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Theorem selberg 20692
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
Dummy variables  d  m  y are mutually distinct and distinct from all other variables.

Proof of Theorem selberg
StepHypRef Expression
1 fveq2 5486 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (Λ `  n )  =  (Λ `  d ) )
2 oveq2 5828 . . . . . . . . . . . . . 14  |-  ( n  =  d  ->  (
x  /  n )  =  ( x  / 
d ) )
32fveq2d 5490 . . . . . . . . . . . . 13  |-  ( n  =  d  ->  (ψ `  ( x  /  n
) )  =  (ψ `  ( x  /  d
) ) )
41, 3oveq12d 5838 . . . . . . . . . . . 12  |-  ( n  =  d  ->  (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  =  ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )
54cbvsumv 12164 . . . . . . . . . . 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 11030 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
7 elfznn 10814 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
87adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
9 vmacl 20351 . . . . . . . . . . . . . . . 16  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
108, 9syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
1110recnd 8857 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
12 elfznn 10814 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  m  e.  NN )
1312adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  NN )
14 vmacl 20351 . . . . . . . . . . . . . . . 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 8857 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  m
)  e.  CC )
176, 11, 16fsummulc2 12241 . . . . . . . . . . . . 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 10385 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
19 rpdivcl 10372 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2018, 19sylan2 462 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2120rpred 10386 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
22 chpval 20355 . . . . . . . . . . . . . . 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 5836 . . . . . . . . . . . . 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 9758 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  m  e.  CC )
267ad2antlr 709 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  NN )
2726nncnd 9758 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  e.  CC )
2826nnne0d 9786 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  d  =/=  0 )
2925, 27, 28divcan3d 9537 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( (
d  x.  m )  /  d )  =  m )
3029fveq2d 5490 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  (Λ `  (
( d  x.  m
)  /  d ) )  =  (Λ `  m
) )
3130oveq2d 5836 . . . . . . . . . . . . . 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 12171 . . . . . . . . . . . . 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 2327 . . . . . . . . . . . 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 12171 . . . . . . . . . . 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 2329 . . . . . . . . . 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 5827 . . . . . . . . . . . . 13  |-  ( n  =  ( d  x.  m )  ->  (
n  /  d )  =  ( ( d  x.  m )  / 
d ) )
3736fveq2d 5490 . . . . . . . . . . . 12  |-  ( n  =  ( d  x.  m )  ->  (Λ `  ( n  /  d
) )  =  (Λ `  ( ( d  x.  m )  /  d
) ) )
3837oveq2d 5836 . . . . . . . . . . 11  |-  ( n  =  ( d  x.  m )  ->  (
(Λ `  d )  x.  (Λ `  ( n  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( ( d  x.  m )  /  d
) ) ) )
39 rpre 10356 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
40 ssrab2 3260 . . . . . . . . . . . . . . . . 17  |-  { y  e.  NN  |  y 
||  n }  C_  NN
41 simprr 735 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  { y  e.  NN  |  y 
||  n } )
4240, 41sseldi 3180 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  NN )
4342anassrs 631 . . . . . . . . . . . . . . 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 10814 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
4645adantl 454 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
47 dvdsdivcl 20416 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  d  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  d )  e.  { y  e.  NN  |  y  ||  n } )
4846, 47sylan 459 . . . . . . . . . . . . . . . 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 3180 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  d )  e.  NN )
50 vmacl 20351 . . . . . . . . . . . . . . 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 8859 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  RR )
5352recnd 8857 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  /\  d  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  d )  x.  (Λ `  ( n  /  d
) ) )  e.  CC )
5453anasss 630 . . . . . . . . . . 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 20423 . . . . . . . . . 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 2320 . . . . . . . . 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 5835 . . . . . . . 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 11030 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
59 vmacl 20351 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
6046, 59syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
6160recnd 8857 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
6245nnrpd 10385 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
63 rpdivcl 10372 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
6462, 63sylan2 462 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
6564rpred 10386 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
66 chpcl 20357 . . . . . . . . . . . 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 8857 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  CC )
6961, 68mulcld 8851 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
7046nnrpd 10385 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
71 relogcl 19927 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
7270, 71syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
7372recnd 8857 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
7461, 73mulcld 8851 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  e.  CC )
7558, 69, 74fsumadd 12206 . . . . . . . 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 11030 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... n )  e. 
