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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremppidif 20401 The difference of the prime pi function at two points counts the number of primes in an interval. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( (π `  N )  -  (π
 `  M ) )  =  ( # `  (
 ( ( M  +  1 ) ... N )  i^i  Prime ) ) )
 
Theoremppi1 20402 The prime pi function at  1. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  1 )  =  0
 
Theoremcht1 20403 The Chebyshev function at  1. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  1 )  =  0
 
Theoremvma1 20404 The von Mangoldt function at  1. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  (Λ `  1 )  =  0
 
Theoremchp1 20405 The second Chebyshev function at  1. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  (ψ `  1 )  =  0
 
Theoremppi1i 20406 Inference form of ppiprm 20389. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (π `  M )  =  K   &    |-  N  e.  Prime   =>    |-  (π `  N )  =  ( K  +  1 )
 
Theoremppi2i 20407 Inference form of ppinprm 20390. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (π `  M )  =  K   &    |-  -.  N  e.  Prime   =>    |-  (π `  N )  =  K
 
Theoremppi2 20408 The prime pi function at  2. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  2 )  =  1
 
Theoremppi3 20409 The prime pi function at  3. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  3 )  =  2
 
Theoremcht2 20410 The Chebyshev function at  2. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  2 )  =  ( log `  2
 )
 
Theoremcht3 20411 The Chebyshev function at  3. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  3 )  =  ( log `  6
 )
 
Theoremppinncl 20412 Closure of the prime pi function in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( ( A  e.  RR  /\  2  <_  A )  ->  (π `  A )  e. 
 NN )
 
Theoremchtrpcl 20413 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  RR  /\  2  <_  A )  ->  ( theta `  A )  e.  RR+ )
 
Theoremppieq0 20414 The prime pi function is zero iff its argument is less than  2. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR  ->  ( (π `  A )  =  0  <->  A  <  2 ) )
 
Theoremppiltx 20415 The prime pi function is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR+  ->  (π
 `  A )  <  A )
 
Theoremprmorcht 20416 Relate the primorial (product of the first  n primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  n ,  1 ) )   =>    |-  ( A  e.  NN  ->  ( exp `  ( theta `  A ) )  =  (  seq  1
 (  x.  ,  F ) `  A ) )
 
Theoremmumullem1 20417 Lemma for mumul 20419. A multiple of a non-squarefree number is non-squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  A )  =  0 )  ->  ( mmu `  ( A  x.  B ) )  =  0 )
 
Theoremmumullem2 20418 Lemma for mumul 20419. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A 
 gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B )  =/=  0
 ) )  ->  ( mmu `  ( A  x.  B ) )  =/=  0 )
 
Theoremmumul 20419 The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  ( mmu `  ( A  x.  B ) )  =  ( ( mmu `  A )  x.  ( mmu `  B ) ) )
 
Theoremsqff1o 20420* There is a bijection from the squarefree divisors of a number  N to the powerset of the prime divisors of  N. Among other things, this implies that a number has  2 ^ k squarefree divisors where  k is the number of prime divisors, and a squarefree number has  2 ^ k divisors (because all divisors of a squarefree number are squarefree). The inverse function to  F takes the product of all the primes in some subset of prime divisors of  N. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  S  =  { x  e.  NN  |  ( ( mmu `  x )  =/=  0  /\  x  ||  N ) }   &    |-  F  =  ( n  e.  S  |->  { p  e.  Prime  |  p  ||  n } )   &    |-  G  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )   =>    |-  ( N  e.  NN  ->  F : S -1-1-onto-> ~P { p  e. 
 Prime  |  p  ||  N } )
 
Theoremdvdsdivcl 20421* The complement of a divisor of  N is also a divisor of  N. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( N  e.  NN  /\  A  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  A )  e.  { x  e. 
 NN  |  x  ||  N } )
 
