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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremm1lgs 20601 The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime  P iff  P  ==  1 mod 4. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( P  e.  ( Prime  \  { 2 } )  ->  ( ( -u 1  / L P )  =  1  <->  ( P  mod  4 )  =  1
 ) )
 
13.4.10  All primes 4n+1 are the sum of two squares
 
Theorem2sqlem1 20602* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   =>    |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^ 2 ) )
 
Theorem2sqlem2 20603* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   =>    |-  ( A  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  A  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) )
 
Theoremmul2sq 20604 Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B )  e.  S )
 
Theorem2sqlem3 20605 Lemma for 2sqlem5 20607. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  ( ph  ->  ( N  x.  P )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )   &    |-  ( ph  ->  P  =  ( ( C ^ 2 )  +  ( D ^ 2 ) ) )   &    |-  ( ph  ->  P 
 ||  ( ( C  x.  B )  +  ( A  x.  D ) ) )   =>    |-  ( ph  ->  N  e.  S )
 
Theorem2sqlem4 20606 Lemma for 2sqlem5 20607. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  ( ph  ->  ( N  x.  P )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )   &    |-  ( ph  ->  P  =  ( ( C ^ 2 )  +  ( D ^ 2 ) ) )   =>    |-  ( ph  ->  N  e.  S )
 
Theorem2sqlem5 20607 Lemma for 2sq 20615. If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( N  x.  P )  e.  S )   &    |-  ( ph  ->  P  e.  S )   =>    |-  ( ph  ->  N  e.  S )
 
Theorem2sqlem6 20608* Lemma for 2sq 20615. If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  A. p  e.  Prime  ( p  ||  B  ->  p  e.  S ) )   &    |-  ( ph  ->  ( A  x.  B )  e.  S )   =>    |-  ( ph  ->  A  e.  S )
 
Theorem2sqlem7 20609* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  Y  =  {
 z  |  E. x  e.  ZZ  E. y  e. 
 ZZ  ( ( x 
 gcd  y )  =  1  /\  z  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) ) }   =>    |-  Y  C_  ( S  i^i  NN )
 
Theorem2sqlem8a 20610* Lemma for 2sqlem8 20611. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  Y  =  {
 z  |  E. x  e.  ZZ  E. y  e. 
 ZZ  ( ( x 
 gcd  y )  =  1  /\  z  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) ) }   &    |-  ( ph  ->  A. b  e.  (
 1 ... ( M  -  1 ) ) A. a  e.  Y  (
 b  ||  a  ->  b  e.  S ) )   &    |-  ( ph  ->  M  ||  N )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  ( A  gcd  B )  =  1 )   &    |-  ( ph  ->  N  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )   &    |-  C  =  ( ( ( A  +  ( M  /  2
 ) )  mod  M )  -  ( M  / 
 2 ) )   &    |-  D  =  ( ( ( B  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   =>    |-  ( ph  ->  ( C  gcd  D )  e.  NN )
 
Theorem2sqlem8 20611* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  Y  =  {
 z  |  E. x  e.  ZZ  E. y  e. 
 ZZ  ( ( x 
 gcd  y )  =  1  /\  z  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) ) }   &    |-  ( ph  ->  A. b  e.  (
 1 ... ( M  -  1 ) ) A. a  e.  Y  (
 b  ||  a  ->  b  e.  S ) )   &    |-  ( ph  ->  M  ||  N )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  ( A  gcd  B )  =  1 )   &    |-  ( ph  ->  N  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )   &    |-  C  =  ( ( ( A  +  ( M  /  2
 ) )  mod  M )  -  ( M  / 
 2 ) )   &    |-  D  =  ( ( ( B  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  E  =  ( C  /  ( C  gcd  D ) )   &    |-  F  =  ( D  /  ( C 
 gcd  D ) )   =>    |-  ( ph  ->  M  e.  S )
 
Theorem2sqlem9 20612* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  Y  =  {
 z  |  E. x  e.  ZZ  E. y  e. 
 ZZ  ( ( x 
 gcd  y )  =  1  /\  z  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) ) }   &    |-  ( ph  ->  A. b  e.  (
 1 ... ( M  -  1 ) ) A. a  e.  Y  (
 b  ||  a  ->  b  e.  S ) )   &    |-  ( ph  ->  M  ||  N )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  Y )   =>    |-  ( ph  ->  M  e.  S )
 
Theorem2sqlem10 20613* Lemma for 2sq 20615. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  Y  =  {
 z  |  E. x  e.  ZZ  E. y  e. 
 ZZ  ( ( x 
 gcd  y )  =  1  /\  z  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) ) }   =>    |-  ( ( A  e.  Y  /\  B  e.  NN  /\  B  ||  A )  ->  B  e.  S )
 
Theorem2sqlem11 20614* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2
 ) )   &    |-  Y  =  {
 z  |  E. x  e.  ZZ  E. y  e. 
 ZZ  ( ( x 
 gcd  y )  =  1  /\  z  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) ) ) }   =>    |-  ( ( P  e.  Prime  /\  ( P 
 mod  4 )  =  1 )  ->  P  e.  S )
 
Theorem2sq 20615* All primes of the form  4 k  +  1 are sums of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( P  e.  Prime  /\  ( P  mod  4 )  =  1
 )  ->  E. x  e.  ZZ  E. y  e. 
 ZZ  P  =  ( ( x ^ 2
 )  +  ( y ^ 2 ) ) )
 
