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Theorem selberg 24465
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 5879 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (Λ `  n )  =  (Λ `  d ) )
2 oveq2 6316 . . . . . . . . . . . . . 14  |-  ( n  =  d  ->  (
x  /  n )  =  ( x  / 
d ) )
32fveq2d 5883 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (ψ `  ( x  /  n
) )  =  (ψ `  ( x  /  d
) ) )
41, 3oveq12d 6326 . . . . . . . . . . . 12  |-  ( n  =  d  ->  (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  =  ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )
54cbvsumv 13839 . . . . . . . . . . 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 12224 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
7 elfznn 11854 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
87adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
9 vmacl 24124 . . . . . . . . . . . . . . . 16  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
108, 9syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
1110recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
12 elfznn 11854 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  m  e.  NN )
1312adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  NN )
14 vmacl 24124 . . . . . . . . . . . . . . . 16  |-  ( m  e.  NN  ->  (Λ `  m )  e.  RR )
1513, 14syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  RR )
1615recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  CC )
176, 11, 16fsummulc2 13922 . . . . . . . . . . . . 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 11362 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
19 rpdivcl 11348 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2018, 19sylan2 482 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2120rpred 11364 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
22 chpval 24128 . . . . . . . . . . . . . . 15  |-  ( ( x  /  d )  e.  RR  ->  (ψ `  ( x  /  d
) )  =  sum_ m  e.  ( 1 ... ( |_ `  (
x  /  d ) ) ) (Λ `  m
) )
2321, 22syl 17 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  =  sum_ m  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) (Λ `  m )
)
2423oveq2d 6324 . . . . . . . . . . . . 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 10647 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  CC )
267ad2antlr 741 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  NN )
2726nncnd 10647 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  CC )
2826nnne0d 10676 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  =/=  0 )
2925, 27, 28divcan3d 10410 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( (
d  x.  m )  /  d )  =  m )
3029fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  (
( d  x.  m
)  /  d ) )  =  (Λ `  m
) )
3130oveq2d 6324 . . . . . . . . . . . . . 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 13846 . . . . . . . . . . . . 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 2515 . . . . . . . . . . . 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 13846 . . . . . . . . . . 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 2517 . . . . . . . . . 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 6315 . . . . . . . . . . . . 13  |-  ( n  =  ( d  x.  m )  ->  (
n  /  d )  =  ( ( d  x.  m )  / 
d ) )
3736fveq2d 5883 . . . . . . . . . . . 12  |-  ( n  =  ( d  x.  m )  ->  (Λ `  ( n  /  d
) )  =  (Λ `  ( ( d  x.  m )  /  d
) ) )
3837oveq2d 6324 . . . . . . . . . . 11  |-  ( n  =  ( d  x.  m )  ->  (
(Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( ( d  x.  m )  /  d
) ) ) )
39 rpre 11331 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
40 ssrab2 3500 . . . . . . . . . . . . . . . . 17  |-  { y  e.  NN  |  y 
||  n }  C_  NN
41 simprr 774 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  { y  e.  NN  |  y 
||  n } )
4240, 41sseldi 3416 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  NN )
4342anassrs 660 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  d  e.  NN )
4443, 9syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  d
)  e.  RR )
45 elfznn 11854 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
4645adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
47 dvdsdivcl 24189 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  d  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  d )  e.  { y  e.  NN  |  y  ||  n } )
4846, 47sylan 479 . . . . . . . . . . . . . . . 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 3416 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  d )  e.  NN )
50 vmacl 24124 . . . . . . . . . . . . . . 15  |-  ( ( n  /  d )  e.  NN  ->  (Λ `  ( n  /  d
) )  e.  RR )
5149, 50syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  (
n  /  d ) )  e.  RR )
5244, 51remulcld 9689 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  RR )
5352recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  CC )
5453anasss 659 . . . . . . . . . . 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 24196 . . . . . . . . . 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 2508 . . . . . . . . 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 6323 . . . . . . . 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 12224 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
59 vmacl 24124 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
6046, 59syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
6160recnd 9687 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
6245nnrpd 11362 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
63 rpdivcl 11348 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
6462, 63sylan2 482 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
6564rpred 11364 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
66 chpcl 24130 . . . . . . . . . . . 12  |-  ( ( x  /  n )  e.  RR  ->  (ψ `  ( x  /  n
) )  e.  RR )
6765, 66syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  RR )
6867recnd 9687 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  CC )
6961, 68mulcld 9681 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
7046nnrpd 11362 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
71 relogcl 23604 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
7270, 71syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
7372recnd 9687 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
7461, 73mulcld 9681 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  e.  CC )
7558, 69, 74fsumadd 13882 . . . . . . . 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 12224 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... n )  e. 