Theoremdvdsflip 20422* An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
 |-  A  =  { x  e.  NN  |  x  ||  N }   &    |-  F  =  ( y  e.  A  |->  ( N  /  y ) )   =>    |-  ( N  e.  NN  ->  F : A -1-1-onto-> A )
 
Theoremfsumdvdsdiaglem 20423* A "diagonal commutation" of divisor sums analogous to fsum0diag 12240. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 ( j  e.  { x  e.  NN  |  x  ||  N }  /\  k  e.  { x  e.  NN  |  x  ||  ( N 
 /  j ) }
 )  ->  ( k  e.  { x  e.  NN  |  x  ||  N }  /\  j  e.  { x  e.  NN  |  x  ||  ( N  /  k
 ) } ) ) )
 
Theoremfsumdvdsdiag 20424* A "diagonal commutation" of divisor sums analogous to fsum0diag 12240. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ( ph  /\  ( j  e.  { x  e.  NN  |  x  ||  N }  /\  k  e.  { x  e.  NN  |  x  ||  ( N 
 /  j ) }
 ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  { x  e.  NN  |  x  ||  N } sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  j
 ) } A  =  sum_
 k  e.  { x  e.  NN  |  x  ||  N } sum_ j  e.  { x  e.  NN  |  x  ||  ( N  /  k
 ) } A )
 
Theoremfsumdvdscom 20425* A double commutation of divisor sums based on fsumdvdsdiag 20424. Note that  A depends on both  j and  k. (Contributed by Mario Carneiro, 13-May-2016.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( j  =  ( k  x.  m )  ->  A  =  B )   &    |-  ( ( ph  /\  (
 j  e.  { x  e.  NN  |  x  ||  N }  /\  k  e. 
 { x  e.  NN  |  x  ||  j }
 ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  { x  e.  NN  |  x  ||  N } sum_ k  e.  { x  e.  NN  |  x  ||  j } A  =  sum_ k  e.  { x  e. 
 NN  |  x  ||  N } sum_ m  e.  { x  e.  NN  |  x  ||  ( N  /  k
 ) } B )
 
Theoremdvdsppwf1o 20426* A bijection from the divisors of a prime power to the integers less than the prime count. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  F  =  ( n  e.  ( 0 ...
 A )  |->  ( P ^ n ) )   =>    |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  F : ( 0 ... A ) -1-1-onto-> { x  e.  NN  |  x  ||  ( P ^ A ) } )
 
Theoremdvdsflf1o 20427* A bijection from the numbers less than  N  /  A to the multiples of  A less than  N. Useful for some sum manipulations. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  N  e.  NN )   &    |-  F  =  ( n  e.  (
 1 ... ( |_ `  ( A  /  N ) ) )  |->  ( N  x.  n ) )   =>    |-  ( ph  ->  F : ( 1 ... ( |_ `  ( A  /  N ) ) ) -1-1-onto-> { x  e.  (
 1 ... ( |_ `  A ) )  |  N  ||  x } )
 
Theoremdvdsflsumcom 20428* A sum commutation from  sum_ n  <_  A ,  sum_ d  ||  n ,  B ( n ,  d ) to  sum_ d  <_  A ,  sum_ m  <_  A  /  d ,  B
( n ,  d m ). (Contributed by Mario Carneiro, 4-May-2016.)
 |-  ( n  =  ( d  x.  m ) 
 ->  B  =  C )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  (
 1 ... ( |_ `  A ) )  /\  d  e. 
 { x  e.  NN  |  x  ||  n }
 ) )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) )
 sum_ d  e.  { x  e.  NN  |  x  ||  n } B  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
 
Theoremfsumfldivdiaglem 20429* Lemma for fsumfldivdiag 20430. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 ( n  e.  (
 1 ... ( |_ `  A ) )  /\  m  e.  ( 1 ... ( |_ `  ( A  /  n ) ) ) )  ->  ( m  e.  ( 1 ... ( |_ `  A ) ) 
 /\  n  e.  (
 1 ... ( |_ `  ( A  /  m ) ) ) ) ) )
 