Theorem2sqblem 20616 The converse to 2sq 20615. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ph  ->  ( P  e.  Prime  /\  P  =/=  2 ) )   &    |-  ( ph  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )   &    |-  ( ph  ->  P  =  ( ( X ^ 2 )  +  ( Y ^ 2 ) ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  ( P  gcd  Y )  =  ( ( P  x.  A )  +  ( Y  x.  B ) ) )   =>    |-  ( ph  ->  ( P  mod  4 )  =  1 )
 
Theorem2sqb 20617* The converse to 2sq 20615. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( P  e.  Prime  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  P  =  ( ( x ^
 2 )  +  (
 y ^ 2 ) )  <->  ( P  =  2  \/  ( P  mod  4 )  =  1
 ) ) )
 
13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem
 
Theoremchebbnd1lem1 20618 Lemma for chebbnd1 20621: show a lower bound on π ( x ) at even integers using similar techniques to those used to prove bpos 20532. (Note that the expression  K is actually equal to  2  x.  N, but proving that is not necessary for the proof, and it's too much work.) The key to the proof is bposlem1 20523, which shows that each term in the expansion  ( (
2  x.  N )  _C  N )  = 
prod_ p  e.  Prime  ( p ^ ( p  pCnt  ( ( 2  x.  N
)  _C  N ) ) ) is at most  2  x.  N, so that the sum really only has nonzero elements up to  2  x.  N, and since each term is at most  2  x.  N, after taking logs we get the inequality π ( 2  x.  N
)  x.  log (
2  x.  N )  <_  log ( ( 2  x.  N )  _C  N ), and bclbnd 20519 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
 |-  K  =  if (
 ( 2  x.  N )  <_  ( ( 2  x.  N )  _C  N ) ,  (
 2  x.  N ) ,  ( ( 2  x.  N )  _C  N ) )   =>    |-  ( N  e.  ( ZZ>= `  4 )  ->  ( log `  (
 ( 4 ^ N )  /  N ) )  <  ( (π `  (
 2  x.  N ) )  x.  ( log `  ( 2  x.  N ) ) ) )
 
Theoremchebbnd1lem2 20619 Lemma for chebbnd1 20621: Show that  log ( N )  /  N does not change too much between  N and  M  =  |_ ( N  /  2
). (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  M  =  ( |_ `  ( N  /  2
 ) )   =>    |-  ( ( N  e.  RR  /\  8  <_  N )  ->  ( ( log `  ( 2  x.  M ) )  /  (
 2  x.  M ) )  <  ( 2  x.  ( ( log `  N )  /  N ) ) )
 
Theoremchebbnd1lem3 20620 Lemma for chebbnd1 20621: get a lower bound on π ( N )  /  ( N  /  log ( N ) ) that is independent of  N. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  =  ( |_ `  ( N  /  2
 ) )   =>    |-  ( ( N  e.  RR  /\  8  <_  N )  ->  ( ( ( log `  2 )  -  ( 1  /  (
 2  x.  _e ) ) )  /  2
 )  <  ( (π `  N )  x.  (
 ( log `  N )  /  N ) ) )
 
Theoremchebbnd1 20621 The Chebyshev bound: The function π ( x ) is eventually lower bounded by a positive constant times  x  /  log ( x ). Alternatively stated, the function  ( x  /  log ( x ) )  / π ( x ) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( ( x  /  ( log `  x ) ) 
 /  (π `  x ) ) )  e.  O ( 1 )
 
Theoremchtppilimlem1 20622 Lemma for chtppilim 20624. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A  <  1 )   &    |-  ( ph  ->  N  e.  (
 2 [,)  +oo ) )   &    |-  ( ph  ->  ( ( N  ^ c  A ) 
 /  (π `  N ) )  <  ( 1  -  A ) )   =>    |-  ( ph  ->  ( ( A ^ 2
 )  x.  ( (π `  N )  x.  ( log `  N ) ) )  <  ( theta `  N ) )
 
Theoremchtppilimlem2 20623* Lemma for chtppilim 20624. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A  <  1 )   =>    |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,)  +oo ) ( z 
 <_  x  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x ) ) )  < 
 ( theta `  x )
 ) )
 
Theoremchtppilim 20624 The  theta function is asymptotic to π ( x ) log ( x ), so it is sufficient to prove 
theta ( x )  /  x 
~~> r  1 to establish the PNT. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( ( theta `  x )  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1
 
Theoremchto1ub 20625 The  theta function is upper bounded by a linear term. Corollary of chtub 20451. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( x  e.  RR+  |->  ( ( theta `  x )  /  x ) )  e.  O ( 1 )
 
Theoremchebbnd2 20626 The Chebyshev bound, part 2: The function π ( x ) is eventually upper bounded by a positive constant times  x  /  log ( x ). Alternatively stated, the function π ( x )  /  (
x  /  log (
x ) ) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( (π `  x )  /  ( x  /  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremchto1lb 20627 The  theta function is lower bounded by a linear term. Corollary of chebbnd1 20621. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O ( 1 )
 
Theoremchpchtlim 20628 The ψ and  theta functions are asymptotic to each other, so is sufficient to prove either 
theta ( x )  /  x 
~~> r  1 or ψ ( x )  /  x  ~~> r  1 to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( (ψ `  x )  /  ( theta `  x )
 ) )  ~~> r  1
 
