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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsgmnncl 20401 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  sigma  B )  e.  NN )
 
Theoremmule1 20402 The Möbius function takes on values in magnitude at most  1. (Together with mucl 20395, this implies that it takes a value in  { -u 1 ,  0 ,  1 } for every natural number.) (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) ) 
 <_  1 )
 
Theoremchtfl 20403 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
 )
 
Theoremchpfl 20404 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( A  e.  RR  ->  (ψ `  ( |_ `  A ) )  =  (ψ `  A )
 )
 
Theoremppiprm 20405 The prime pi function at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime ) 
 ->  (π `  ( A  +  1 ) )  =  ( (π `  A )  +  1 ) )
 
Theoremppinprm 20406 The prime pi function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e. 
 Prime )  ->  (π `  ( A  +  1 )
 )  =  (π `  A ) )
 
Theoremchtprm 20407 The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime ) 
 ->  ( theta `  ( A  +  1 ) )  =  ( ( theta `  A )  +  ( log `  ( A  +  1 ) ) ) )
 
Theoremchtnprm 20408 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e. 
 Prime )  ->  ( theta `  ( A  +  1 ) )  =  (
 theta `  A ) )
 
Theoremchpp1 20409 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
 |-  ( A  e.  NN0  ->  (ψ `  ( A  +  1 ) )  =  ( (ψ `  A )  +  (Λ `  ( A  +  1 )
 ) ) )
 
Theoremchtwordi 20410 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  <_  ( theta `  B )
 )
 
Theoremchpwordi 20411 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (ψ `  A )  <_  (ψ `  B )
 )
 
Theoremchtdif 20412* The difference of the Chebyshev function at two points sums the logarithms of the primes in an interval. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ( theta `  N )  -  ( theta `  M ) )  =  sum_ p  e.  ( ( ( M  +  1 )
 ... N )  i^i 
 Prime ) ( log `  p ) )
 
Theoremefchtdvds 20413 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A ) ) 
 ||  ( exp `  ( theta `  B ) ) )
 
Theoremppifl 20414 The prime pi function does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
 
Theoremppip1le 20415 The prime pi function cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  ( (π `  A )  +  1 ) )
 
Theoremppiwordi 20416 The prime pi function is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  <_  (π
 `  B ) )
 
Theoremppidif 20417 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 20418 The prime pi function at  1. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  1 )  =  0
 
Theoremcht1 20419 The Chebyshev function at  1. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  1 )  =  0
 
Theoremvma1 20420 The von Mangoldt function at  1. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  (Λ `  1 )  =  0
 
Theoremchp1 20421 The second Chebyshev function at  1. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  (ψ `  1 )  =  0
 
Theoremppi1i 20422 Inference form of ppiprm 20405. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (π `  M )  =  K   &    |-  N  e.  Prime   =>    |-  (π `  N )  =  ( K  +  1 )
 
Theoremppi2i 20423 Inference form of ppinprm 20406. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (π `  M )  =  K   &    |-  -.  N  e.  Prime   =>    |-  (π `  N )  =  K
 
Theoremppi2 20424 The prime pi function at  2. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  2 )  =  1
 
Theoremppi3 20425 The prime pi function at  3. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  3 )  =  2
 
Theoremcht2 20426 The Chebyshev function at  2. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  2 )  =  ( log `  2
 )
 
Theoremcht3 20427 The Chebyshev function at  3. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  3 )  =  ( log `  6
 )
 
Theoremppinncl 20428 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 20429 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 20430 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 20431 The prime pi function is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR+  ->  (π
 `  A )  <  A )
 
Theoremprmorcht 20432 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 20433 Lemma for mumul 20435. 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 20434 Lemma for mumul 20435. 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 20435 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 20436* 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 20437* 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 20438* 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 20439* A "diagonal commutation" of divisor sums analogous to fsum0diag 12256. (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 20440* A "diagonal commutation" of divisor sums analogous to fsum0diag 12256. (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 20441* A double commutation of divisor sums based on fsumdvdsdiag 20440. 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 20442* 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 20443* 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 20444* 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 20445* Lemma for fsumfldivdiag 20446. (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 20446* The right-hand side of dvdsflsumcom 20444 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 20447* 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 20449. (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 20448* 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 20449* 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 20450* 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 20451* 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 20452* 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 20453 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 20454 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 20455 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 20456 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 20457 Lemma for ppiub 20459. (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 20458 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 20459 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 20460 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 20461 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  ( theta `  A )  <_  (ψ `  A )
 )
 
Theoremchprpcl 20462 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 20463 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 20464 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 20465 Upper bound on the  theta function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR+  ->  ( theta `  A )  <_  ( (π `  A )  x.  ( log `  A ) ) )
 
Theoremchtublem 20466 Lemma for chtub 20467. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN  ->  ( theta `  ( (
 2  x.  N )  -  1 ) ) 
 <_  ( ( theta `  N )  +  ( ( log `  4 )  x.  ( N  -  1
 ) ) ) )
 
Theoremchtub 20467 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 20468* 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 20469* Apply fsumvma 20468 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 20470* The logarithmic analogue of pcprod 12959. 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 20471* 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 20472* 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 20473* 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 20474* 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 20475 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 20476 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 20477 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 20478 Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 20479. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  ( ( A  e.  RR+  /\  1  <_  A ) 
 ->  ( abs `  (
 ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  (
 ( log `  A )  -  1 ) ) ) )  <_  (
 ( log `  A )  +  1 ) )
 
Theoremlogfacrlim 20479 Combine the estimates logfacubnd 20476 and logfaclbnd 20477, 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 20480* 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 20481* Write out logfacrlim 20479 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 20482 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 20483 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 20484 Lemma for perfect 20486. (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 20485 Lemma for perfect 20486. (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 20486* 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 20487 Extend class notation with the group of Dirichlet characters.
 class DChr
 
Definitiondf-dchr 20488* 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 20489* 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 20490* 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 20491 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 20492* 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 20493* 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 20494* 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 20495 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 20496 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 20497 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 20498* 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 20499 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 20500 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 )
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