Theoremfsumfldivdiag 20430* The right hand side of dvdsflsumcom 20428 is commutative in the variables, because it can be written as the manifestly symmetric sum over those  <. m ,  n >. such that  m  x.  n  <_  A. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ( ph  /\  ( n  e.  (
 1 ... ( |_ `  A ) )  /\  m  e.  ( 1 ... ( |_ `  ( A  /  n ) ) ) ) )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) )
 sum_ m  e.  (
 1 ... ( |_ `  ( A  /  n ) ) ) B  =  sum_ m  e.  ( 1 ... ( |_ `  A ) ) sum_ n  e.  ( 1 ... ( |_ `  ( A  /  m ) ) ) B )
 
Theoremmusum 20431* The sum of the Möbius function over the divisors of  N gives one if  N  =  1, but otherwise always sums to zero. This makes the Möbius function useful for inverting divisor sums; see also muinv 20433. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( N  e.  NN  -> 
 sum_ k  e.  { n  e.  NN  |  n  ||  N }  ( mmu `  k )  =  if ( N  =  1 ,  1 ,  0 ) )
 
Theoremmusumsum 20432* Evaluate a collapsing sum over the Möbius function. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  ( m  =  1 
 ->  B  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 C_  NN )   &    |-  ( ph  ->  1  e.  A )   &    |-  (
 ( ph  /\  m  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ m  e.  A  sum_ k  e.  { n  e.  NN  |  n  ||  m }  ( ( mmu `  k )  x.  B )  =  C )
 
Theoremmuinv 20433* The Möbius inversion formula. If  G ( n )  =  sum_ k  ||  n F ( k ) for every  n  e.  NN, then  F ( n )  = 
sum_ k  ||  n  mmu ( k ) G ( n  /  k )  = 
sum_ k  ||  n mmu ( n  /  k
) G ( k ), i.e. the Möbius function is the Dirichlet convolution inverse of the constant function  1. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  G  =  ( n  e.  NN  |->  sum_ k  e.  { x  e. 
 NN  |  x  ||  n }  ( F `  k ) ) )   =>    |-  ( ph  ->  F  =  ( m  e.  NN  |->  sum_
 j  e.  { x  e.  NN  |  x  ||  m }  ( ( mmu `  j )  x.  ( G `  ( m  /  j ) ) ) ) )
 
Theoremdvdsmulf1o 20434* If  M and  N are two coprime integers, multiplication forms a bijection from the set of pairs  <. j ,  k >. where  j  ||  M and  k  ||  N, to the set of divisors of  M  x.  N. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  X  =  { x  e.  NN  |  x  ||  M }   &    |-  Y  =  { x  e.  NN  |  x  ||  N }   &    |-  Z  =  { x  e.  NN  |  x  ||  ( M  x.  N ) }   =>    |-  ( ph  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y )
 -1-1-onto-> Z )
 
Theoremfsumdvdsmul 20435* Product of two divisor sums. (This is also the main part of the proof that " sum_ k  ||  N F ( k ) is a multiplicative function if  F is".) (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  X  =  { x  e.  NN  |  x  ||  M }   &    |-  Y  =  { x  e.  NN  |  x  ||  N }   &    |-  Z  =  { x  e.  NN  |  x  ||  ( M  x.  N ) }   &    |-  ( ( ph  /\  j  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Y )  ->  B  e.  CC )   &    |-  ( ( ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  ( A  x.  B )  =  D )   &    |-  ( i  =  ( j  x.  k
 )  ->  C  =  D )   =>    |-  ( ph  ->  ( sum_ j  e.  X  A  x.  sum_ k  e.  Y  B )  =  sum_ i  e.  Z  C )
 
