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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremppival2 20901 Value of the prime pi function. (Contributed by Mario Carneiro, 18-Sep-2014.)
π

Theoremppival2g 20902 Value of the prime pi function. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremppif 20903 Domain and range of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Theoremppicl 20904 Real closure of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Theoremmuval 20905* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmuval1 20906 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)

Theoremmuval2 20907* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremisnsqf 20908* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremissqf 20909* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremsqfpc 20910 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremdvdssqf 20911 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremsqf11 20912* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremmuf 20913 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmucl 20914 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremsgmval 20915* The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)

Theoremsgmval2 20916* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theorem0sgm 20917* The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremsgmf 20918 The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)

Theoremsgmcl 20919 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremsgmnncl 20920 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremmule1 20921 The Möbius function takes on values in magnitude at most . (Together with mucl 20914, this implies that it takes a value in for every natural number.) (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchtfl 20922 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchpfl 20923 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ ψ

Theoremppiprm 20924 The prime pi function at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
π π

Theoremppinprm 20925 The prime pi function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
π π

Theoremchtprm 20926 The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchtnprm 20927 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)

Theoremchpp1 20928 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ ψ Λ

Theoremchtwordi 20929 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchpwordi 20930 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ ψ

Theoremchtdif 20931* 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.)

Theoremefchtdvds 20932 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppifl 20933 The prime pi function does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
π π

Theoremppip1le 20934 The prime pi function cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
π π

Theoremppiwordi 20935 The prime pi function is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
π π

Theoremppidif 20936 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.)
π π

Theoremppi1 20937 The prime pi function at . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremcht1 20938 The Chebyshev function at . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremvma1 20939 The von Mangoldt function at . (Contributed by Mario Carneiro, 9-Apr-2016.)
Λ

Theoremchp1 20940 The second Chebyshev function at . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theoremppi1i 20941 Inference form of ppiprm 20924. (Contributed by Mario Carneiro, 21-Sep-2014.)
π               π

Theoremppi2i 20942 Inference form of ppinprm 20925. (Contributed by Mario Carneiro, 21-Sep-2014.)
π               π

Theoremppi2 20943 The prime pi function at . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremppi3 20944 The prime pi function at . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremcht2 20945 The Chebyshev function at . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremcht3 20946 The Chebyshev function at . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppinncl 20947 Closure of the prime pi function in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchtrpcl 20948 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppieq0 20949 The prime pi function is zero iff its argument is less than . (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremppiltx 20950 The prime pi function is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremprmorcht 20951 Relate the primorial (product of the first primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmumullem1 20952 Lemma for mumul 20954. A multiple of a non-squarefree number is non-squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremmumullem2 20953 Lemma for mumul 20954. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremmumul 20954 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.)

Theoremsqff1o 20955* There is a bijection from the squarefree divisors of a number to the powerset of the prime divisors of . Among other things, this implies that a number has squarefree divisors where is the number of prime divisors, and a squarefree number has divisors (because all divisors of a squarefree number are squarefree). The inverse function to takes the product of all the primes in some subset of prime divisors of . (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremdvdsdivcl 20956* The complement of a divisor of is also a divisor of . (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremdvdsflip 20957* An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)

Theoremfsumdvdsdiaglem 20958* A "diagonal commutation" of divisor sums analogous to fsum0diag 12551. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)

Theoremfsumdvdsdiag 20959* A "diagonal commutation" of divisor sums analogous to fsum0diag 12551. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)

Theoremfsumdvdscom 20960* A double commutation of divisor sums based on fsumdvdsdiag 20959. Note that depends on both and . (Contributed by Mario Carneiro, 13-May-2016.)

Theoremdvdsppwf1o 20961* A bijection from the divisors of a prime power to the integers less than the prime count. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremdvdsflf1o 20962* A bijection from the numbers less than to the multiples of less than . Useful for some sum manipulations. (Contributed by Mario Carneiro, 3-May-2016.)

Theoremdvdsflsumcom 20963* A sum commutation from to . (Contributed by Mario Carneiro, 4-May-2016.)

Theoremfsumfldivdiaglem 20964* Lemma for fsumfldivdiag 20965. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremfsumfldivdiag 20965* The right-hand side of dvdsflsumcom 20963 is commutative in the variables, because it can be written as the manifestly symmetric sum over those such that . (Contributed by Mario Carneiro, 4-May-2016.)

Theoremmusum 20966* The sum of the Möbius function over the divisors of gives one if , but otherwise always sums to zero. This makes the Möbius function useful for inverting divisor sums; see also muinv 20968. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremmusumsum 20967* Evaluate a collapsing sum over the Möbius function. (Contributed by Mario Carneiro, 4-May-2016.)

Theoremmuinv 20968* The Möbius inversion formula. If for every , then , i.e. the Möbius function is the Dirichlet convolution inverse of the constant function . (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremdvdsmulf1o 20969* If and are two coprime integers, multiplication forms a bijection from the set of pairs where and , to the set of divisors of . (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremfsumdvdsmul 20970* Product of two divisor sums. (This is also the main part of the proof that " is a multiplicative function if is".) (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremsgmppw 20971* The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)

Theorem0sgmppw 20972 A prime power has divisors. (Contributed by Mario Carneiro, 17-May-2016.)

Theorem1sgmprm 20973 The sum of divisors for a prime is because the only divisors are and . (Contributed by Mario Carneiro, 17-May-2016.)

Theorem1sgm2ppw 20974 The sum of the divisors of . (Contributed by Mario Carneiro, 17-May-2016.)

Theoremsgmmul 20975 The divisor function for fixed parameter is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremppiublem1 20976 Lemma for ppiub 20978. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremppiublem2 20977 A prime greater than does not divide or , so its residue is or . (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremppiub 20978 An upper bound on the Gauss prime function, which counts the number of primes less than . (Contributed by Mario Carneiro, 13-Mar-2014.)
π

Theoremvmalelog 20979 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremchtlepsi 20980 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchprpcl 20981 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpeq0 20982 The second Chebyshev function is zero iff its argument is less than . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theoremchteq0 20983 The first Chebyshev function is zero iff its argument is less than . (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremchtleppi 20984 Upper bound on the function. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtublem 20985 Lemma for chtub 20986. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremchtub 20986 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)

Theoremfsumvma 20987* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
Λ

Theoremfsumvma2 20988* Apply fsumvma 20987 for the common case of all numbers less than a real number . (Contributed by Mario Carneiro, 30-Apr-2016.)
Λ

Theorempclogsum 20989* The logarithmic analogue of pcprod 13254. The sum of the logarithms of the primes dividing multiplied by their powers yields the logarithm of . (Contributed by Mario Carneiro, 15-Apr-2016.)

Theoremvmasum 20990* The sum of the von Mangoldt function over the divisors of . Equation 9.2.4 of [Shapiro], p. 328. (Contributed by Mario Carneiro, 15-Apr-2016.)
Λ

Theoremlogfac2 20991* 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.)
Λ

Theoremchpval2 20992* Express the second Chebyshev function directly as a sum over the primes less than (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpchtsum 20993* 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.)
ψ

Theoremchpub 20994 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremlogfacubnd 20995 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfaclbnd 20996 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfacbnd3 20997 Show the stronger statement alluded to in logfacrlim 20998. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremlogfacrlim 20998 Combine the estimates logfacubnd 20995 and logfaclbnd 20996, to get . Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement, . (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)

Theoremlogexprlim 20999* The sum has the asymptotic expansion . (More precisely, the omitted term has order .) (Contributed by Mario Carneiro, 22-May-2016.)

Theoremlogfacrlim2 21000* Write out logfacrlim 20998 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.)

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