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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlgsdir 20401 The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 20417 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  / L N )  =  ( ( A  / L N )  x.  ( B  / L N ) ) )
 
Theoremlgsdilem2 20402* Lemma for lgsdi 20403. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  M  =/=  0 )   &    |-  ( ph  ->  N  =/=  0 )   &    |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n  pCnt  M ) ) ,  1 ) )   =>    |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `  ( abs `  M ) )  =  (  seq  1 (  x.  ,  F ) `  ( abs `  ( M  x.  N ) ) ) )
 
Theoremlgsdi 20403 The Legendre symbol is completely multiplicative in its right argument. (Contributed by Mario Carneiro, 5-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )
 
Theoremlgsne0 20404 The Legendre symbol is nonzero (and hence equal to  1 or  -u 1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( A 
 / L N )  =/=  0  <->  ( A  gcd  N )  =  1 ) )
 
Theoremlgsabs1 20405 The Legendre symbol is nonzero (and hence equal to  1 or  -u 1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  ( A  / L N ) )  =  1  <->  ( A  gcd  N )  =  1 ) )
 
Theoremlgssq 20406 The Legendre symbol at a square is equal to  1. Together with lgsmod 20392 this implies that the Legendre symbol takes value  1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.)
 |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( A ^
 2 )  / L N )  =  1
 )
 
Theoremlgssq2 20407 The Legendre symbol at a square is equal to  1. (Contributed by Mario Carneiro, 5-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L
 ( N ^ 2
 ) )  =  1 )
 
Theorem1lgs 20408 The Legendre symbol at  1. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( N  e.  ZZ  ->  ( 1  / L N )  =  1
 )
 
Theoremlgs1 20409 The Legendre symbol at  1. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( A  e.  ZZ  ->  ( A  / L
 1 )  =  1 )
 
Theoremlgsdirnn0 20410 Variation on lgsdir 20401 valid for all  A ,  B but only for positive  N. (The exact location of the failure of this law is for  A  =  0,  B  <  0,  N  =  -u 1 in which case  ( 0  / L -u 1
)  =  1 but  ( B  / L -u 1 )  = 
-u 1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( ( A  x.  B )  / L N )  =  ( ( A  / L N )  x.  ( B  / L N ) ) )
 
Theoremlgsdinn0 20411 Variation on lgsdi 20403 valid for all  M ,  N but only for positive  A. (The exact location of the failure of this law is for  A  =  -u
1,  M  =  0, and some  N in which case  ( -u 1  / L 0 )  =  1 but  ( -u 1  / L N )  = 
-u 1 when  -u 1 is not a quadratic residue mod  N.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L
 ( M  x.  N ) )  =  (
 ( A  / L M )  x.  ( A  / L N ) ) )
 
Theoremlgsqrlem1 20412 Lemma for lgsqr 20417. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  P )   &    |-  S  =  (Poly1 `  Y )   &    |-  B  =  ( Base `  S )   &    |-  D  =  ( deg1  `  Y )   &    |-  O  =  (eval1 `  Y )   &    |-  .^  =  (.g `  (mulGrp `  S ) )   &    |-  X  =  (var1 `  Y )   &    |-  .-  =  ( -g `  S )   &    |-  .1.  =  ( 1r `  S )   &    |-  T  =  ( (
 ( ( P  -  1 )  /  2
 )  .^  X )  .- 
 .1.  )   &    |-  L  =  ( ZRHom `  Y )   &    |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  ( ( A ^ ( ( P  -  1 )  / 
 2 ) )  mod  P )  =  ( 1 
 mod  P ) )   =>    |-  ( ph  ->  ( ( O `  T ) `  ( L `  A ) )  =  ( 0g `  Y ) )
 
Theoremlgsqrlem2 20413* Lemma for lgsqr 20417. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  P )   &    |-  S  =  (Poly1 `  Y )   &    |-  B  =  ( Base `  S )   &    |-  D  =  ( deg1  `  Y )   &    |-  O  =  (eval1 `  Y )   &    |-  .^  =  (.g `  (mulGrp `  S ) )   &    |-  X  =  (var1 `  Y )   &    |-  .-  =  ( -g `  S )   &    |-  .1.  =  ( 1r `  S )   &    |-  T  =  ( (
 ( ( P  -  1 )  /  2
 )  .^  X )  .- 
 .1.  )   &    |-  L  =  ( ZRHom `  Y )   &    |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  G  =  ( y  e.  ( 1
 ... ( ( P  -  1 )  / 
 2 ) )  |->  ( L `  ( y ^ 2 ) ) )   =>    |-  ( ph  ->  G : ( 1 ... ( ( P  -  1 )  /  2
 ) ) -1-1-> ( `' ( O `  T ) " { ( 0g
 `  Y ) }
 ) )
 
Theoremlgsqrlem3 20414* Lemma for lgsqr 20417. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  P )   &    |-  S  =  (Poly1 `  Y )   &    |-  B  =  ( Base `  S )   &    |-  D  =  ( deg1  `  Y )   &    |-  O  =  (eval1 `  Y )   &    |-  .^  =  (.g `  (mulGrp `  S ) )   &    |-  X  =  (var1 `  Y )   &    |-  .-  =  ( -g `  S )   &    |-  .1.  =  ( 1r `  S )   &    |-  T  =  ( (
 ( ( P  -  1 )  /  2
 )  .^  X )  .- 
 .1.  )   &    |-  L  =  ( ZRHom `  Y )   &    |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  G  =  ( y  e.  ( 1
 ... ( ( P  -  1 )  / 
 2 ) )  |->  ( L `  ( y ^ 2 ) ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  ( A  / L P )  =  1 )   =>    |-  ( ph  ->  ( L `  A )  e.  ( `' ( O `
  T ) " { ( 0g `  Y ) } )
 )
 
Theoremlgsqrlem4 20415* Lemma for lgsqr 20417. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  P )   &    |-  S  =  (Poly1 `  Y )   &    |-  B  =  ( Base `  S )   &    |-  D  =  ( deg1  `  Y )   &    |-  O  =  (eval1 `  Y )   &    |-  .^  =  (.g `  (mulGrp `  S ) )   &    |-  X  =  (var1 `  Y )   &    |-  .-  =  ( -g `  S )   &    |-  .1.  =  ( 1r `  S )   &    |-  T  =  ( (
 ( ( P  -  1 )  /  2
 )  .^  X )  .- 
 .1.  )   &    |-  L  =  ( ZRHom `  Y )   &    |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  G  =  ( y  e.  ( 1
 ... ( ( P  -  1 )  / 
 2 ) )  |->  ( L `  ( y ^ 2 ) ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  ( A  / L P )  =  1 )   =>    |-  ( ph  ->  E. x  e.  ZZ  P  ||  (
 ( x ^ 2
 )  -  A ) )
 
Theoremlgsqrlem5 20416* Lemma for lgsqr 20417. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  ( A 
 / L P )  =  1 )  ->  E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A ) )
 
Theoremlgsqr 20417* The Legendre symbol for odd primes is 
1 iff the number is not a multiple of the prime (in which case it is  0, see lgsne0 20404) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 20405). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  (
 ( A  / L P )  =  1  <->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A ) ) ) )
 
Theoremlgsdchrval 20418* The Legendre symbol function  X ( m )  =  ( m  / L N ), where  N is an odd positive number, is a Dirichlet character modulo  N. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  L  =  ( ZRHom `  Z )   &    |-  X  =  ( y  e.  B  |->  ( iota
 h E. m  e. 
 ZZ  ( y  =  ( L `  m )  /\  h  =  ( m  / L N ) ) ) )   =>    |-  ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  ->  ( X `  ( L `  A ) )  =  ( A  / L N ) )
 
Theoremlgsdchr 20419* The Legendre symbol function  X ( m )  =  ( m  / L N ), where  N is an odd positive number, is a real Dirichlet character modulo  N. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  L  =  ( ZRHom `  Z )   &    |-  X  =  ( y  e.  B  |->  ( iota
 h E. m  e. 
 ZZ  ( y  =  ( L `  m )  /\  h  =  ( m  / L N ) ) ) )   =>    |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  ( X  e.  D  /\  X : B --> RR ) )
 
13.4.9  Quadratic Reciprocity
 
Theoremlgseisenlem1 20420* Lemma for lgseisen 20424. If  R ( u )  =  ( Q  x.  u )  mod  P and  M ( u )  =  ( -u
1 ^ R ( u ) )  x.  R ( u ), then for any even  1  <_  u  <_  P  -  1,  M ( u ) is also an even integer  1  <_  M
( u )  <_  P  -  1. To simplify these statements, we divide all the even numbers by  2, so that it becomes the statement that  M ( x  /  2 )  =  ( -u 1 ^ R ( x  / 
2 ) )  x.  R ( x  / 
2 )  /  2 is an integer between  1 and  ( P  -  1 )  / 
2. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  R  =  ( ( Q  x.  ( 2  x.  x ) )  mod  P )   &    |-  M  =  ( x  e.  ( 1 ... (
 ( P  -  1
 )  /  2 )
 )  |->  ( ( ( ( -u 1 ^ R )  x.  R )  mod  P )  /  2 ) )   =>    |-  ( ph  ->  M : ( 1 ... ( ( P  -  1 )  /  2
 ) ) --> ( 1
 ... ( ( P  -  1 )  / 
 2 ) ) )
 
Theoremlgseisenlem2 20421* Lemma for lgseisen 20424. The function  M is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  R  =  ( ( Q  x.  ( 2  x.  x ) )  mod  P )   &    |-  M  =  ( x  e.  ( 1 ... (
 ( P  -  1
 )  /  2 )
 )  |->  ( ( ( ( -u 1 ^ R )  x.  R )  mod  P )  /  2 ) )   &    |-  S  =  ( ( Q  x.  (
 2  x.  y ) )  mod  P )   =>    |-  ( ph  ->  M :
 ( 1 ... (
 ( P  -  1
 )  /  2 )
 )
 -1-1-onto-> ( 1 ... (
 ( P  -  1
 )  /  2 )
 ) )
 
Theoremlgseisenlem3 20422* Lemma for lgseisen 20424. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  R  =  ( ( Q  x.  ( 2  x.  x ) )  mod  P )   &    |-  M  =  ( x  e.  ( 1 ... (
 ( P  -  1
 )  /  2 )
 )  |->  ( ( ( ( -u 1 ^ R )  x.  R )  mod  P )  /  2 ) )   &    |-  S  =  ( ( Q  x.  (
 2  x.  y ) )  mod  P )   &    |-  Y  =  (ℤ/n `  P )   &    |-  G  =  (mulGrp `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( x  e.  ( 1 ... ( ( P  -  1 )  /  2
 ) )  |->  ( L `
  ( ( -u 1 ^ R )  x.  Q ) ) ) )  =  ( 1r
 `  Y ) )
 
Theoremlgseisenlem4 20423* Lemma for lgseisen 20424. The function  M is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  R  =  ( ( Q  x.  ( 2  x.  x ) )  mod  P )   &    |-  M  =  ( x  e.  ( 1 ... (
 ( P  -  1
 )  /  2 )
 )  |->  ( ( ( ( -u 1 ^ R )  x.  R )  mod  P )  /  2 ) )   &    |-  S  =  ( ( Q  x.  (
 2  x.  y ) )  mod  P )   &    |-  Y  =  (ℤ/n `  P )   &    |-  G  =  (mulGrp `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( ph  ->  ( ( Q ^ ( ( P  -  1 )  / 
 2 ) )  mod  P )  =  ( (
 -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2
 ) ) ( |_ `  ( ( Q  /  P )  x.  (
 2  x.  x ) ) ) )  mod  P ) )
 
Theoremlgseisen 20424* Eisenstein's lemma, an expression for 
( P  / L Q ) when  P ,  Q are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  ( Q  / L P )  =  ( -u 1 ^ sum_ x  e.  (
 1 ... ( ( P  -  1 )  / 
 2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) ) )
 
Theoremlgsquadlem1 20425* Lemma for lgsquad 20428. Count the members of  S with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  M  =  ( ( P  -  1 )  /  2
 )   &    |-  N  =  ( ( Q  -  1 ) 
 /  2 )   &    |-  S  =  { <. x ,  y >.  |  ( ( x  e.  ( 1 ...
 M )  /\  y  e.  ( 1 ... N ) )  /\  ( y  x.  P )  < 
 ( x  x.  Q ) ) }   =>    |-  ( ph  ->  (
 -u 1 ^ sum_ u  e.  ( ( ( |_ `  ( M 
 /  2 ) )  +  1 ) ... M ) ( |_ `  (
 ( Q  /  P )  x.  ( 2  x.  u ) ) ) )  =  ( -u 1 ^ ( # `  { z  e.  S  |  -.  2  ||  ( 1st `  z
 ) } ) ) )
 