Theoremsgmppw 20436* The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ( A  e.  CC  /\  P  e.  Prime  /\  N  e.  NN0 )  ->  ( A  sigma  ( P ^ N ) )  =  sum_ k  e.  (
 0 ... N ) ( ( P  ^ c  A ) ^ k
 ) )
 
Theorem0sgmppw 20437 A prime power  P ^ K has  K  +  1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ( P  e.  Prime  /\  K  e.  NN0 )  ->  ( 0  sigma  ( P ^ K ) )  =  ( K  +  1 ) )
 
Theorem1sgmprm 20438 The sum of divisors for a prime is 
P  +  1 because the only divisors are  1 and  P. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( P  e.  Prime  ->  ( 1  sigma  P )  =  ( P  +  1 ) )
 
Theorem1sgm2ppw 20439 The sum of the divisors of  2 ^ ( N  -  1 ). (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( N  e.  NN  ->  ( 1  sigma  ( 2 ^ ( N  -  1 ) ) )  =  ( ( 2 ^ N )  -  1 ) )
 
Theoremsgmmul 20440 The divisor function for fixed parameter  A is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( A  e.  CC  /\  ( M  e.  NN  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )
 )  ->  ( A  sigma  ( M  x.  N ) )  =  (
 ( A  sigma  M )  x.  ( A  sigma  N ) ) )
 
Theoremppiublem1 20441 Lemma for ppiub 20443. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  ( N  <_  6  /\  ( ( P  e.  Prime  /\  4  <_  P )  ->  ( ( P 
 mod  6 )  e.  ( N ... 5
 )  ->  ( P  mod  6 )  e.  {
 1 ,  5 } ) ) )   &    |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (
 2  ||  M  \/  3  ||  M  \/  M  e.  { 1 ,  5 } )   =>    |-  ( M  <_  6  /\  ( ( P  e.  Prime  /\  4  <_  P )  ->  ( ( P 
 mod  6 )  e.  ( M ... 5
 )  ->  ( P  mod  6 )  e.  {
 1 ,  5 } ) ) )
 
Theoremppiublem2 20442 A prime greater than  3 does not divide  2 or  3, so its residue  mod  6 is  1 or  5. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  ( ( P  e.  Prime  /\  4  <_  P )  ->  ( P  mod  6 )  e.  { 1 ,  5 } )
 
Theoremppiub 20443 An upper bound on the Gauss prime 
pi function, which counts the number of primes less than 
N. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ( N  e.  RR  /\  0  <_  N )  ->  (π `  N )  <_  ( ( N  / 
 3 )  +  2 ) )
 
Theoremvmalelog 20444 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  (Λ `  A )  <_  ( log `  A ) )
 
Theoremchtlepsi 20445 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  ( theta `  A )  <_  (ψ `  A )
 )
 
Theoremchprpcl 20446 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( ( A  e.  RR  /\  2  <_  A )  ->  (ψ `  A )  e.  RR+ )
 
Theoremchpeq0 20447 The second Chebyshev function is zero iff its argument is less than  2. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( A  e.  RR  ->  ( (ψ `  A )  =  0  <->  A  <  2 ) )
 
Theoremchteq0 20448 The first Chebyshev function is zero iff its argument is less than  2. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( A  e.  RR  ->  ( ( theta `  A )  =  0  <->  A  <  2 ) )
 
Theoremchtleppi 20449 Upper bound on the  theta function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR+  ->  ( theta `  A )  <_  ( (π `  A )  x.  ( log `  A ) ) )
 
Theoremchtublem 20450 Lemma for chtub 20451. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN  ->  ( theta `  ( (
 2  x.  N )  -  1 ) ) 
 <_  ( ( theta `  N )  +  ( ( log `  4 )  x.  ( N  -  1
 ) ) ) )
 
Theoremchtub 20451 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)
 |-  ( ( N  e.  RR  /\  2  <  N )  ->  ( theta `  N )  <  ( ( log `  2 )  x.  (
 ( 2  x.  N )  -  3 ) ) )
 
