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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.
StepHypRef Expression
1 fveq2 5687 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (Λ `  n )  =  (Λ `  d ) )
2 oveq2 6048 . . . . . . . . . . . . . 14  |-  ( n  =  d  ->  (
x  /  n )  =  ( x  / 
d ) )
32fveq2d 5691 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (ψ `  ( x  /  n
) )  =  (ψ `  ( x  /  d
) ) )
41, 3oveq12d 6058 . . . . . . . . . . . 12  |-  ( n  =  d  ->  (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  =  ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )
54cbvsumv 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 )
87adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
9 vmacl 20854 . . . . . . . . . . . . . . . 16  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
108, 9syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
1110recnd 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 )
1312adantl 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 )
1513, 14syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  RR )
1615recnd 9070 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  CC )
176, 11, 16fsummulc2 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 ) ) )
187nnrpd 10603 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
19 rpdivcl 10590 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2018, 19sylan2 461 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2120rpred 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
) )
2321, 22syl 16 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  =  sum_ m  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) (Λ `  m )
)
2423oveq2d 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 )
) )
2513nncnd 9972 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  CC )
267ad2antlr 708 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  NN )
2726nncnd 9972 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  CC )
2826nnne0d 10000 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  =/=  0 )
2925, 27, 28divcan3d 9751 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( (
d  x.  m )  /  d )  =  m )
3029fveq2d 5691 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  (
( d  x.  m
)  /  d ) )  =  (Λ `  m
) )
3130oveq2d 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
) ) )
3231sumeq2dv 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 ) ) )
3317, 24, 323eqtr4d 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 ) ) ) )
3433sumeq2dv 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 ) ) ) )
355, 34syl5eq 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 ) )
3736fveq2d 5691 . . . . . . . . . . . 12  |-  ( n  =  ( d  x.  m )  ->  (Λ `  ( n  /  d
) )  =  (Λ `  ( ( d  x.  m )  /  d
) ) )
3837oveq2d 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 } )
4240, 41sseldi 3306 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  NN )
4342anassrs 630 . . . . . . . . . . . . . . 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 11036 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
4645adantl 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 } )
4846, 47sylan 458 . . . . . . . . . . . . . . . 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 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 )
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 9072 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  RR )
5352recnd 9070 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  CC )
5453anasss 629 . . . . . . . . . . 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 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 ) ) ) )
5635, 55eqtr4d 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
) ) ) )
5756oveq1d 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 )
6046, 59syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
6160recnd 9070 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
6245nnrpd 10603 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
63 rpdivcl 10590 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
6462, 63sylan2 461 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
6564rpred 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 )
6765, 66syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  RR )
6867recnd 9070 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  CC )
6961, 68mulcld 9064 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
7046nnrpd 10603 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
71 relogcl 20426 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
7270, 71syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
7372recnd 9070 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
7461, 73mulcld 9064 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  e.  CC )
7558, 69, 74fsumadd 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
) )
7846, 77syl 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 )
8076, 78, 79syl2anc 643 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
8180, 52fsumrecl 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 )
8281recnd 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 )
8358, 82, 74fsumadd 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 ) ) ) )
8457, 75, 833eqtr4d 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 ) ) ) )
8573, 68addcomd 9224 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) )  =  ( (ψ `  ( x  /  n
) )  +  ( log `  n ) ) )
8685oveq2d 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 ) ) ) )
8761, 68, 73adddid 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 ) ) ) )
8886, 87eqtrd 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 ) ) ) )
8988sumeq2dv 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 ) ) ) )
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 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 ) ) ) )
9384, 89, 923eqtr4d 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 ) ) )
9436fveq2d 5691 . . . . . . . . 9  |-  ( n  =  ( d  x.  m )  ->  ( log `  ( n  / 
d ) )  =  ( log `  (
( d  x.  m
)  /  d ) ) )
9594oveq1d 6055 . . . . . . . 8  |-  ( n  =  ( d  x.  m )  ->  (
( log `  (
n  /  d ) ) ^ 2 )  =  ( ( log `  ( ( d  x.  m )  /  d
) ) ^ 2 ) )
9695oveq2d 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 )
9842, 97syl 16 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  ZZ )
9998zcnd 10332 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  CC )
10062ad2antrl 709 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  ->  n  e.  RR+ )
10142nnrpd 10603 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  RR+ )
102100, 101rpdivcld 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 )
104103recnd 9070 . . . . . . . . . 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 11476 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( ( log `  (
n  /  d ) ) ^ 2 )  e.  CC )
10799, 106mulcld 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 )
10896, 39, 107dvdsflsumcom 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 ) ) )
10929fveq2d 5691 . . . . . . . . . 10  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  ( ( d  x.  m )  /  d
) )  =  ( log `  m ) )
110109oveq1d 6055 . . . . . . . . 9  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  ( ( d  x.  m )  / 
d ) ) ^
2 )  =  ( ( log `  m
) ^ 2 ) )
111110oveq2d 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 ) ) )
112111sumeq2dv 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 ) ) )
113112sumeq2dv 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 ) ) )
11493, 108, 1133eqtrd 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 ) ) )
115114oveq1d 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 ) )
116115oveq1d 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 ) ) ) )
117116mpteq2ia 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
)
119118selberglem2 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 )
120117, 119eqeltri 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 )
Colors of variables: wff set class
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
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