Theoremchpo1ub 20629 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
 |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  e.  O ( 1 )
 
Theoremchpo1ubb 20630* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
 |- 
 E. c  e.  RR+  A. x  e.  RR+  (ψ `  x )  <_  ( c  x.  x )
 
Theoremvmadivsum 20631* The sum of the von Mangoldt function over  n is asymptotic to  log x  +  O ( 1 ). Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)
 |-  ( x  e.  RR+  |->  ( sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  -  ( log `  x ) ) )  e.  O ( 1 )
 
Theoremvmadivsumb 20632* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
 |- 
 E. c  e.  RR+  A. x  e.  ( 1 [,)  +oo ) ( abs `  ( sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  -  ( log `  x ) ) )  <_  c
 
Theoremrplogsumlem1 20633* Lemma for rplogsum 20676. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( A  e.  NN  -> 
 sum_ n  e.  (
 2 ... A ) ( ( log `  n )  /  ( n  x.  ( n  -  1
 ) ) )  <_ 
 2 )
 
Theoremrplogsumlem2 20634* Lemma for rplogsum 20676. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( A  e.  ZZ  -> 
 sum_ n  e.  (
 1 ... A ) ( ( (Λ `  n )  -  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )  /  n )  <_  2 )
 
Theoremdchrisum0lem1a 20635 Lemma for dchrisum0lem1 20665. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1
 ... ( |_ `  X ) ) )  ->  ( X  <_  ( ( X ^ 2 ) 
 /  D )  /\  ( |_ `  ( ( X ^ 2 ) 
 /  D ) )  e.  ( ZZ>= `  ( |_ `  X ) ) ) )
 
Theoremrpvmasumlem 20636* Lemma for rpvmasum 20675. Calculate the "trivial case" estimate  sum_ n  <_  x (  .1.  (
n )Λ ( n )  /  n )  =  log x  +  O
( 1 ), where  .1.  ( x ) is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (  .1.  `  ( L `  n ) )  x.  ( (Λ `  n )  /  n ) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
 
Theoremdchrisumlema 20637* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( n  =  x  ->  A  =  B )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ( ph  /\  n  e.  RR+ )  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  RR+  /\  x  e.  RR+ )  /\  ( M  <_  n  /\  n  <_  x ) )  ->  B  <_  A )   &    |-  ( ph  ->  ( n  e.  RR+  |->  A )  ~~> r  0 )   &    |-  F  =  ( n  e.  NN  |->  ( ( X `  ( L `  n ) )  x.  A ) )   =>    |-  ( ph  ->  (
 ( I  e.  RR+  ->  [_ I  /  n ]_ A  e.  RR )  /\  ( I  e.  ( M [,)  +oo )  ->  0  <_ 
 [_ I  /  n ]_ A ) ) )
 
Theoremdchrisumlem1 20638* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( n  =  x  ->  A  =  B )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ( ph  /\  n  e.  RR+ )  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  RR+  /\  x  e.  RR+ )  /\  ( M  <_  n  /\  n  <_  x ) )  ->  B  <_  A )   &    |-  ( ph  ->  ( n  e.  RR+  |->  A )  ~~> r  0 )   &    |-  F  =  ( n  e.  NN  |->  ( ( X `  ( L `  n ) )  x.  A ) )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  A. u  e.  (
 0..^ N ) ( abs `  sum_ n  e.  ( 0..^ u ) ( X `  ( L `  n ) ) )  <_  R )   =>    |-  (
 ( ph  /\  U  e.  NN0 )  ->  ( abs ` 
 sum_ n  e.  (
 0..^ U ) ( X `  ( L `
  n ) ) )  <_  R )
 
Theoremdchrisumlem2 20639* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( n  =  x  ->  A  =  B )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ( ph  /\  n  e.  RR+ )  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  RR+  /\  x  e.  RR+ )  /\  ( M  <_  n  /\  n  <_  x ) )  ->  B  <_  A )   &    |-  ( ph  ->  ( n  e.  RR+  |->  A )  ~~> r  0 )   &    |-  F  =  ( n  e.  NN  |->  ( ( X `  ( L `  n ) )  x.  A ) )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  A. u  e.  (
 0..^ N ) ( abs `  sum_ n  e.  ( 0..^ u ) ( X `  ( L `  n ) ) )  <_  R )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  M  <_  U )   &    |-  ( ph  ->  U 
 <_  ( I  +  1 ) )   &    |-  ( ph  ->  I  e.  NN )   &    |-  ( ph  ->  J  e.  ( ZZ>=
 `  I ) )   =>    |-  ( ph  ->  ( abs `  ( (  seq  1
 (  +  ,  F ) `  J )  -  (  seq  1 (  +  ,  F ) `  I
 ) ) )  <_  ( ( 2  x.  R )  x.  [_ U  /  n ]_ A ) )
 
Theoremdchrisumlem3 20640* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( n  =  x  ->  A  =  B )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ( ph  /\  n  e.  RR+ )  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  RR+  /\  x  e.  RR+ )  /\  ( M  <_  n  /\  n  <_  x ) )  ->  B  <_  A )   &    |-  ( ph  ->  ( n  e.  RR+  |->  A )  ~~> r  0 )   &    |-  F  =  ( n  e.  NN  |->  ( ( X `  ( L `  n ) )  x.  A ) )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  A. u  e.  (
 0..^ N ) ( abs `  sum_ n  e.  ( 0..^ u ) ( X `  ( L `  n ) ) )  <_  R )   =>    |-  ( ph  ->  E. t E. c  e.  ( 0 [,)  +oo ) (  seq  1 (  +  ,  F )  ~~>  t  /\  A. x  e.  ( M [,)  +oo ) ( abs `  (
 (  seq  1 (  +  ,  F ) `  ( |_ `  x ) )  -  t
 ) )  <_  (
 c  x.  B ) ) )
 