Theoremlgsquadlem2 20426* Lemma for lgsquad 20428. Count the members of  S with even coordinates, and combine with lgsquadlem1 20425 to get the total count of lattice points in  S (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  M  =  ( ( P  -  1 )  /  2
 )   &    |-  N  =  ( ( Q  -  1 ) 
 /  2 )   &    |-  S  =  { <. x ,  y >.  |  ( ( x  e.  ( 1 ...
 M )  /\  y  e.  ( 1 ... N ) )  /\  ( y  x.  P )  < 
 ( x  x.  Q ) ) }   =>    |-  ( ph  ->  ( Q  / L P )  =  ( -u 1 ^ ( # `  S ) ) )
 
Theoremlgsquadlem3 20427* Lemma for lgsquad 20428. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  M  =  ( ( P  -  1 )  /  2
 )   &    |-  N  =  ( ( Q  -  1 ) 
 /  2 )   &    |-  S  =  { <. x ,  y >.  |  ( ( x  e.  ( 1 ...
 M )  /\  y  e.  ( 1 ... N ) )  /\  ( y  x.  P )  < 
 ( x  x.  Q ) ) }   =>    |-  ( ph  ->  ( ( P  / L Q )  x.  ( Q  / L P ) )  =  ( -u 1 ^ ( M  x.  N ) ) )
 
Theoremlgsquad 20428 The Law of Quadratic Reciprocity. If  P and  Q are distinct odd primes, then the product of the Legendre symbols  ( P  / L Q ) and  ( Q  / L P ) is the parity of  ( ( P  -  1 )  /  2 )  x.  ( ( Q  - 
1 )  /  2
). This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ( P  e.  ( Prime  \  { 2 } )  /\  Q  e.  ( Prime  \  { 2 } )  /\  P  =/=  Q )  ->  ( ( P  / L Q )  x.  ( Q  / L P ) )  =  ( -u 1 ^ (
 ( ( P  -  1 )  /  2
 )  x.  ( ( Q  -  1 ) 
 /  2 ) ) ) )
 
Theoremlgsquad2lem1 20429 Lemma for lgsquad2 20431. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  -.  2  ||  M )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  -.  2  ||  N )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  ( A  x.  B )  =  M )   &    |-  ( ph  ->  ( ( A 
 / L N )  x.  ( N  / L A ) )  =  ( -u 1 ^ (
 ( ( A  -  1 )  /  2
 )  x.  ( ( N  -  1 ) 
 /  2 ) ) ) )   &    |-  ( ph  ->  ( ( B  / L N )  x.  ( N  / L B ) )  =  ( -u 1 ^ ( ( ( B  -  1 ) 
 /  2 )  x.  ( ( N  -  1 )  /  2
 ) ) ) )   =>    |-  ( ph  ->  ( ( M  / L N )  x.  ( N  / L M ) )  =  ( -u 1 ^ (
 ( ( M  -  1 )  /  2
 )  x.  ( ( N  -  1 ) 
 /  2 ) ) ) )
 
Theoremlgsquad2lem2 20430* Lemma for lgsquad2 20431. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  -.  2  ||  M )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  -.  2  ||  N )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  (
 ( ph  /\  ( m  e.  ( Prime  \  {
 2 } )  /\  ( m  gcd  N )  =  1 ) ) 
 ->  ( ( m  / L N )  x.  ( N  / L m ) )  =  ( -u 1 ^ ( ( ( m  -  1 ) 
 /  2 )  x.  ( ( N  -  1 )  /  2
 ) ) ) )   &    |-  ( ps  <->  A. x  e.  (
 1 ... k ) ( ( x  gcd  (
 2  x.  N ) )  =  1  ->  ( ( x  / L N )  x.  ( N  / L x ) )  =  ( -u 1 ^ ( ( ( x  -  1 ) 
 /  2 )  x.  ( ( N  -  1 )  /  2
 ) ) ) ) )   =>    |-  ( ph  ->  (
 ( M  / L N )  x.  ( N  / L M ) )  =  ( -u 1 ^ ( ( ( M  -  1 ) 
 /  2 )  x.  ( ( N  -  1 )  /  2
 ) ) ) )
 