Theoremfsumvma 20452* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( x  =  ( p ^ k ) 
 ->  B  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 C_  NN )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  <->  ( ( p  e.  Prime  /\  k  e. 
 NN )  /\  ( p ^ k )  e.  A ) ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  ( x  e.  A  /\  (Λ `  x )  =  0 ) ) 
 ->  B  =  0 )   =>    |-  ( ph  ->  sum_ x  e.  A  B  =  sum_ p  e.  P  sum_ k  e.  K  C )
 
Theoremfsumvma2 20453* Apply fsumvma 20452 for the common case of all numbers less than a real number  A. (Contributed by Mario Carneiro, 30-Apr-2016.)
 |-  ( x  =  ( p ^ k ) 
 ->  B  =  C )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  ( 1
 ... ( |_ `  A ) ) )  ->  B  e.  CC )   &    |-  (
 ( ph  /\  ( x  e.  ( 1 ... ( |_ `  A ) )  /\  (Λ `  x )  =  0 )
 )  ->  B  =  0 )   =>    |-  ( ph  ->  sum_ x  e.  ( 1 ... ( |_ `  A ) ) B  =  sum_ p  e.  ( ( 0 [,]
 A )  i^i  Prime )
 sum_ k  e.  (
 1 ... ( |_ `  (
 ( log `  A )  /  ( log `  p ) ) ) ) C )
 
Theorempclogsum 20454* The logarithmic analogue of pcprod 12943. The sum of the logarithms of the primes dividing  A multiplied by their powers yields the logarithm of  A. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( A  e.  NN  -> 
 sum_ p  e.  (
 ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A ) )
 
Theoremvmasum 20455* The sum of the von Mangoldt function over the divisors of  n. Equation 9.2.4 of [Shapiro], p. 328. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( A  e.  NN  -> 
 sum_ n  e.  { x  e.  NN  |  x  ||  A }  (Λ `  n )  =  ( log `  A ) )
 
Theoremlogfac2 20456* Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  = 
 sum_ k  e.  (
 1 ... ( |_ `  A ) ) ( (Λ `  k )  x.  ( |_ `  ( A  /  k ) ) ) )
 
Theoremchpval2 20457* Express the second Chebyshev function directly as a sum over the primes less than  A (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  (
 ( 0 [,] A )  i^i  Prime ) ( ( log `  p )  x.  ( |_ `  (
 ( log `  A )  /  ( log `  p ) ) ) ) )
 
Theoremchpchtsum 20458* The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ k  e.  (
 1 ... ( |_ `  A ) ) ( theta `  ( A  ^ c  ( 1  /  k
 ) ) ) )
 
Theoremchpub 20459 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  (ψ `  A )  <_  ( ( theta `  A )  +  ( ( sqr `  A )  x.  ( log `  A ) ) ) )
 
Theoremlogfacubnd 20460 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)
 |-  ( ( A  e.  RR+  /\  1  <_  A ) 
 ->  ( log `  ( ! `  ( |_ `  A ) ) )  <_  ( A  x.  ( log `  A ) ) )
 
Theoremlogfaclbnd 20461 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)
 |-  ( A  e.  RR+  ->  ( A  x.  (
 ( log `  A )  -  2 ) ) 
 <_  ( log `  ( ! `  ( |_ `  A ) ) ) )
 
Theoremlogfacbnd3 20462 Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 20463. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  ( ( A  e.  RR+  /\  1  <_  A ) 
 ->  ( abs `  (
 ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  (
 ( log `  A )  -  1 ) ) ) )  <_  (
 ( log `  A )  +  1 ) )
 
Theoremlogfacrlim 20463 Combine the estimates logfacubnd 20460 and logfaclbnd 20461, to get  log ( x ! )  =  x log x  +  O
( x ). Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement,  log ( x ! )  =  x log x  -  x  +  O ( log x
). (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)
 |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1
 