Theoremdchrisum 20641* If  n  e.  [ M ,  +oo )  |->  A ( n ) is a positive decreasing function approaching zero, then the infinite sum  sum_ n ,  X
( n ) A ( n ) is convergent, with the partial sum  sum_ n  <_  x ,  X ( n ) A ( n ) within  O ( A ( M ) ) of the limit  T. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( n  =  x  ->  A  =  B )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ( ph  /\  n  e.  RR+ )  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  RR+  /\  x  e.  RR+ )  /\  ( M  <_  n  /\  n  <_  x ) )  ->  B  <_  A )   &    |-  ( ph  ->  ( n  e.  RR+  |->  A )  ~~> r  0 )   &    |-  F  =  ( n  e.  NN  |->  ( ( X `  ( L `  n ) )  x.  A ) )   =>    |-  ( ph  ->  E. t E. c  e.  (
 0 [,)  +oo ) ( 
 seq  1 (  +  ,  F )  ~~>  t  /\  A. x  e.  ( M [,)  +oo ) ( abs `  ( (  seq  1
 (  +  ,  F ) `  ( |_ `  x ) )  -  t
 ) )  <_  (
 c  x.  B ) ) )
 
Theoremdchrmusumlema 20642* Lemma for dchrmusum 20673 and dchrisumn0 20670. Apply dchrisum 20641 for the function  1  /  y. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   =>    |-  ( ph  ->  E. t E. c  e.  (
 0 [,)  +oo ) ( 
 seq  1 (  +  ,  F )  ~~>  t  /\  A. y  e.  ( 1 [,)  +oo ) ( abs `  ( (  seq  1
 (  +  ,  F ) `  ( |_ `  y
 ) )  -  t
 ) )  <_  (
 c  /  y )
 ) )
 
Theoremdchrmusum2 20643* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by  n, is bounded, provided that  T  =/=  0. Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  T ) ) 
 <_  ( C  /  y
 ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `
  d ) )  x.  ( ( mmu `  d )  /  d
 ) )  x.  T ) )  e.  O ( 1 ) )
 
Theoremdchrvmasumlem1 20644* An alternative expression for a Dirichlet-weighted von Mangoldt sum in terms of the Möbius function. Equation 9.4.11 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n ) )  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
 ) )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) ( ( X `
  ( L `  m ) )  x.  ( ( log `  m )  /  m ) ) ) )
 
Theoremdchrvmasum2lem 20645* Give an expression for  log x remarkably similar to  sum_ n  <_  x
( X ( n )Λ ( n )  /  n ) given in dchrvmasumlem1 20644. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  1  <_  A )   =>    |-  ( ph  ->  ( log `  A )  = 
 sum_ d  e.  (
 1 ... ( |_ `  A ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
 ) )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) ( ( X `
  ( L `  m ) )  x.  ( ( log `  (
 ( A  /  d
 )  /  m )
 )  /  m )
 ) ) )
 
Theoremdchrvmasum2if 20646* Combine the results of dchrvmasumlem1 20644 and dchrvmasum2lem 20645 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  1  <_  A )   =>    |-  ( ph  ->  ( sum_ n  e.  ( 1
 ... ( |_ `  A ) ) ( ( X `  ( L `
  n ) )  x.  ( (Λ `  n )  /  n ) )  +  if ( ps ,  ( log `  A ) ,  0 )
 )  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) ( ( ( X `
  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
 ) )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) ( ( X `
  ( L `  m ) )  x.  ( ( log `  if ( ps ,  ( A 
 /  d ) ,  m ) )  /  m ) ) ) )
 
Theoremdchrvmasumlem2 20647* Lemma for dchrvmasum 20674. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  (
 ( ph  /\  m  e.  RR+ )  ->  F  e.  CC )   &    |-  ( m  =  ( x  /  d
 )  ->  F  =  K )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  T  e.  CC )   &    |-  (
 ( ph  /\  m  e.  ( 3 [,)  +oo ) )  ->  ( abs `  ( F  -  T ) )  <_  ( C  x.  ( ( log `  m )  /  m ) ) )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  A. m  e.  ( 1 [,) 3
 ) ( abs `  ( F  -  T ) ) 
 <_  R )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T ) ) 
 /  d ) )  e.  O ( 1 ) )
 
Theoremdchrvmasumlem3 20648* Lemma for dchrvmasum 20674. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  (
 ( ph  /\  m  e.  RR+ )  ->  F  e.  CC )   &    |-  ( m  =  ( x  /  d
 )  ->  F  =  K )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  T  e.  CC )   &    |-  (
 ( ph  /\  m  e.  ( 3 [,)  +oo ) )  ->  ( abs `  ( F  -  T ) )  <_  ( C  x.  ( ( log `  m )  /  m ) ) )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  A. m  e.  ( 1 [,) 3
 ) ( abs `  ( F  -  T ) ) 
 <_  R )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
 ) )  x.  ( K  -  T ) ) )  e.  O ( 1 ) )
 
Theoremdchrvmasumlema 20649* Lemma for dchrvmasum 20674 and dchrvmasumif 20652. Apply dchrisum 20641 for the function  log ( y )  /  y, which is decreasing above  _e (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) )   =>    |-  ( ph  ->  E. t E. c  e.  (
 0 [,)  +oo ) ( 
 seq  1 (  +  ,  F )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
 (  +  ,  F ) `  ( |_ `  y
 ) )  -  t
 ) )  <_  (
 c  x.  ( ( log `  y )  /  y ) ) ) )
 
Theoremdchrvmasumiflem1 20650* Lemma for dchrvmasumif 20652. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  y
 ) )   &    |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a
 ) ) )   &    |-  ( ph  ->  E  e.  (
 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  K )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
 (  +  ,  K ) `  ( |_ `  y
 ) )  -  T ) )  <_  ( E  x.  ( ( log `  y )  /  y
 ) ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
 ) )  x.  ( sum_ k  e.  ( 1
 ... ( |_ `  ( x  /  d ) ) ) ( ( X `
  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
 ) ,  k ) )  /  k ) )  -  if ( S  =  0 , 
 0 ,  T ) ) ) )  e.  O ( 1 ) )
 
Theoremdchrvmasumiflem2 20651* Lemma for dchrvmasum 20674. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  y
 ) )   &    |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a
 ) ) )   &    |-  ( ph  ->  E  e.  (
 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  K )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
 (  +  ,  K ) `  ( |_ `  y
 ) )  -  T ) )  <_  ( E  x.  ( ( log `  y )  /  y
 ) ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `
  n ) )  x.  ( (Λ `  n )  /  n ) )  +  if ( S  =  0 ,  ( log `  x ) ,  0 ) ) )  e.  O ( 1 ) )
 
Theoremdchrvmasumif 20652* An asymptotic approximation for the sum of  X ( n )Λ (
n )  /  n conditional on the value of the infinite sum  S. (We will later show that the case  S  =  0 is impossible, and hence establish dchrvmasum 20674.) (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  y
 ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `
  n ) )  x.  ( (Λ `  n )  /  n ) )  +  if ( S  =  0 ,  ( log `  x ) ,  0 ) ) )  e.  O ( 1 ) )
 
Theoremdchrvmaeq0 20653* The set  W is the collection of all non-principal Dirichlet characters such that the sum  sum_ n  e.  NN ,  X ( n )  /  n is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  y
 ) )   &    |-  W  =  {
 y  e.  ( D 
 \  {  .1.  }
 )  |  sum_ m  e.  NN  ( ( y `
  ( L `  m ) )  /  m )  =  0 }   =>    |-  ( ph  ->  ( X  e.  W  <->  S  =  0
 ) )
 
Theoremdchrisum0fval 20654* Value of the function  F, the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   =>    |-  ( A  e.  NN  ->  ( F `  A )  =  sum_ t  e. 
 { q  e.  NN  |  q  ||  A }  ( X `  ( L `
  t ) ) )
 
Theoremdchrisum0fmul 20655* The function  F, the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  ( A  gcd  B )  =  1 )   =>    |-  ( ph  ->  ( F `  ( A  x.  B ) )  =  ( ( F `
  A )  x.  ( F `  B ) ) )
 
Theoremdchrisum0ff 20656* The function  F is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X : (
 Base `  Z ) --> RR )   =>    |-  ( ph  ->  F : NN --> RR )
 
Theoremdchrisum0flblem1 20657* Lemma for dchrisum0flb 20659. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X : (
 Base `  Z ) --> RR )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  ( F `  ( P ^ A ) ) )
 
Theoremdchrisum0flblem2 20658* Lemma for dchrisum0flb 20659. Induction over relatively prime factors, with the prime power case handled in dchrisum0flblem1 . (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X : (
 Base `  Z ) --> RR )   &    |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  P 
 ||  A )   &    |-  ( ph  ->  A. y  e.  (
 1..^ A ) if ( ( sqr `  y
 )  e.  NN , 
 1 ,  0 ) 
 <_  ( F `  y
 ) )   =>    |-  ( ph  ->  if ( ( sqr `  A )  e.  NN ,  1 ,  0 )  <_  ( F `  A ) )
 
Theoremdchrisum0flb 20659* The divisor sum of a real Dirichlet character, is lower bounded by zero everywhere and one at the squares. Equation 9.4.29 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X : (
 Base `  Z ) --> RR )   &    |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  if ( ( sqr `  A )  e.  NN ,  1 ,  0 )  <_  ( F `  A ) )
 
Theoremdchrisum0fno1 20660* The sum  sum_ k  <_  x ,  F (
x )  /  sqr k is divergent (i.e. not eventually bounded). Equation 9.4.30 of [Shapiro], p. 383. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `
  v ) ) )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X : (
 Base `  Z ) --> RR )   &    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ k  e.  (
 1 ... ( |_ `  x ) ) ( ( F `  k ) 
 /  ( sqr `  k
 ) ) )  e.  O ( 1 ) )   =>    |- 
 -.  ph
 
