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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2sqb 21101* The converse to 2sq 21099. (Contributed by Mario Carneiro, 20-Jun-2015.)

13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem

Theoremchebbnd1lem1 21102 Lemma for chebbnd1 21105: show a lower bound on π at even integers using similar techniques to those used to prove bpos 21016. (Note that the expression is actually equal to , but proving that is not necessary for the proof, and it's too much work.) The key to the proof is bposlem1 21007, which shows that each term in the expansion is at most , so that the sum really only has nonzero elements up to , and since each term is at most , after taking logs we get the inequality π , and bclbnd 21003 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
π

Theoremchebbnd1lem2 21103 Lemma for chebbnd1 21105: Show that does not change too much between and . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd1lem3 21104 Lemma for chebbnd1 21105: get a lower bound on π that is independent of . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchebbnd1 21105 The Chebyshev bound: The function π is eventually lower bounded by a positive constant times . Alternatively stated, the function π is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilimlem1 21106 Lemma for chtppilim 21108. (Contributed by Mario Carneiro, 22-Sep-2014.)
π        π

Theoremchtppilimlem2 21107* Lemma for chtppilim 21108. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilim 21108 The function is asymptotic to π, so it is sufficient to prove to establish the PNT. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchto1ub 21109 The function is upper bounded by a linear term. Corollary of chtub 20935. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd2 21110 The Chebyshev bound, part 2: The function π is eventually upper bounded by a positive constant times . Alternatively stated, the function π is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchto1lb 21111 The function is lower bounded by a linear term. Corollary of chebbnd1 21105. (Contributed by Mario Carneiro, 8-Apr-2016.)

Theoremchpchtlim 21112 The ψ and functions are asymptotic to each other, so is sufficient to prove either or ψ to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpo1ub 21113 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ

Theoremchpo1ubb 21114* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
ψ

Theoremvmadivsum 21115* The sum of the von Mangoldt function over is asymptotic to . Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)
Λ

Theoremvmadivsumb 21116* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremrplogsumlem1 21117* Lemma for rplogsum 21160. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremrplogsumlem2 21118* Lemma for rplogsum 21160. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Λ

Theoremdchrisum0lem1a 21119 Lemma for dchrisum0lem1 21149. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremrpvmasumlem 21120* Lemma for rpvmasum 21159. Calculate the "trivial case" estimate Λ , where is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                     Λ

Theoremdchrisumlema 21121* Lemma for dchrisum 21125. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumlem1 21122* Lemma for dchrisum 21125. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^        ..^

Theoremdchrisumlem2 21123* Lemma for dchrisum 21125. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisumlem3 21124* Lemma for dchrisum 21125. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisum 21125* If is a positive decreasing function approaching zero, then the infinite sum is convergent, with the partial sum within of the limit . Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlema 21126* Lemma for dchrmusum 21157 and dchrisumn0 21154. Apply dchrisum 21125 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 21127* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded, provided that . Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem1 21128* An alternative expression for a Dirichlet-weighted von Mangoldt sum in terms of the Möbius function. Equation 9.4.11 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr                                          Λ

Theoremdchrvmasum2lem 21129* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 21128. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 21130* Combine the results of dchrvmasumlem1 21128 and dchrvmasum2lem 21129 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 21131* Lemma for dchrvmasum 21158. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 21132* Lemma for dchrvmasum 21158. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 21133* Lemma for dchrvmasum 21158 and dchrvmasumif 21136. Apply dchrisum 21125 for the function , which is decreasing above (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem1 21134* Lemma for dchrvmasumif 21136. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 21135* Lemma for dchrvmasum 21158. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 21136* An asymptotic approximation for the sum of Λ conditional on the value of the infinite sum . (We will later show that the case is impossible, and hence establish dchrvmasum 21158.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrvmaeq0 21137* The set is the collection of all non-principal Dirichlet characters such that the sum is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fval 21138* Value of the function , the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fmul 21139* The function , the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0ff 21140* The function is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem1 21141* Lemma for dchrisum0flb 21143. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem2 21142* Lemma for dchrisum0flb 21143. Induction over relatively prime factors, with the prime power case handled in dchrisum0flblem1 . (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               ..^

