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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremamgm 20301 Inequality of arithmetic and geometric means. Here g calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements together), and fld g calculates the group sum in the additive group (i.e. the sum of the elements). (Contributed by Mario Carneiro, 20-Jun-2015.)
mulGrpfld       g fld g

13.3.12  Euler-Mascheroni constant

Syntaxcem 20302 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)

Definitiondf-em 20303 Define the Euler-Macheroni constant, 0.577... . This is the limit of the series , with a proof that the limit exists in emcl 20312. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremlogdifbnd 20304 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)

Theorememcllem1 20305* Lemma for emcl 20312. The series and are sequences of real numbers that approach from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem2 20306* Lemma for emcl 20312. is increasing, and is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem3 20307* Lemma for emcl 20312. The function is the difference between and . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem4 20308* Lemma for emcl 20312. The difference between series and tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem5 20309* Lemma for emcl 20312. The partial sums of the series , which is used in the definition df-em 20303, is in fact the same as . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem6 20310* Lemma for emcl 20312. By the previous lemmas, and must approach a common limit, which is by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem7 20311* Lemma for emcl 20312 and harmonicbnd 20313. Derive bounds on as and . (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

Theorememcl 20312 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd 20313* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremharmonicbnd2 20314* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theorememre 20315 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememgt0 20316 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd3 20317* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmoniclbnd 20318* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicubnd 20319* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicbnd4 20320* The asymptotic behavior of . (Contributed by Mario Carneiro, 14-May-2016.)

Theoremfsumharmonic 20321* Bound a finite sum based on the harmonic series, where the "strong" bound only applies asymptotically, and there is a "weak" bound for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)

13.4  Basic number theory

13.4.1  Wilson's theorem

Theoremwilthlem1 20322 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in are and . (Note that from prmdiveq 12870, is the modular inverse of in . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremwilthlem2 20323* Lemma for wilth 20325: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from to in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except and , and so each pair multiplies to , and and multiply to , so the full product is equal to . Here we make this precise by doing the product pair by pair.

The induction hypothesis says that every subset of that is closed under inverse (i.e. all pairs are matched up) and contains multiplies to . Given such a set, we take out one element . If there are no such elements, then which forms the base case. Otherwise, is also closed under inverse and contains , so the induction hypothesis says that this equals ; and the remaining two elements are either equal to each other, in which case wilthlem1 20322 gives that or , and we've already excluded the second case, so the product gives ; or and their product is . In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.)

mulGrpfld                            g        g

Theoremwilthlem3 20324* Lemma for wilth 20325. Here we round out the argument of wilthlem2 20323 with the final step of the induction. The induction argument shows that every subset of that is closed under inverse and contains multiplies to , and clearly itself is such a set. Thus, the product of all the elements is , and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.)
mulGrpfld

Theoremwilth 20325 Wilson's theorem. A number is prime iff it is greater or equal to and is congruent to , , or alternatively if divides . In this part of the proof we show the relatively simple reverse implication; see wilthlem3 20324 for the forward implication. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

13.4.2  The Fundamental Theorem of Algebra

Theoremftalem1 20326* Lemma for fta 20333: "growth lemma". There exists some such that is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem2 20327* Lemma for fta 20333. There exists some such that has magnitude greater than outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem3 20328* Lemma for fta 20333. There exists a global minimum of the function . The proof uses a circle of radius where is the value coming from ftalem1 20326; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly                     fld

Theoremftalem4 20329* Lemma for fta 20333: Closure of the auxiliary variables for ftalem5 20330. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem5 20330* Lemma for fta 20333: Main proof. We have already shifted the minimum found in ftalem3 20328 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let be the lowest term in the polynomial that is nonzero, and let be a -th root of . Then an evaluation of where is a sufficiently small positive number yields for the first term and for the -th term, and all higher terms are bounded because is small. Thus, , in contradiction to our choice of as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem6 20331* Lemma for fta 20333: Discharge the auxiliary variables in ftalem5 20330. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem7 20332* Lemma for fta 20333. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremfta 20333* The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. (Contributed by Mario Carneiro, 15-Sep-2014.)
Poly deg

13.4.3  The Basel problem (ζ(2) = π2/6)

Theorembasellem1 20334 Lemma for basel 20343. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.)