Theoremrpvmasum2 20661* A partial result along the lines of rpvmasum 20675. The sum of the von Mangoldt function over those integers  n  ==  A (mod  N) is asymptotic to  ( 1  -  M
) ( log x  /  phi ( x ) )  +  O ( 1 ), where  M is the number of non-principal Dirichlet characters with  sum_ n  e.  NN ,  X ( n )  /  n  =  0. Our goal is to show this set is empty. Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  A  e.  U )   &    |-  T  =  ( `' L " { A } )   &    |-  (
 ( ph  /\  f  e.  W )  ->  A  =  ( 1r `  Z ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ n  e.  (
 ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  n )  /  n ) )  -  ( ( log `  x )  x.  ( 1  -  ( # `  W ) ) ) ) )  e.  O ( 1 ) )
 
Theoremdchrisum0re 20662* Suppose  X is a non-principal Dirichlet character with  sum_ n  e.  NN ,  X ( n )  /  n  =  0. Then  X is a real character. Part of Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   =>    |-  ( ph  ->  X : ( Base `  Z )
 --> RR )
 
Theoremdchrisum0lema 20663* Lemma for dchrisum0 20669. Apply dchrisum 20641 for the function  1  /  sqr y. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  ( sqr `  a ) ) )   =>    |-  ( ph  ->  E. t E. c  e.  (
 0 [,)  +oo ) ( 
 seq  1 (  +  ,  F )  ~~>  t  /\  A. y  e.  ( 1 [,)  +oo ) ( abs `  ( (  seq  1
 (  +  ,  F ) `  ( |_ `  y
 ) )  -  t
 ) )  <_  (
 c  /  ( sqr `  y ) ) ) )
 
Theoremdchrisum0lem1b 20664* Lemma for dchrisum0lem1 20665. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  ( sqr `  a ) ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  ( sqr `  y ) ) )   =>    |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  (
 1 ... ( |_ `  x ) ) )  ->  ( abs `  sum_ m  e.  ( ( ( |_ `  x )  +  1 ) ... ( |_ `  ( ( x ^
 2 )  /  d
 ) ) ) ( ( X `  ( L `  m ) ) 
 /  ( sqr `  m ) ) )  <_  ( ( 2  x.  C )  /  ( sqr `  x ) ) )
 
Theoremdchrisum0lem1 20665* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  ( sqr `  a ) ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  ( sqr `  y ) ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ m  e.  ( ( ( |_ `  x )  +  1 ) ... ( |_ `  ( x ^ 2
 ) ) ) sum_ d  e.  ( 1 ... ( |_ `  (
 ( x ^ 2
 )  /  m )
 ) ) ( ( ( X `  ( L `  m ) ) 
 /  ( sqr `  m ) )  /  ( sqr `  d ) ) )  e.  O ( 1 ) )
 
Theoremdchrisum0lem2a 20666* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  ( sqr `  a ) ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  ( sqr `  y ) ) )   &    |-  H  =  ( y  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  y
 ) ) ( 1 
 /  ( sqr `  d
 ) )  -  (
 2  x.  ( sqr `  y ) ) ) )   &    |-  ( ph  ->  H  ~~> r  U )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ m  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
  ( L `  m ) )  /  ( sqr `  m )
 )  x.  ( H `
  ( ( x ^ 2 )  /  m ) ) ) )  e.  O ( 1 ) )
 
Theoremdchrisum0lem2 20667* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  ( sqr `  a ) ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  ( sqr `  y ) ) )   &    |-  H  =  ( y  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  y
 ) ) ( 1 
 /  ( sqr `  d
 ) )  -  (
 2  x.  ( sqr `  y ) ) ) )   &    |-  ( ph  ->  H  ~~> r  U )   &    |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) ) 
 /  a ) )   &    |-  ( ph  ->  E  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  K )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  K ) `  ( |_ `  y ) )  -  T ) ) 
 <_  ( E  /  y
 ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ m  e.  ( 1 ... ( |_ `  x ) )
 sum_ d  e.  (
 1 ... ( |_ `  (
 ( x ^ 2
 )  /  m )
 ) ) ( ( ( X `  ( L `  m ) ) 
 /  ( sqr `  m ) )  /  ( sqr `  d ) ) )  e.  O ( 1 ) )
 
Theoremdchrisum0lem3 20668* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  ( sqr `  a ) ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) ) 
 <_  ( C  /  ( sqr `  y ) ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ m  e.  ( 1 ... ( |_ `  ( x ^
 2 ) ) )
 sum_ d  e.  (
 1 ... ( |_ `  (
 ( x ^ 2
 )  /  m )
 ) ) ( ( X `  ( L `
  m ) ) 
 /  ( sqr `  ( m  x.  d ) ) ) )  e.  O ( 1 ) )
 
Theoremdchrisum0 20669* The sum  sum_ n  e.  NN ,  X ( n )  /  n is nonzero for all non-principal Dirichlet characters (i.e. the assumption  X  e.  W is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 20643 and dchrvmasumif 20652. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  W  =  { y  e.  ( D  \  {  .1.  } )  |  sum_ m  e.  NN  ( ( y `  ( L `
  m ) ) 
 /  m )  =  0 }   &    |-  ( ph  ->  X  e.  W )   =>    |-  -.  ph
 
Theoremdchrisumn0 20670* The sum  sum_ n  e.  NN ,  X ( n )  /  n is nonzero for all non-principal Dirichlet characters (i.e. the assumption  X  e.  W is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 20643 and dchrvmasumif 20652. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  T ) ) 
 <_  ( C  /  y
 ) )   =>    |-  ( ph  ->  T  =/=  0 )
 
Theoremdchrmusumlem 20671* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by  n, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  T ) ) 
 <_  ( C  /  y
 ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( ( mmu `  n )  /  n ) ) )  e.  O ( 1 ) )
 
Theoremdchrvmasumlem 20672* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by  n, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   &    |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  T )   &    |-  ( ph  ->  A. y  e.  (
 1 [,)  +oo ) ( abs `  ( (  seq  1 (  +  ,  F ) `  ( |_ `  y ) )  -  T ) ) 
 <_  ( C  /  y
 ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n ) ) )  e.  O ( 1 ) )
 
Theoremdchrmusum 20673* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by  n, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( ( mmu `  n )  /  n ) ) )  e.  O ( 1 ) )
 
Theoremdchrvmasum 20674* The sum of the von Mangoldt function multiplied by a non-principal Dirichlet character, divided by  n, is bounded. Equation 9.4.8 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  G  =  (DChr `  N )   &    |-  D  =  ( Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  X  =/=  .1.  )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n ) ) )  e.  O ( 1 ) )
 
Theoremrpvmasum 20675* The sum of the von Mangoldt function over those integers  n  ==  A (mod  N) is asymptotic to  log x  /  phi ( x )  +  O ( 1 ). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  A  e.  U )   &    |-  T  =  ( `' L " { A } )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ n  e.  (
 ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  n )  /  n ) )  -  ( log `  x )
 ) )  e.  O ( 1 ) )
 
Theoremrplogsum 20676* The sum of  log p  /  p over the primes 
p  ==  A (mod  N) is asymptotic to  log x  /  phi ( x )  +  O ( 1 ). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 16-Apr-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  A  e.  U )   &    |-  T  =  ( `' L " { A } )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ p  e.  (
 ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p ) )  -  ( log `  x )
 ) )  e.  O ( 1 ) )
 
Theoremdirith2 20677 Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to  N. Theorem 9.4.1 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  Z  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  A  e.  U )   &    |-  T  =  ( `' L " { A } )   =>    |-  ( ph  ->  ( Prime  i^i  T )  ~~  NN )
 
Theoremdirith 20678* Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to  N. Theorem 9.4.1 of [Shapiro], p. 375. See http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html for an informal exposition. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  { p  e.  Prime  |  N  ||  ( p  -  A ) }  ~~  NN )
 
13.4.12  The Prime Number Theorem
 
Theoremmudivsum 20679* Asymptotic formula for  sum_ n  <_  x ,  mmu ( n )  /  n  =  O ( 1 ). Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)
 |-  ( x  e.  RR+  |->  sum_
 n  e.  ( 1
 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n ) )  e.  O ( 1 )
 
Theoremmulogsumlem 20680* Lemma for mulogsum 20681. (Contributed by Mario Carneiro, 14-May-2016.)
 |-  ( x  e.  RR+  |->  sum_
 n  e.  ( 1
 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n )  x.  ( sum_ m  e.  (
 1 ... ( |_ `  ( x  /  n ) ) ) ( 1  /  m )  -  ( log `  ( x  /  n ) ) ) ) )  e.  O ( 1 )
 
Theoremmulogsum 20681* Asymptotic formula for  sum_ n  <_  x ,  ( mmu ( n )  /  n ) log (
x  /  n )  =  O ( 1 ). Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)
 |-  ( x  e.  RR+  |->  sum_
 n  e.  ( 1
 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n )  x.  ( log `  ( x  /  n ) ) ) )  e.  O ( 1 )
 
Theoremlogdivsum 20682* Asymptotic analysis of  sum_ n  <_  x ,  log n  /  n  =  ( log x ) ^ 2  /  2  +  L  +  O ( log x  /  x ). (Contributed by Mario Carneiro, 18-May-2016.)
 |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_ `  y
 ) ) ( ( log `  i )  /  i )  -  (
 ( ( log `  y
 ) ^ 2 ) 
 /  2 ) ) )   =>    |-  ( F : RR+ --> RR 
 /\  F  e.  dom  ~~> r 
 /\  ( ( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  (
 ( F `  A )  -  L ) ) 
 <_  ( ( log `  A )  /  A ) ) )
 
Theoremmulog2sumlem1 20683* Asymptotic formula for  sum_ n  <_  x ,  log (
x  /  n )  /  n  =  ( 1  /  2 ) log ^ 2 ( x )  +  gamma  x.  log x  -  L  +  O ( log x  /  x ), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_ `  y
 ) ) ( ( log `  i )  /  i )  -  (
 ( ( log `  y
 ) ^ 2 ) 
 /  2 ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  _e 
 <_  A )   =>    |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) ) 
 /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
 2 )  +  (
 ( gamma  x.  ( log `  A ) )  -  L ) ) ) )  <_  ( 2  x.  ( ( log `  A )  /  A ) ) )
 
Theoremmulog2sumlem2 20684* Lemma for mulog2sum 20686. (Contributed by Mario Carneiro, 19-May-2016.)
 |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_ `  y
 ) ) ( ( log `  i )  /  i )  -  (
 ( ( log `  y
 ) ^ 2 ) 
 /  2 ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   &    |-  T  =  ( ( ( ( log `  ( x  /  n ) ) ^ 2
 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )   &    |-  R  =  ( (
 ( 1  /  2
 )  +  ( gamma  +  ( abs `  L ) ) )  +  sum_
 m  e.  ( 1
 ... 2 ) ( ( log `  ( _e  /  m ) ) 
 /  m ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n )  x.  T )  -  ( log `  x )
 ) )  e.  O ( 1 ) )
 
Theoremmulog2sumlem3 20685* Lemma for mulog2sum 20686. (Contributed by Mario Carneiro, 13-May-2016.)
 |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_ `  y
 ) ) ( ( log `  i )  /  i )  -  (
 ( ( log `  y
 ) ^ 2 ) 
 /  2 ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n )  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 ) )
 
Theoremmulog2sum 20686* Asymptotic formula for  sum_ n  <_  x ,  ( mmu ( n )  /  n ) log ^
2 ( x  /  n )  =  2 log x  +  O
( 1 ). Equation 10.2.8 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 19-May-2016.)
 |-  ( x  e.  RR+  |->  ( sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n )  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremvmalogdivsum2 20687* The sum  sum_ n  <_  x , Λ ( n ) log ( x  /  n )  /  n is asymptotic to  log ^ 2 ( x )  / 
2  +  O ( log x ). Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  /  n )  x.  ( log `  ( x  /  n ) ) ) 
 /  ( log `  x ) )  -  (
 ( log `  x )  /  2 ) ) )  e.  O ( 1 )
 
Theoremvmalogdivsum 20688* The sum  sum_ n  <_  x , Λ ( n ) log n  /  n is asymptotic to  log ^ 2 ( x )  / 
2  +  O ( log x ). Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  /  n )  x.  ( log `  n ) ) 
 /  ( log `  x ) )  -  (
 ( log `  x )  /  2 ) ) )  e.  O ( 1 )
 
Theorem2vmadivsumlem 20689* Lemma for 2vmadivsum 20690. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo ) ( abs `  ( sum_ i  e.  (
 1 ... ( |_ `  y
 ) ) ( (Λ `  i )  /  i
 )  -  ( log `  y ) ) ) 
 <_  A )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( (
 sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  /  n )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( x  /  n ) ) ) ( (Λ `  m )  /  m ) ) 
 /  ( log `  x ) )  -  (
 ( log `  x )  /  2 ) ) )  e.  O ( 1 ) )
 
Theorem2vmadivsum 20690* The sum  sum_ m n  <_  x , Λ (
m )Λ ( n )  /  m n is asymptotic to  log ^ 2 ( x )  / 
2  +  O ( log x ). (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  /  n )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( x  /  n ) ) ) ( (Λ `  m )  /  m ) ) 
 /  ( log `  x ) )  -  (
 ( log `  x )  /  2 ) ) )  e.  O ( 1 )
 
Theoremlogsqvma 20691* A formula for  log ^ 2 ( N ) in terms of the primes. Equation 10.4.6 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
 |-  ( N  e.  NN  -> 
 sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( sum_ u  e.  { x  e. 
 NN  |  x  ||  d }  ( (Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  +  ( (Λ `  d )  x.  ( log `  d
 ) ) )  =  ( ( log `  N ) ^ 2 ) )
 
Theoremlogsqvma2 20692* The Möbius inverse of logsqvma 20691. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
 |-  ( N  e.  NN  -> 
 sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( ( mmu `  d )  x.  ( ( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
 
Theoremlog2sumbnd 20693* Bound on the difference between 
sum_ n  <_  A ,  log ^ 2 ( n ) and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  ( ( A  e.  RR+  /\  1  <_  A ) 
 ->  ( abs `  ( sum_ n  e.  ( 1
 ... ( |_ `  A ) ) ( ( log `  n ) ^ 2 )  -  ( A  x.  (
 ( ( log `  A ) ^ 2 )  +  ( 2  -  (
 2  x.  ( log `  A ) ) ) ) ) ) ) 
 <_  ( ( ( log `  A ) ^ 2
 )  +  2 ) )
 
Theoremselberglem1 20694* Lemma for selberg 20697. Estimation of the asymptotic part of selberglem3 20696. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  T  =  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
 2  x.  ( log `  ( x  /  n ) ) ) ) )  /  n )   =>    |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  x.  T )  -  (
 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremselberglem2 20695* Lemma for selberg 20697. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  T  =  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
 2  x.  ( log `  ( x  /  n ) ) ) ) )  /  n )   =>    |-  ( x  e.  RR+  |->  ( (
 sum_ n  e.  (
 1 ... ( |_ `  x ) ) sum_ m  e.  ( 1 ... ( |_ `  ( x  /  n ) ) ) ( ( mmu `  n )  x.  (
 ( log `  m ) ^ 2 ) ) 
 /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremselberglem3 20696* Lemma for selberg 20697. Estimation of the left hand side of logsqvma2 20692. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) )
 sum_ d  e.  { y  e.  NN  |  y  ||  n }  ( ( mmu `  d )  x.  ( ( log `  ( n  /  d ) ) ^ 2 ) ) 
 /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremselberg 20697* 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.)
 |-  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) ) ) 
 /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremselbergb 20698* Convert eventual boundedness in selberg 20697 to boundedness on  [ 1 , 
+oo ). (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 30-May-2016.)
 |- 
 E. c  e.  RR+  A. x  e.  ( 1 [,)  +oo ) ( abs `  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) ) ) 
 /  x )  -  ( 2  x.  ( log `  x ) ) ) )  <_  c
 
Theoremselberg2lem 20699* Lemma for selberg2 20700. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n ) )  -  (
 (ψ `  x )  x.  ( log `  x ) ) )  /  x ) )  e.  O ( 1 )
 
Theoremselberg2 20700* Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
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