Theoremdchrisum0flb 21143* The divisor sum of a real Dirichlet character, is lower bounded by zero everywhere and one at the squares. Equation 9.4.29 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fno1 21144* The sum is divergent (i.e. not eventually bounded). Equation 9.4.30 of [Shapiro], p. 383. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremrpvmasum2 21145* A partial result along the lines of rpvmasum 21159. The sum of the von Mangoldt function over those integers (mod ) is asymptotic to , where is the number of non-principal Dirichlet characters with . Our goal is to show this set is empty. Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                            Unit                            Λ

Theoremdchrisum0re 21146* Suppose is a non-principal Dirichlet character with . Then is a real character. Part of Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lema 21147* Lemma for dchrisum0 21153. Apply dchrisum 21125 for the function . (Contributed by Mario Carneiro, 10-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1b 21148* Lemma for dchrisum0lem1 21149. (Contributed by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1 21149* Lemma for dchrisum0 21153. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2a 21150* Lemma for dchrisum0 21153. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2 21151* Lemma for dchrisum0 21153. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem3 21152* Lemma for dchrisum0 21153. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0 21153* The sum is nonzero for all non-principal Dirichlet characters (i.e. the assumption is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 21127 and dchrvmasumif 21136. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumn0 21154* The sum is nonzero for all non-principal Dirichlet characters (i.e. the assumption is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 21127 and dchrvmasumif 21136. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlem 21155* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem 21156* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrmusum 21157* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum 21158* The sum of the von Mangoldt function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.8 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr                                   Λ

Theoremrpvmasum 21159* The sum of the von Mangoldt function over those integers (mod ) is asymptotic to . Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
ℤ/n       RHom              Unit                     Λ

Theoremrplogsum 21160* The sum of over the primes (mod ) is asymptotic to . Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 16-Apr-2016.)
ℤ/n       RHom              Unit

Theoremdirith2 21161 Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to . Theorem 9.4.1 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
ℤ/n       RHom              Unit

Theoremdirith 21162* Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to . Theorem 9.4.1 of [Shapiro], p. 375. See http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html for an informal exposition. (Contributed by Mario Carneiro, 12-May-2016.)

13.4.12  The Prime Number Theorem

Theoremmudivsum 21163* Asymptotic formula for . Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsumlem 21164* Lemma for mulogsum 21165. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsum 21165* Asymptotic formula for . Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremlogdivsum 21166* Asymptotic analysis of . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem1 21167* Asymptotic formula for , with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem2 21168* Lemma for mulog2sum 21170. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremmulog2sumlem3 21169* Lemma for mulog2sum 21170. (Contributed by Mario Carneiro, 13-May-2016.)

Theoremmulog2sum 21170* Asymptotic formula for . Equation 10.2.8 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremvmalogdivsum2 21171* The sum Λ is asymptotic to . Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremvmalogdivsum 21172* The sum Λ is asymptotic to . Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Λ        Λ Λ

Theorem2vmadivsum 21174* The sum ΛΛ is asymptotic to . (Contributed by Mario Carneiro, 30-May-2016.)
Λ Λ

Theoremlogsqvma 21175* A formula for in terms of the primes. Equation 10.4.6 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Λ Λ Λ

Theoremlogsqvma2 21176* The Möbius inverse of logsqvma 21175. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Λ Λ Λ

Theoremlog2sumbnd 21177* Bound on the difference between and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem1 21178* Lemma for selberg 21181. Estimation of the asymptotic part of selberglem3 21180. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem2 21179* Lemma for selberg 21181. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberglem3 21180* Lemma for selberg 21181. Estimation of the left-hand side of logsqvma2 21176. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberg 21181* Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Λ ΛