveq2 5679 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (Λ `  n )  =  (Λ `  d ) )
2 oveq2 6088 . . . . . . . . . . . . . 14  |-  ( n  =  d  ->  (
x  /  n )  =  ( x  / 
d ) )
32fveq2d 5683 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (ψ `  ( x  /  n
) )  =  (ψ `  ( x  /  d
) ) )
41, 3oveq12d 6098 . . . . . . . . . . . 12  |-  ( n  =  d  ->  (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  =  ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )
54cbvsumv 13157 . . . . . . . . . . 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 11779 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
7 elfznn 11465 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
87adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
9 vmacl 22341 . . . . . . . . . . . . . . . 16  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
108, 9syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
1110recnd 9400 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
12 elfznn 11465 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  m  e.  NN )
1312adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  NN )
14 vmacl 22341 . . . . . . . . . . . . . . . 16  |-  ( m  e.  NN  ->  (Λ `  m )  e.  RR )
1513, 14syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  RR )
1615recnd 9400 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  CC )
176, 11, 16fsummulc2 13234 . . . . . . . . . . . . 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 ) ) )
187nnrpd 11014 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
19 rpdivcl 11001 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2018, 19sylan2 471 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2120rpred 11015 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
22 chpval 22345 . . . . . . . . . . . . . . 15  |-  ( ( x  /  d )  e.  RR  ->  (ψ `  ( x  /  d
) )  =  sum_ m  e.  ( 1 ... ( |_ `  (
x  /  d ) ) ) (Λ `  m
) )
2321, 22syl 16 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  =  sum_ m  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) (Λ `  m )
)
2423oveq2d 6096 . . . . . . . . . . . . 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 )
) )
2513nncnd 10326 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  CC )
267ad2antlr 719 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  NN )
2726nncnd 10326 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  CC )
2826nnne0d 10354 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  =/=  0 )
2925, 27, 28divcan3d 10100 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( (
d  x.  m )  /  d )  =  m )
3029fveq2d 5683 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  (
( d  x.  m
)  /  d ) )  =  (Λ `  m
) )
3130oveq2d 6096 . . . . . . . . . . . . . 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
) ) )
3231sumeq2dv 13164 . . . . . . . . . . . . 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 ) ) )
3317, 24, 323eqtr4d 2475 . . . . . . . . . . . 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 ) ) ) )
3433sumeq2dv 13164 . . . . . . . . . . 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 ) ) ) )
355, 34syl5eq 2477 . . . . . . . . . 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 6087 . . . . . . . . . . . . 13  |-  ( n  =  ( d  x.  m )  ->  (
n  /  d )  =  ( ( d  x.  m )  / 
d ) )
3736fveq2d 5683 . . . . . . . . . . . 12  |-  ( n  =  ( d  x.  m )  ->  (Λ `  ( n  /  d
) )  =  (Λ `  ( ( d  x.  m )  /  d
) ) )
3837oveq2d 6096 . . . . . . . . . . 11  |-  ( n  =  ( d  x.  m )  ->  (
(Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( ( d  x.  m )  /  d
) ) ) )
39 rpre 10985 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
40 ssrab2 3425 . . . . . . . . . . . . . . . . 17  |-  { y  e.  NN  |  y 
||  n }  C_  NN
41 simprr 749 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  { y  e.  NN  |  y 
||  n } )
4240, 41sseldi 3342 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  NN )
4342anassrs 641 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  d  e.  NN )
4443, 9syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  d
)  e.  RR )
45 elfznn 11465 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
4645adantl 463 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
47 dvdsdivcl 22406 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  d  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  d )  e.  { y  e.  NN  |  y  ||  n } )
4846, 47sylan 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  d )  e. 
{ y  e.  NN  |  y  ||  n }
)
4940, 48sseldi 3342 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  d )  e.  NN )
50 vmacl 22341 . . . . . . . . . . . . . . 15  |-  ( ( n  /  d )  e.  NN  ->  (Λ `  ( n  /  d
) )  e.  RR )
5149, 50syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  (
n  /  d ) )  e.  RR )
5244, 51remulcld 9402 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  RR )
5352recnd 9400 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  CC )
5453anasss 640 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( (Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  e.  CC )
5538, 39, 54dvdsflsumcom 22413 . . . . . . . . . 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 ) ) ) )
5635, 55eqtr4d 2468 . . . . . . . . 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
) ) ) )
5756oveq1d 6095 . . . . . . . 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 11779 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
59 vmacl 22341 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
6046, 59syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
6160recnd 9400 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
6245nnrpd 11014 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
63 rpdivcl 11001 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
6462, 63sylan2 471 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
6564rpred 11015 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
66 chpcl 22347 . . . . . . . . . . . 12  |-  ( ( x  /  n )  e.  RR  ->  (ψ `  ( x  /  n
) )  e.  RR )
6765, 66syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  RR )
6867recnd 9400 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  CC )
6961, 68mulcld 9394 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
7046nnrpd 11014 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
71 relogcl 21912 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
7270, 71syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
7372recnd 9400 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
7461, 73mulcld 9394 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  e.  CC )
7558, 69, 74fsumadd 13199 . . . . . . . 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 11779 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... n )  e. 
