MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  selberg Structured version   Unicode version

Theorem selberg 23931
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 5848 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (Λ `  n )  =  (Λ `  d ) )
2 oveq2 6278 . . . . . . . . . . . . . 14  |-  ( n  =  d  ->  (
x  /  n )  =  ( x  / 
d ) )
32fveq2d 5852 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (ψ `  ( x  /  n
) )  =  (ψ `  ( x  /  d
) ) )
41, 3oveq12d 6288 . . . . . . . . . . . 12  |-  ( n  =  d  ->  (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  =  ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )
54cbvsumv 13600 . . . . . . . . . . 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 12065 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
7 elfznn 11717 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
87adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
9 vmacl 23590 . . . . . . . . . . . . . . . 16  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
108, 9syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
1110recnd 9611 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
12 elfznn 11717 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  m  e.  NN )
1312adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  NN )
14 vmacl 23590 . . . . . . . . . . . . . . . 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 9611 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  CC )
176, 11, 16fsummulc2 13681 . . . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
19 rpdivcl 11244 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2018, 19sylan2 472 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2120rpred 11259 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
22 chpval 23594 . . . . . . . . . . . . . . 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 6286 . . . . . . . . . . . . 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 10547 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  CC )
267ad2antlr 724 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  NN )
2726nncnd 10547 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  CC )
2826nnne0d 10576 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  =/=  0 )
2925, 27, 28divcan3d 10321 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( (
d  x.  m )  /  d )  =  m )
3029fveq2d 5852 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  (
( d  x.  m
)  /  d ) )  =  (Λ `  m
) )
3130oveq2d 6286 . . . . . . . . . . . . . 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 13607 . . . . . . . . . . . . 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 2505 . . . . . . . . . . . 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 13607 . . . . . . . . . . 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 2507 . . . . . . . . . 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 6277 . . . . . . . . . . . . 13  |-  ( n  =  ( d  x.  m )  ->  (
n  /  d )  =  ( ( d  x.  m )  / 
d ) )
3736fveq2d 5852 . . . . . . . . . . . 12  |-  ( n  =  ( d  x.  m )  ->  (Λ `  ( n  /  d
) )  =  (Λ `  ( ( d  x.  m )  /  d
) ) )
3837oveq2d 6286 . . . . . . . . . . 11  |-  ( n  =  ( d  x.  m )  ->  (
(Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( ( d  x.  m )  /  d
) ) ) )
39 rpre 11227 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
40 ssrab2 3571 . . . . . . . . . . . . . . . . 17  |-  { y  e.  NN  |  y 
||  n }  C_  NN
41 simprr 755 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  { y  e.  NN  |  y 
||  n } )
4240, 41sseldi 3487 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  NN )
4342anassrs 646 . . . . . . . . . . . . . . 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 11717 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
4645adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
47 dvdsdivcl 23655 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  d  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  d )  e.  { y  e.  NN  |  y  ||  n } )
4846, 47sylan 469 . . . . . . . . . . . . . . . 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 3487 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  d )  e.  NN )
50 vmacl 23590 . . . . . . . . . . . . . . 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 9613 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  RR )
5352recnd 9611 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  CC )
5453anasss 645 . . . . . . . . . . 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 23662 . . . . . . . . . 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 2498 . . . . . . . . 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 6285 . . . . . . . 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 12065 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
59 vmacl 23590 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
6046, 59syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
6160recnd 9611 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
6245nnrpd 11257 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
63 rpdivcl 11244 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
6462, 63sylan2 472 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
6564rpred 11259 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
66 chpcl 23596 . . . . . . . . . . . 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 9611 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  CC )
6961, 68mulcld 9605 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
7046nnrpd 11257 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
71 relogcl 23129 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
7270, 71syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
7372recnd 9611 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
7461, 73mulcld 9605 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  e.  CC )
7558, 69, 74fsumadd 13643 . . . . . . . 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 12065 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... n )  e. 
Fin )
77 sgmss 23578 . . . . . . . . . . . . 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 7733 . . . . . . . . . . . 12  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
8076, 78, 79syl2anc 659 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
8180, 52fsumrecl 13638 . . . . . . . . . 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 9611 . . . . . . . . 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 13643 . . . . . . . 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 2505 . . . . . . 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 9771 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) )  =  ( (ψ `  ( x  /  n
) )  +  ( log `  n ) ) )
8685oveq2d 6286 . . . . . . . . 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 9609 . . . . . . . . 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 2495 . . . . . . . 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 13607 . . . . . . 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 23926 . . . . . . . . 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 13607 . . . . . . 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 2505 . . . . . 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 5852 . . . . . . . . 9  |-  ( n  =  ( d  x.  m )  ->  ( log `  ( n  / 
d ) )  =  ( log `  (
( d  x.  m
)  /  d ) ) )
9594oveq1d 6285 . . . . . . . 8  |-  ( n  =  ( d  x.  m )  ->  (
( log `  (
n  /  d ) ) ^ 2 )  =  ( ( log `  ( ( d  x.  m )  /  d
) ) ^ 2 ) )
9695oveq2d 6286 . . . . . . 7  |-  ( n  =  ( d  x.  m )  ->  (
( mmu `  d
)  x.  ( ( log `  ( n  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
( d  x.  m
)  /  d ) ) ^ 2 ) ) )
97 mucl 23613 . . . . . . . . . 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 10966 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  CC )
10062ad2antrl 725 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  ->  n  e.  RR+ )
10142nnrpd 11257 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  RR+ )
102100, 101rpdivcld 11276 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( n  /  d
)  e.  RR+ )
103 relogcl 23129 . . . . . . . . . . 11  |-  ( ( n  /  d )  e.  RR+  ->  ( log `  ( n  /  d
) )  e.  RR )
104103recnd 9611 . . . . . . . . . 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 12290 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( ( log `  (
n  /  d ) ) ^ 2 )  e.  CC )
10799, 106mulcld 9605 . . . . . . 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 23662 . . . . . 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 5852 . . . . . . . . . 10  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  ( ( d  x.  m )  /  d
) )  =  ( log `  m ) )
110109oveq1d 6285 . . . . . . . . 9  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  ( ( d  x.  m )  / 
d ) ) ^
2 )  =  ( ( log `  m
) ^ 2 ) )
111110oveq2d 6286 . . . . . . . 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 13607 . . . . . . 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 13607 . . . . . 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 2499 . . . . 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 6285 . . . 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 6285 . . 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 4521 . 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 2454 . . 3  |-  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)  =  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)
119118selberglem2 23929 . 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 2538 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 367    = wceq 1398    e. wcel 1823   {crab 2808    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202   NNcn 10531   2c2 10581   ZZcz 10860   RR+crp 11221   ...cfz 11675   |_cfl 11908   ^cexp 12148   O(1)co1 13391   sum_csu 13590    || cdvds 14070   logclog 23108  Λcvma 23563  ψcchp 23564   mmucmu 23566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-o1 13395  df-lo1 13396  df-sum 13591  df-ef 13885  df-e 13886  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-cmp 20054  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cxp 23111  df-em 23520  df-vma 23569  df-chp 23570  df-mu 23572
This theorem is referenced by:  selbergb  23932  selberg2  23934  selbergs  23957
  Copyright terms: Public domain W3C validator