Theorembasellem2 20335* Lemma for basel 20343. Show that is a polynomial of degree , and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
Poly deg coeff

Theorembasellem3 20336* Lemma for basel 20343. Using the binomial theorem and de Moivre's formula, we have the identity , so taking imaginary parts yields , where . (Contributed by Mario Carneiro, 30-Jul-2014.)

Theorembasellem4 20337* Lemma for basel 20343. By basellem3 20336, the expression goes to zero whenever for some , so this function enumerates distinct roots of a degree- polynomial, which must therefore be all the roots by fta1 19704. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem5 20338* Lemma for basel 20343. Using vieta1 19708, we can calculate the sum of the roots of as the quotient of the top two coefficients, and since the function enumerates the roots, we are left with an equation that sums the function at the different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theorembasellem6 20339 Lemma for basel 20343. The function goes to zero because it is bounded by . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem7 20340 Lemma for basel 20343. The function for any fixed goes to . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem8 20341* Lemma for basel 20343. The function of partial sums of the inverse squares is bounded below by and above by , obtained by summing the inequality over the roots of the polynomial , and applying the identity basellem5 20338. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theorembasellem9 20342* Lemma for basel 20343. Since by basellem8 20341 is bounded by two expressions that tend to , must also go to by the squeeze theorem climsqz 12130. But the series is exactly the partial sums of , so it follows that this is also the value of the infinite sum . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasel 20343 The sum of the inverse squares is . This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). (Contributed by Mario Carneiro, 30-Jul-2014.)

13.4.4  Number-theoretical functions

Syntaxccht 20344 Extend class notation with the first Chebyshev function.

Syntaxcvma 20345 Extend class notation with the von Mangoldt function.
Λ

Syntaxcchp 20346 Extend class notation with the second Chebyshev function.
ψ

Syntaxcppi 20347 Extend class notation with the prime Pi function.
π

Syntaxcmu 20348 Extend class notation with the Möbius function.

Syntaxcsgm 20349 Extend class notation with the divisor function.

Definitiondf-cht 20350* Define the first Chebyshev function, which adds up the logarithms of all primes less than . The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Definitiondf-vma 20351* Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Definitiondf-chp 20352* Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than . (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ Λ

Definitiondf-ppi 20353 Define the prime π function, which counts the number of primes less than or equal to . (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Definitiondf-mu 20354* Define the Möbius function, which is zero for non-squarefree numbers and is or for squarefree numbers according as to the number of prime divisors of the number is even or odd. (Contributed by Mario Carneiro, 22-Sep-2014.)

Definitiondf-sgm 20355* Define the divisor function, which counts the number of divisors of , to the power . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremefnnfsumcl 20356* Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)

Theoremppisval 20357 The set of primes less than expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppisval2 20358 The set of primes less than expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppifi 20359 The set of primes less than is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremsgmss 20360* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremprmdvdsfi 20361* The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)

Theoremchtf 20362 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremchtcl 20363 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremchtval 20364* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremefchtcl 20365 The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)

Theoremchtge0 20366 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremvmaval 20367* Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremisppw 20368* Two ways to say that is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremisppw2 20369* Two ways to say that is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmappw 20370 Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmaprm 20371 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmacl 20372 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmaf 20373 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremefvmacl 20374 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmage0 20375 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremchpval 20376* Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ Λ

Theoremchpf 20377 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchpcl 20378 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremefchpcl 20379 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchpge0 20380 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremppival 20381 Value of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Theoremppival2 20382 Value of the prime pi function. (Contributed by Mario Carneiro, 18-Sep-2014.)
π

Theoremppival2g 20383 Value of the prime pi function. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremppif 20384 Domain and range of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Theoremppicl 20385 Real closure of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Theoremmuval 20386* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmuval1 20387 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)

Theoremmuval2 20388* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremisnsqf 20389* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremissqf 20390* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremsqfpc 20391 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremdvdssqf 20392 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremsqf11 20393* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)

Theoremmuf 20394 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremmucl 20395 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremsgmval 20396* The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)

Theoremsgmval2 20397* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theorem0sgm 20398* 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 20399 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 20400 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)

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