Fin )
77 sgmss 22329 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  { y  e.  NN  |  y 
||  n }  C_  ( 1 ... n
) )
7846, 77syl 16 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  C_  ( 1 ... n ) )
79 ssfi 7521 . . . . . . . . . . . 12  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
8076, 78, 79syl2anc 654 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
8180, 52fsumrecl 13195 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  sum_ d  e. 
{ y  e.  NN  |  y  ||  n } 
( (Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  e.  RR )
8281recnd 9400 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  sum_ d  e. 
{ y  e.  NN  |  y  ||  n } 
( (Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  e.  CC )
8358, 82, 74fsumadd 13199 . . . . . . . 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 ) ) ) )
8457, 75, 833eqtr4d 2475 . . . . . . 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 ) ) ) )
8573, 68addcomd 9559 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) )  =  ( (ψ `  ( x  /  n
) )  +  ( log `  n ) ) )
8685oveq2d 6096 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  =  ( (Λ `  n
)  x.  ( (ψ `  ( x  /  n
) )  +  ( log `  n ) ) ) )
8761, 68, 73adddid 9398 . . . . . . . . 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 ) ) ) )
8886, 87eqtrd 2465 . . . . . . . 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 ) ) ) )
8988sumeq2dv 13164 . . . . . . 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 22677 . . . . . . . . 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 ) ) ) )
9146, 90syl 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 ) ) ) )
9291sumeq2dv 13164 . . . . . . 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 ) ) ) )
9384, 89, 923eqtr4d 2475 . . . . . 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 ) ) )
9436fveq2d 5683 . . . . . . . . 9  |-  ( n  =  ( d  x.  m )  ->  ( log `  ( n  / 
d ) )  =  ( log `  (
( d  x.  m
)  /  d ) ) )
9594oveq1d 6095 . . . . . . . 8  |-  ( n  =  ( d  x.  m )  ->  (
( log `  (
n  /  d ) ) ^ 2 )  =  ( ( log `  ( ( d  x.  m )  /  d
) ) ^ 2 ) )
9695oveq2d 6096 . . . . . . 7  |-  ( n  =  ( d  x.  m )  ->  (
( mmu `  d
)  x.  ( ( log `  ( n  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
( d  x.  m
)  /  d ) ) ^ 2 ) ) )
97 mucl 22364 . . . . . . . . . 10  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
9842, 97syl 16 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  ZZ )
9998zcnd 10736 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  CC )
10062ad2antrl 720 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  ->  n  e.  RR+ )
10142nnrpd 11014 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  RR+ )
102100, 101rpdivcld 11032 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( n  /  d
)  e.  RR+ )
103 relogcl 21912 . . . . . . . . . . 11  |-  ( ( n  /  d )  e.  RR+  ->  ( log `  ( n  /  d
) )  e.  RR )
104103recnd 9400 . . . . . . . . . 10  |-  ( ( n  /  d )  e.  RR+  ->  ( log `  ( n  /  d
) )  e.  CC )
105102, 104syl 16 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( log `  (
n  /  d ) )  e.  CC )
106105sqcld 11990 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( ( log `  (
n  /  d ) ) ^ 2 )  e.  CC )
10799, 106mulcld 9394 . . . . . . 7  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( ( mmu `  d )  x.  (
( log `  (
n  /  d ) ) ^ 2 ) )  e.  CC )
10896, 39, 107dvdsflsumcom 22413 . . . . . 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 ) ) )
10929fveq2d 5683 . . . . . . . . . 10  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  ( ( d  x.  m )  /  d
) )  =  ( log `  m ) )
110109oveq1d 6095 . . . . . . . . 9  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  ( ( d  x.  m )  / 
d ) ) ^
2 )  =  ( ( log `  m
) ^ 2 ) )
111110oveq2d 6096 . . . . . . . 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 ) ) )
112111sumeq2dv 13164 . . . . . . 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 ) ) )
113112sumeq2dv 13164 . . . . . 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 ) ) )
11493, 108, 1133eqtrd 2469 . . . . 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 ) ) )
115114oveq1d 6095 . . . 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 ) )
116115oveq1d 6095 . . 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 ) ) ) )
117116mpteq2ia 4362 . 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 2433 . . 3  |-  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)  =  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)
119118selberglem2 22680 . 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)
120117, 119eqeltri 2503 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)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1362    e. wcel 1755   {crab 2709    C_ wss 3316   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9268   RRcr 9269   1c1 9271    + caddc 9273    x. cmul 9275    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   ZZcz 10634   RR+crp 10979   ...cfz 11424   |_cfl 11624   ^cexp 11849   O(1)co1 12948   sum_csu 13147    || cdivides 13518   logclog 21891  Λcvma 22314  ψcchp 22315   mmucmu 22317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-o1 12952  df-lo1 12953  df-sum 13148  df-ef 13336  df-e 13337  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894  df-em 22271  df-vma 22320  df-chp 22321  df-mu 22323
This theorem is referenced by:  selbergb  22683  selberg2  22685  selbergs  22708
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