Fin )
77 sgmss 24112 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  { y  e.  NN  |  y 
||  n }  C_  ( 1 ... n
) )
7846, 77syl 17 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  C_  ( 1 ... n ) )
79 ssfi 7810 . . . . . . . . . . . 12  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
8076, 78, 79syl2anc 673 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
8180, 52fsumrecl 13877 . . . . . . . . . 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 9687 . . . . . . . . 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 13882 . . . . . . . 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 2515 . . . . . . 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 9853 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) )  =  ( (ψ `  ( x  /  n
) )  +  ( log `  n ) ) )
8685oveq2d 6324 . . . . . . . . 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 9685 . . . . . . . . 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 2505 . . . . . . . 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 13846 . . . . . . 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 24460 . . . . . . . . 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 17 . . . . . . . 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 13846 . . . . . . 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 2515 . . . . . 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 5883 . . . . . . . . 9  |-  ( n  =  ( d  x.  m )  ->  ( log `  ( n  / 
d ) )  =  ( log `  (
( d  x.  m
)  /  d ) ) )
9594oveq1d 6323 . . . . . . . 8  |-  ( n  =  ( d  x.  m )  ->  (
( log `  (
n  /  d ) ) ^ 2 )  =  ( ( log `  ( ( d  x.  m )  /  d
) ) ^ 2 ) )
9695oveq2d 6324 . . . . . . 7  |-  ( n  =  ( d  x.  m )  ->  (
( mmu `  d
)  x.  ( ( log `  ( n  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
( d  x.  m
)  /  d ) ) ^ 2 ) ) )
97 mucl 24147 . . . . . . . . . 10  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
9842, 97syl 17 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  ZZ )
9998zcnd 11064 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  CC )
10062ad2antrl 742 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  ->  n  e.  RR+ )
10142nnrpd 11362 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  RR+ )
102100, 101rpdivcld 11381 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( n  /  d
)  e.  RR+ )
103 relogcl 23604 . . . . . . . . . . 11  |-  ( ( n  /  d )  e.  RR+  ->  ( log `  ( n  /  d
) )  e.  RR )
104103recnd 9687 . . . . . . . . . 10  |-  ( ( n  /  d )  e.  RR+  ->  ( log `  ( n  /  d
) )  e.  CC )
105102, 104syl 17 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( log `  (
n  /  d ) )  e.  CC )
106105sqcld 12452 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( ( log `  (
n  /  d ) ) ^ 2 )  e.  CC )
10799, 106mulcld 9681 . . . . . . 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 24196 . . . . . 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 5883 . . . . . . . . . 10  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  ( ( d  x.  m )  /  d
) )  =  ( log `  m ) )
110109oveq1d 6323 . . . . . . . . 9  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  ( ( d  x.  m )  / 
d ) ) ^
2 )  =  ( ( log `  m
) ^ 2 ) )
111110oveq2d 6324 . . . . . . . 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 13846 . . . . . . 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 13846 . . . . . 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 2509 . . . . 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 6323 . . . 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 6323 . . 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 4478 . 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 2471 . . 3  |-  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)  =  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)
119118selberglem2 24463 . 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 2545 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 376    = wceq 1452    e. wcel 1904   {crab 2760    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   ZZcz 10961   RR+crp 11325   ...cfz 11810   |_cfl 12059   ^cexp 12310   O(1)co1 13627   sum_csu 13829    || cdvds 14382   logclog 23583  Λcvma 24097  ψcchp 24098   mmucmu 24100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-o1 13631  df-lo1 13632  df-sum 13830  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-em 23997  df-vma 24103  df-chp 24104  df-mu 24106
This theorem is referenced by:  selbergb  24466  selberg2  24468  selbergs  24491
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