Theoremlogexprlim 20464* The sum  sum_ n  <_  x ,  log ^ N
( x  /  n
) has the asymptotic expansion  ( N ! ) x  +  o ( x ). (More precisely, the omitted term has order  O ( log
^ N ( x )  /  x ).) (Contributed by Mario Carneiro, 22-May-2016.)
 |-  ( N  e.  NN0  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( log `  ( x  /  n ) ) ^ N )  /  x ) )  ~~> r  ( ! `
  N ) )
 
Theoremlogfacrlim2 20465* Write out logfacrlim 20463 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.)
 |-  ( x  e.  RR+  |->  sum_
 n  e.  ( 1
 ... ( |_ `  x ) ) ( ( log `  ( x  /  n ) )  /  x ) )  ~~> r  1
 
13.4.5  Perfect Number Theorem
 
Theoremmersenne 20466 A Mersenne prime is a prime number of the form  2 ^ P  -  1. This theorem shows that the  P in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime ) 
 ->  P  e.  Prime )
 
Theoremperfect1 20467 Euclid's contribution to the Euclid-Euler theorem. A number of the form  2 ^ (
p  -  1 )  x.  ( 2 ^ p  -  1 ) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime ) 
 ->  ( 1  sigma  ( ( 2 ^ ( P  -  1 ) )  x.  ( ( 2 ^ P )  -  1 ) ) )  =  ( ( 2 ^ P )  x.  ( ( 2 ^ P )  -  1
 ) ) )
 
Theoremperfectlem1 20468 Lemma for perfect 20470. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  -.  2  ||  B )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( ( 2 ^ ( A  +  1 )
 )  e.  NN  /\  ( ( 2 ^
 ( A  +  1 ) )  -  1
 )  e.  NN  /\  ( B  /  (
 ( 2 ^ ( A  +  1 )
 )  -  1 ) )  e.  NN )
 )
 
Theoremperfectlem2 20469 Lemma for perfect 20470. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  -.  2  ||  B )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( B  e.  Prime  /\  B  =  ( ( 2 ^
 ( A  +  1 ) )  -  1
 ) ) )
 
Theoremperfect 20470* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer  N is a perfect number (that is, its divisor sum is  2 N) if and only if it is of the form  2 ^ ( p  - 
1 )  x.  (
2 ^ p  - 
1 ), where  2 ^ p  -  1 is prime (a Mersenne prime). (It follows from this that  p is also prime.) (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ( N  e.  NN  /\  2  ||  N )  ->  ( ( 1 
 sigma  N )  =  ( 2  x.  N )  <->  E. p  e.  ZZ  ( ( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) ) )
 
13.4.6  Characters of Z/nZ
 
Syntaxcdchr 20471 Extend class notation with the group of Dirichlet characters.
 class DChr
 
Definitiondf-dchr 20472* The group of Dirichlet characters 
mod  n is the set of monoid homomorphisms from  ZZ  /  n ZZ to the multiplicative monoid of the complexes, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |- DChr  =  ( n  e.  NN  |->  [_ (ℤ/n `  n )  /  z ]_ [_ { x  e.  ( (mulGrp `  z
 ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
 )  \  (Unit `  z
 ) )  X.  {
 0 } )  C_  x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( b  X.  b ) ) >. } )
 
Theoremdchrval 20473* Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  D  =  { x  e.  (
 (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U )  X.  { 0 } )  C_  x }
 )   =>    |-  ( ph  ->  G  =  { <. ( Base `  ndx ) ,  D >. , 
 <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) ) >. } )
 
Theoremdchrbas 20474* Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  D  =  { x  e.  (
 (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U )  X.  { 0 } )  C_  x }
 )
 
Theoremdchrelbas 20475 A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  ( ( B  \  U )  X.  { 0 } )  C_  X )
 ) )
 
