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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlgsval3 20501 The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A  / L P )  =  ( ( ( ( A ^ (
 ( P  -  1
 )  /  2 )
 )  +  1 ) 
 mod  P )  -  1
 ) )
 
Theoremlgsvalmod 20502 The Legendre symbol is equivalent to 
a ^ ( ( p  -  1 )  /  2 ),  mod  p. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  (
 ( A  / L P )  mod  P )  =  ( ( A ^ ( ( P  -  1 )  / 
 2 ) )  mod  P ) )
 
Theoremlgsval4 20503* Restate lgsval 20487 for nonzero  N, where the function  F has been abbreviated into a self-referential expression taking the value of  / L on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( A  / L N )  =  ( if ( ( N  <  0 
 /\  A  <  0
 ) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  F ) `
  ( abs `  N ) ) ) )
 
Theoremlgsfcl3 20504* Closure of the function  F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  F : NN --> ZZ )
 
Theoremlgsval4a 20505* Same as lgsval4 20503 for positive  N. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  ( A  / L N )  =  ( 
 seq  1 (  x. 
 ,  F ) `  N ) )
 
Theoremlgsneg 20506 The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( A  / L -u N )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( A 
 / L N ) ) )
 
Theoremlgsneg1 20507 The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  NN0  /\  N  e.  ZZ )  ->  ( A  / L -u N )  =  ( A  / L N ) )
 
Theoremlgsmod 20508 The Legendre (Jacobi) symbol is preserved under reduction  mod  n when  n is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\ 
 -.  2  ||  N )  ->  ( ( A 
 mod  N )  / L N )  =  ( A  / L N ) )
 
Theoremlgsdilem 20509 Lemma for lgsdi 20519 and lgsdir 20517: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  if ( ( N  <  0 
 /\  ( A  x.  B )  <  0 ) ,  -u 1 ,  1 )  =  ( if ( ( N  <  0 
 /\  A  <  0
 ) ,  -u 1 ,  1 )  x. 
 if ( ( N  <  0  /\  B  <  0 ) ,  -u 1 ,  1 ) ) )
 
Theoremlgsdir2lem1 20510 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( 1 
 mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  (
 ( 3  mod  8
 )  =  3  /\  ( -u 3  mod  8
 )  =  5 ) )
 
Theoremlgsdir2lem2 20511 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( K  e.  ZZ  /\  2  ||  ( K  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ...
 K )  ->  ( A  mod  8 )  e.  S ) ) )   &    |-  M  =  ( K  +  1 )   &    |-  N  =  ( M  +  1 )   &    |-  N  e.  S   =>    |-  ( N  e.  ZZ  /\  2  ||  ( N  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  (
 0 ... N )  ->  ( A  mod  8 )  e.  S ) ) )
 
Theoremlgsdir2lem3 20512 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( A 
 mod  8 )  e.  ( { 1 ,  7 }  u.  {
 3 ,  5 } ) )
 
Theoremlgsdir2lem4 20513 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  e. 
 { 1 ,  7 } )  ->  (
 ( ( A  x.  B )  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  { 1 ,  7 } )
 )
 
Theoremlgsdir2lem5 20514 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  (
 ( A  mod  8
 )  e.  { 3 ,  5 }  /\  ( B  mod  8 )  e.  { 3 ,  5 } ) ) 
 ->  ( ( A  x.  B )  mod  8 )  e.  { 1 ,  7 } )
 
Theoremlgsdir2 20515 The Legendre symbol is completely multiplicative at  2. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  x.  B )  / L 2 )  =  ( ( A  / L 2 )  x.  ( B  / L
 2 ) ) )
 
Theoremlgsdirprm 20516 The Legendre symbol is completely multiplicative at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  ->  ( ( A  x.  B )  / L P )  =  ( ( A  / L P )  x.  ( B  / L P ) ) )
 
Theoremlgsdir 20517 The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 20533 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 20518* Lemma for lgsdi 20519. (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 20519 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 20520 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 20521 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 20522 The Legendre symbol at a square is equal to  1. Together with lgsmod 20508 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 20523 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 20524 The Legendre symbol at  1. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( N  e.  ZZ  ->  ( 1  / L N )  =  1
 )
 
