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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.)
π

Theoremmuval 20202* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmuval1 20203 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)

Theoremmuval2 20204* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremisnsqf 20205* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremissqf 20206* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremsqfpc 20207 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremdvdssqf 20208 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremsqf11 20209* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremmuf 20210 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmucl 20211 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremsgmval 20212* The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)

Theoremsgmval2 20213* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)

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.)

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.)

Theoremsgmcl 20216 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremsgmnncl 20217 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremmule1 20218 The Möbius function takes on values in magnitude at most . (Together with mucl 20211, this implies that it takes a value in for every natural number.) (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchtfl 20219 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchpfl 20220 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ ψ

Theoremppiprm 20221 The prime pi function at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
π π

Theoremppinprm 20222 The prime pi function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
π π

Theoremchtprm 20223 The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchtnprm 20224 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)

Theoremchpp1 20225 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ ψ Λ

Theoremchtwordi 20226 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchpwordi 20227 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ ψ

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.)

Theoremefchtdvds 20229 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppifl 20230 The prime pi function does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
π π

Theoremppip1le 20231 The prime pi function cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
π π

Theoremppiwordi 20232 The prime pi function is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
π π

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.)
π π

Theoremppi1 20234 The prime pi function at . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremcht1 20235 The Chebyshev function at . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremvma1 20236 The von Mangoldt function at . (Contributed by Mario Carneiro, 9-Apr-2016.)
Λ

Theoremchp1 20237 The second Chebyshev function at . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theoremppi1i 20238 Inference form of ppiprm 20221. (Contributed by Mario Carneiro, 21-Sep-2014.)
π               π

Theoremppi2i 20239 Inference form of ppinprm 20222. (Contributed by Mario Carneiro, 21-Sep-2014.)
π               π

Theoremppi2 20240 The prime pi function at . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremppi3 20241 The prime pi function at . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremcht2 20242 The Chebyshev function at . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremcht3 20243 The Chebyshev function at . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppinncl 20244 Closure of the prime pi function in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchtrpcl 20245 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppieq0 20246 The prime pi function is zero iff its argument is less than . (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremppiltx 20247 The prime pi function is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremprmorcht 20248 Relate the primorial (product of the first primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmumullem1 20249 Lemma for mumul 20251. A multiple of a non-squarefree number is non-squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremmumullem2 20250 Lemma for mumul 20251. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

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.)

Theoremsqff1o 20252* 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 20253* The complement of a divisor of is also a divisor of . (Contributed by Mario Carneiro, 2-Jul-2015.)

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.)

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.)

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.)

Theoremfsumdvdscom 20257* A double commutation of divisor sums based on fsumdvdsdiag 20256. Note that depends on both and . (Contributed by Mario Carneiro, 13-May-2016.)

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.)

Theoremdvdsflf1o 20259* 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 20260* A sum commutation from to . (Contributed by Mario Carneiro, 4-May-2016.)

Theoremfsumfldivdiaglem 20261* Lemma for fsumfldivdiag 20262. (Contributed by Mario Carneiro, 10-May-2016.)

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 such that . (Contributed by Mario Carneiro, 4-May-2016.)

Theoremmusum 20263* 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 20265. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremmusumsum 20264* Evaluate a collapsing sum over the Möbius function. (Contributed by Mario Carneiro, 4-May-2016.)

Theoremmuinv 20265* 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 20266* 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 20267* 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 20268* The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)

Theorem0sgmppw 20269 A prime power has divisors. (Contributed by Mario Carneiro, 17-May-2016.)

Theorem1sgmprm 20270 The sum of divisors for a prime is because the only divisors are and . (Contributed by Mario Carneiro, 17-May-2016.)

Theorem1sgm2ppw 20271 The sum of the divisors of . (Contributed by Mario Carneiro, 17-May-2016.)

Theoremsgmmul 20272 The divisor function for fixed parameter is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremppiublem1 20273 Lemma for ppiub 20275. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremppiublem2 20274 A prime greater than does not divide or , so its residue is or . (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremppiub 20275 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 20276 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremchtlepsi 20277 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchprpcl 20278 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpeq0 20279 The second Chebyshev function is zero iff its argument is less than . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theoremchteq0 20280 The first Chebyshev function is zero iff its argument is less than . (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremchtleppi 20281 Upper bound on the function. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtublem 20282 Lemma for chtub 20283. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremchtub 20283 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)

Theoremfsumvma 20284* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
Λ

Theoremfsumvma2 20285* Apply fsumvma 20284 for the common case of all numbers less than a real number . (Contributed by Mario Carneiro, 30-Apr-2016.)
Λ

Theorempclogsum 20286* The logarithmic analogue of pcprod 12817. 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 20287* 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 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.)
Λ

Theoremchpval2 20289* 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 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.)
ψ

Theoremchpub 20291 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremlogfacubnd 20292 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfaclbnd 20293 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfacbnd3 20294 Show the stronger statement alluded to in logfacrlim 20295. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremlogfacrlim 20295 Combine the estimates logfacubnd 20292 and logfaclbnd 20293, 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 20296* The sum has the asymptotic expansion . (More precisely, the omitted term has order .) (Contributed by Mario Carneiro, 22-May-2016.)

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.)

13.4.5  Perfect Number Theorem

Theoremmersenne 20298 A Mersenne prime is a prime number of the form . This theorem shows that the in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremperfect1 20299 Euclid's contribution to the Euclid-Euler theorem. A number of the form is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremperfectlem1 20300 Lemma for perfect 20302. (Contributed by Mario Carneiro, 7-Jun-2016.)

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