Theoremdchrelbas2 20476* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  B  ( ( X `
  x )  =/=  0  ->  x  e.  U ) ) ) )
 
Theoremdchrelbas3 20477* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  ( X  e.  D  <->  ( X : B
 --> CC  /\  ( A. x  e.  U  A. y  e.  U  ( X `  ( x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) )  /\  ( X `  ( 1r
 `  Z ) )  =  1  /\  A. x  e.  B  (
 ( X `  x )  =/=  0  ->  x  e.  U ) ) ) ) )
 
Theoremdchrelbasd 20478* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   &    |-  ( k  =  x  ->  X  =  A )   &    |-  ( k  =  y  ->  X  =  C )   &    |-  ( k  =  ( x ( .r
 `  Z ) y )  ->  X  =  E )   &    |-  ( k  =  ( 1r `  Z )  ->  X  =  Y )   &    |-  ( ( ph  /\  k  e.  U )  ->  X  e.  CC )   &    |-  ( ( ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  E  =  ( A  x.  C ) )   &    |-  ( ph  ->  Y  =  1 )   =>    |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  X ,  0 ) )  e.  D )
 
Theoremdchrrcl 20479 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  ( X  e.  D  ->  N  e.  NN )
 
Theoremdchrmhm 20480 A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  D  C_  (
 (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
 
Theoremdchrf 20481 A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  X : B --> CC )
 
Theoremdchrelbas4 20482* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   =>    |-  ( X  e.  D  <->  ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
 1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) )
 
Theoremdchrzrh1 20483 Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  ( X `  ( L `  1 ) )  =  1 )
 
Theoremdchrzrhcl 20484 A Dirichlet character takes values in the complexes. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  ( X `  ( L `
  A ) )  e.  CC )
 
Theoremdchrzrhmul 20485 A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   =>    |-  ( ph  ->  ( X `  ( L `  ( A  x.  C ) ) )  =  ( ( X `  ( L `  A ) )  x.  ( X `
  ( L `  C ) ) ) )
 
Theoremdchrplusg 20486 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  .x.  =  (  o F  x.  |`  ( D  X.  D ) ) )
 
Theoremdchrmul 20487 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  ( X  .x.  Y )  =  ( X  o F  x.  Y ) )
 
Theoremdchrmulcl 20488 Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  D )
 
Theoremdchrn0 20489 A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  (
 ( X `  A )  =/=  0  <->  A  e.  U ) )
 
Theoremdchr1cl 20490* Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  .1. 
 e.  D )
 
Theoremdchrmulid2 20491* Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (  .1.  .x.  X )  =  X )
 
Theoremdchrinvcl 20492* Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  K  =  ( k  e.  B  |->  if (
 k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )   =>    |-  ( ph  ->  ( K  e.  D  /\  ( K 
 .x.  X )  =  .1.  ) )
 
Theoremdchrabl 20493 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   =>    |-  ( N  e.  NN  ->  G  e.  Abel )
 
Theoremdchrfi 20494 The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  ( N  e.  NN  ->  D  e.  Fin )
 
Theoremdchrghm 20495 A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  U  =  (Unit `  Z )   &    |-  H  =  ( (mulGrp `  Z )s  U )   &    |-  M  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )
 
Theoremdchr1 20496 Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  .1.  =  ( 0g `  G )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  (  .1.  `  A )  =  1 )
 
Theoremdchreq 20497* A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  ( X  =  Y  <->  A. k  e.  U  ( X `  k )  =  ( Y `  k ) ) )
 
Theoremdchrresb 20498 A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  |`  U )  =  ( Y  |`  U )  <->  X  =  Y )
 )
 
Theoremdchrabs 20499 A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  Z  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( abs `  ( X `  A ) )  =  1 )
 
Theoremdchrinv 20500 The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of  X are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  I  =  ( inv
 g `  G )   =>    |-  ( ph  ->  ( I `  X )  =  ( *  o.  X ) )
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