Theoremlgs1 20525 The Legendre symbol at  1. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( A  e.  ZZ  ->  ( A  / L
 1 )  =  1 )
 
Theoremlgsdirnn0 20526 Variation on lgsdir 20517 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 20527 Variation on lgsdi 20519 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 20528 Lemma for lgsqr 20533. (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 20529* Lemma for lgsqr 20533. (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 20530* Lemma for lgsqr 20533. (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 20531* Lemma for lgsqr 20533. (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 20532* Lemma for lgsqr 20533. (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 20533* 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 20520) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 20521). 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 20534* 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 20535* 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 20536* Lemma for lgseisen 20540. 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 20537* Lemma for lgseisen 20540. 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 20538* Lemma for lgseisen 20540. (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 20539* Lemma for lgseisen 20540. 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 20540* 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 20541* Lemma for lgsquad 20544. 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 20542* Lemma for lgsquad 20544. Count the members of  S with even coordinates, and combine with lgsquadlem1 20541 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 20543* Lemma for lgsquad 20544. (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 20544 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 20545 Lemma for lgsquad2 20547. (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 20546* Lemma for lgsquad2 20547. (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 20547 Extend lgsquad 20544 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 20548 Extend lgsquad2 20547 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 20549 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 20550* Lemma for 2sq 20563. (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 20551* Lemma for 2sq 20563. (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 20552 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 20553 Lemma for 2sqlem5 20555. (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 20554 Lemma for 2sqlem5 20555. (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 20555 Lemma for 2sq 20563. 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 20556* Lemma for 2sq 20563. 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 20557* Lemma for 2sq 20563. (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 20558* Lemma for 2sqlem8 20559. (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 20559* Lemma for 2sq 20563. (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 20560* Lemma for 2sq 20563. (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 20561* Lemma for 2sq 20563. 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 20562* Lemma for 2sq 20563. (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 20563* 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 20564 The converse to 2sq 20563. (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 20565* The converse to 2sq 20563. (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 20566 Lemma for chebbnd1 20569: show a lower bound on π ( x ) at even integers using similar techniques to those used to prove bpos 20480. (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 20471, 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 20467 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 20567 Lemma for chebbnd1 20569: 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 20568 Lemma for chebbnd1 20569: 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 20569 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 20570 Lemma for chtppilim 20572. (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 20571* Lemma for chtppilim 20572. (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 20572 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 20573 The  theta function is upper bounded by a linear term. Corollary of chtub 20399. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( x  e.  RR+  |->  ( ( theta `  x )  /  x ) )  e.  O ( 1 )
 
Theoremchebbnd2 20574 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 20575 The  theta function is lower bounded by a linear term. Corollary of chebbnd1 20569. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( x  e.  (
 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O ( 1 )
 
Theoremchpchtlim 20576 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 20577 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 20578* 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 20579* 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 20580* 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 20581* Lemma for rplogsum 20624. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( A  e.  NN  -> 
 sum_ n  e.  (
 2 ... A ) ( ( log `  n )  /  ( n  x.  ( n  -  1
 ) ) )  <_ 
 2 )
 
Theoremrplogsumlem2 20582* Lemma for rplogsum 20624. 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 20583 Lemma for dchrisum0lem1 20613. (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 20584* Lemma for rpvmasum 20623. 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 20585* Lemma for dchrisum 20589. 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 20586* Lemma for dchrisum 20589. 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 20587* Lemma for dchrisum 20589. 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 20588* Lemma for dchrisum 20589. 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 20589* 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 20590* Lemma for dchrmusum 20621 and dchrisumn0 20618. Apply dchrisum 20589 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 20591* 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 20592* 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 20593* Give an expression for  log x remarkably similar to  sum_ n  <_  x
( X ( n )Λ ( n )  /  n ) given in dchrvmasumlem1 20592. 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 20594* Combine the results of dchrvmasumlem1 20592 and dchrvmasum2lem 20593 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 20595* Lemma for dchrvmasum 20622. (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 20596* Lemma for dchrvmasum 20622. (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 20597* Lemma for dchrvmasum 20622 and dchrvmasumif 20600. Apply dchrisum 20589 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 20598* Lemma for dchrvmasumif 20600. (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 20599* Lemma for dchrvmasum 20622. (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 20600* 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 20622.) (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 ) )
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