Fin )
77 sgmss 20339 . . . . . . . . . . . . 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 7079 . . . . . . . . . . . 12  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
8076, 78, 79syl2anc 644 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
8180, 52fsumrecl 12202 . . . . . . . . . 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 8857 . . . . . . . . 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 12206 . . . . . . . 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 2327 . . . . . . 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 9010 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) )  =  ( (ψ `  ( x  /  n
) )  +  ( log `  n ) ) )
8685oveq2d 5836 . . . . . . . . 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 8855 . . . . . . . . 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 2317 . . . . . . . 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 12171 . . . . . . 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 20687 . . . . . . . . 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 12171 . . . . . . 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 2327 . . . . . 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 5490 . . . . . . . . 9  |-  ( n  =  ( d  x.  m )  ->  ( log `  ( n  / 
d ) )  =  ( log `  (
( d  x.  m
)  /  d ) ) )
9594oveq1d 5835 . . . . . . . 8  |-  ( n  =  ( d  x.  m )  ->  (
( log `  (
n  /  d ) ) ^ 2 )  =  ( ( log `  ( ( d  x.  m )  /  d
) ) ^ 2 ) )
9695oveq2d 5836 . . . . . . 7  |-  ( n  =  ( d  x.  m )  ->  (
( mmu `  d
)  x.  ( ( log `  ( n  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
( d  x.  m
)  /  d ) ) ^ 2 ) ) )
97 mucl 20374 . . . . . . . . . 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 10114 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( mmu `  d
)  e.  CC )
10062ad2antrl 710 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  ->  n  e.  RR+ )
10142nnrpd 10385 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
d  e.  RR+ )
102100, 101rpdivcld 10403 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( n  /  d
)  e.  RR+ )
103 relogcl 19927 . . . . . . . . . . 11  |-  ( ( n  /  d )  e.  RR+  ->  ( log `  ( n  /  d
) )  e.  RR )
104103recnd 8857 . . . . . . . . . 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 11238 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  (
n  e.  ( 1 ... ( |_ `  x ) )  /\  d  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( ( log `  (
n  /  d ) ) ^ 2 )  e.  CC )
10799, 106mulcld 8851 . . . . . . 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 20423 . . . . . 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 5490 . . . . . . . . . 10  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  ( ( d  x.  m )  /  d
) )  =  ( log `  m ) )
110109oveq1d 5835 . . . . . . . . 9  |-  ( ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  m  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  ( ( d  x.  m )  / 
d ) ) ^
2 )  =  ( ( log `  m
) ^ 2 ) )
111110oveq2d 5836 . . . . . . . 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 12171 . . . . . . 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 12171 . . . . . 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 2321 . . . . 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 5835 . . . 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 5835 . . 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 4104 . 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 2285 . . 3  |-  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)  =  ( ( ( ( log `  (
x  /  d ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  / 
d ) ) ) ) )  /  d
)
119118selberglem2 20690 . 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 2355 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 360    = wceq 1624    e. wcel 1685   {crab 2549    C_ wss 3154   class class class wbr 4025    e. cmpt 4079   ` cfv 5222  (class class class)co 5820   Fincfn 6859   CCcc 8731   RRcr 8732   1c1 8734    + caddc 8736    x. cmul 8738    - cmin 9033    / cdiv 9419   NNcn 9742   2c2 9791   ZZcz 10020   RR+crp 10350   ...cfz 10777   |_cfl 10919   ^cexp 11099   O ( 1 )co1 11955   sum_csu 12153    || cdivides 12526   logclog 19907  Λcvma 20324  ψcchp 20325   mmucmu 20327
This theorem is referenced by:  selbergb  20693  selberg2  20695  selbergs  20718
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10655  df-ioc 10656  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-shft 11557  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-limsup 11940  df-clim 11957  df-rlim 11958  df-o1 11959  df-lo1 11960  df-sum 12154  df-ef 12344  df-e 12345  df-sin 12346  df-cos 12347  df-pi 12349  df-dvds 12527  df-gcd 12681  df-prm 12754  df-pc 12885  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-starv 13218  df-sca 13219  df-vsca 13220  df-tset 13222  df-ple 13223  df-ds 13225  df-hom 13227  df-cco 13228  df-rest 13322  df-topn 13323  df-topgen 13339  df-pt 13340  df-prds 13343  df-xrs 13398  df-0g 13399  df-gsum 13400  df-qtop 13405  df-imas 13406  df-xps 13408  df-mre 13483  df-mrc 13484  df-acs 13486  df-mnd 14362  df-submnd 14411  df-mulg 14487  df-cntz 14788  df-cmn 15086  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-cnfld 16373  df-top 16631  df-bases 16633  df-topon 16634  df-topsp 16635  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-lp 16863  df-perf 16864  df-cn 16952  df-cnp 16953  df-haus 17038  df-cmp 17109  df-tx 17252  df-hmeo 17441  df-fbas 17515  df-fg 17516  df-fil 17536  df-fm 17628  df-flim 17629  df-flf 17630  df-xms 17880  df-ms 17881  df-tms 17882  df-cncf 18377  df-limc 19211  df-dv 19212  df-log 19909  df-cxp 19910  df-em 20282  df-vma 20330  df-chp 20331  df-mu 20333
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