Theoremlgsquad2 20431 Extend lgsquad 20428 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  -.  2  ||  M )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  -.  2  ||  N )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   =>    |-  ( ph  ->  ( ( M  / L N )  x.  ( N  / L M ) )  =  ( -u 1 ^ ( ( ( M  -  1 ) 
 /  2 )  x.  ( ( N  -  1 )  /  2
 ) ) ) )
 
Theoremlgsquad3 20432 Extend lgsquad2 20431 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\ 
 -.  2  ||  N ) )  ->  ( M 
 / L N )  =  ( ( -u 1 ^ ( ( ( M  -  1 ) 
 /  2 )  x.  ( ( N  -  1 )  /  2
 ) ) )  x.  ( N  / L M ) ) )
 
Theoremm1lgs 20433 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 20434* Lemma for 2sq 20447. (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 20435* Lemma for 2sq 20447. (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 20436 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 20437 Lemma for 2sqlem5 20439. (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 20438 Lemma for 2sqlem5 20439. (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 20439 Lemma for 2sq 20447. 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 20440* Lemma for 2sq 20447. 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 20441* Lemma for 2sq 20447. (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 20442* Lemma for 2sqlem8 20443. (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 20443* Lemma for 2sq 20447. (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 20444* Lemma for 2sq 20447. (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 20445* Lemma for 2sq 20447. 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 20446* Lemma for 2sq 20447. (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 20447* 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 20448 The converse to 2sq 20447. (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 20449* The converse to 2sq 20447. (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 20450 Lemma for chebbnd1 20453: show a lower bound on π ( x ) at even integers using similar techniques to those used to prove bpos 20364. (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 20355, 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 20351 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 20451 Lemma for chebbnd1 20453: 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 20452 Lemma for chebbnd1 20453: 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 20453 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 20454 Lemma for chtppilim 20456. (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 20455* Lemma for chtppilim 20456. (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 20456 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 20457 The  theta function is upper bounded by a linear term. Corollary of chtub 20283. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( x  e.  RR+  |->  ( ( theta `  x )  /  x ) )  e.  O ( 1 )
 
Theoremchebbnd2 20458 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 20459 The  theta function is lower bounded by a linear term. Corollary of chebbnd1 20453. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O ( 1 )
 
Theoremchpchtlim 20460 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 20461 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 20462* 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 20463* 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 20464* 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 20465* Lemma for rplogsum 20508. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( A  e.  NN  -> 
 sum_ n  e.  (
 2 ... A ) ( ( log `  n )  /  ( n  x.  ( n  -  1
 ) ) )  <_ 
 2 )
 
Theoremrplogsumlem2 20466* Lemma for rplogsum 20508. 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 20467 Lemma for dchrisum0lem1 20497. (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 20468* Lemma for rpvmasum 20507. 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 20469* Lemma for dchrisum 20473. 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 20470* Lemma for dchrisum 20473. 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 20471* Lemma for dchrisum 20473. 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 20472* Lemma for dchrisum 20473. 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 20473* 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 20474* Lemma for dchrmusum 20505 and dchrisumn0 20502. Apply dchrisum 20473 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 20475* 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 20476* 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 20477* Give an expression for  log x remarkably similar to  sum_ n  <_  x
( X ( n )Λ ( n )  /  n ) given in dchrvmasumlem1 20476. 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 20478* Combine the results of dchrvmasumlem1 20476 and dchrvmasum2lem 20477 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 20479* Lemma for dchrvmasum 20506. (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 20480* Lemma for dchrvmasum 20506. (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 20481* Lemma for dchrvmasum 20506 and dchrvmasumif 20484. Apply dchrisum 20473 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 20482* Lemma for dchrvmasumif 20484. (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 20483* Lemma for dchrvmasum 20506. (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 20484* 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 20506.) (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 20485* 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 20486* 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 20487* 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 20488* 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 20489* Lemma for dchrisum0flb 20491. 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 20490* Lemma for dchrisum0flb 20491. 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 20491* 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 20492* 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 20493* A partial result along the lines of rpvmasum 20507. 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 20494* 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 20495* Lemma for dchrisum0 20501. Apply dchrisum 20473 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 20496* Lemma for dchrisum0lem1 20497. (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 20497* Lemma for dchrisum0 20501. (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 20498* Lemma for dchrisum0 20501. (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 20499* Lemma for dchrisum0 20501. (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 20500* Lemma for dchrisum0 20501. (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 ) )
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