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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremppicl 20201 Real closure of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  A )  e. 
 NN0 )
 
Theoremmuval 20202* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  ( mmu `  A )  =  if ( E. p  e.  Prime  ( p ^ 2 ) 
 ||  A ,  0 ,  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A } ) ) ) )
 
Theoremmuval1 20203 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>=
 `  2 )  /\  ( P ^ 2 ) 
 ||  A )  ->  ( mmu `  A )  =  0 )
 
Theoremmuval2 20204* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
 )  ->  ( mmu `  A )  =  (
 -u 1 ^ ( # `
  { p  e. 
 Prime  |  p  ||  A } ) ) )
 
Theoremisnsqf 20205* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( A  e.  NN  ->  ( ( mmu `  A )  =  0  <->  E. p  e.  Prime  ( p ^ 2 )  ||  A ) )
 
Theoremissqf 20206* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( A  e.  NN  ->  ( ( mmu `  A )  =/=  0  <->  A. p  e.  Prime  ( p  pCnt  A )  <_ 
 1 ) )
 
Theoremsqfpc 20207 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0  /\  P  e.  Prime )  ->  ( P  pCnt  A ) 
 <_  1 )
 
Theoremdvdssqf 20208 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  B  ||  A )  ->  ( ( mmu `  A )  =/=  0  ->  ( mmu `  B )  =/=  0 ) )
 
Theoremsqf11 20209* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0 ) )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  ||  A  <->  p  ||  B ) ) )
 
Theoremmuf 20210 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |- 
 mmu : NN --> ZZ
 
Theoremmucl 20211 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  ( mmu `  A )  e.  ZZ )
 
Theoremsgmval 20212* The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  NN )  ->  ( A  sigma  B )  =  sum_ k  e.  { p  e.  NN  |  p  ||  B }  ( k  ^ c  A ) )
 
Theoremsgmval2 20213* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  sigma  B )  =  sum_ k  e.  { p  e.  NN  |  p  ||  B }  ( k ^ A ) )
 
Theorem0sgm 20214* The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( A  e.  NN  ->  ( 0  sigma  A )  =  ( # `  { p  e.  NN  |  p  ||  A } ) )
 
Theoremsgmf 20215 The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
 |- 
 sigma  : ( CC  X.  NN ) --> CC
 
Theoremsgmcl 20216 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  NN )  ->  ( A  sigma  B )  e.  CC )
 
Theoremsgmnncl 20217 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  sigma  B )  e.  NN )
 
Theoremmule1 20218 The Möbius function takes on values in magnitude at most  1. (Together with mucl 20211, 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 20219 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
 )
 
Theoremchpfl 20220 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 20221 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 20222 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 20223 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 20224 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 20225 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 20226 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 20227 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 20228* 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 20229 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 20230 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 20231 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 20232 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 20233 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 20234 The prime pi function at  1. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  1 )  =  0
 
Theoremcht1 20235 The Chebyshev function at  1. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  1 )  =  0
 
Theoremvma1 20236 The von Mangoldt function at  1. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  (Λ `  1 )  =  0
 
Theoremchp1 20237 The second Chebyshev function at  1. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  (ψ `  1 )  =  0
 
Theoremppi1i 20238 Inference form of ppiprm 20221. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (π `  M )  =  K   &    |-  N  e.  Prime   =>    |-  (π `  N )  =  ( K  +  1 )
 
Theoremppi2i 20239 Inference form of ppinprm 20222. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  M  e.  NN0   &    |-  N  =  ( M  +  1 )   &    |-  (π `  M )  =  K   &    |-  -.  N  e.  Prime   =>    |-  (π `  N )  =  K
 
Theoremppi2 20240 The prime pi function at  2. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  2 )  =  1
 
Theoremppi3 20241 The prime pi function at  3. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  (π `  3 )  =  2
 
Theoremcht2 20242 The Chebyshev function at  2. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  2 )  =  ( log `  2
 )
 
Theoremcht3 20243 The Chebyshev function at  3. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( theta `  3 )  =  ( log `  6
 )
 
Theoremppinncl 20244 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 20245 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 20246 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 20247 The prime pi function is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR+  ->  (π
 `  A )  <  A )
 
Theoremprmorcht 20248 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 20249 Lemma for mumul 20251. 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 20250 Lemma for mumul 20251. 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 20251 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 20252* 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 20253* 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 20254* 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 20255* A "diagonal commutation" of divisor sums analogous to fsum0diag 12117. (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 20256* A "diagonal commutation" of divisor sums analogous to fsum0diag 12117. (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 20257* A double commutation of divisor sums based on fsumdvdsdiag 20256. 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 20258* 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 20259* 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 20260* 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 20261* Lemma for fsumfldivdiag 20262. (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 20262* The right hand side of dvdsflsumcom 20260 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 20263* 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 20265. (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 20264* 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 20265* 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 20266* 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 20267* 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 20268* 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 20269 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 20270 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 20271 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 20272 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 20273 Lemma for ppiub 20275. (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 20274 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 20275 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 20276 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 20277 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  ( theta `  A )  <_  (ψ `  A )
 )
 
Theoremchprpcl 20278 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 20279 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 20280 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 20281 Upper bound on the  theta function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR+  ->  ( theta `  A )  <_  ( (π `  A )  x.  ( log `  A ) ) )
 
Theoremchtublem 20282 Lemma for chtub 20283. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN  ->  ( theta `  ( (
 2  x.  N )  -  1 ) ) 
 <_  ( ( theta `  N )  +  ( ( log `  4 )  x.  ( N  -  1
 ) ) ) )
 
Theoremchtub 20283 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 20284* 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 20285* Apply fsumvma 20284 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 20286* The logarithmic analogue of pcprod 12817. 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 20287* 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 20288* 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 20289* 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 20290* 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 20291 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 20292 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 20293 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 20294 Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 20295. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  ( ( A  e.  RR+  /\  1  <_  A ) 
 ->  ( abs `  (
 ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  (
 ( log `  A )  -  1 ) ) ) )  <_  (
 ( log `  A )  +  1 ) )
 
Theoremlogfacrlim 20295 Combine the estimates logfacubnd 20292 and logfaclbnd 20293, 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 20296* 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 20297* Write out logfacrlim 20295 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 20298 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 20299 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 20300 Lemma for perfect 20302. (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 )
 )
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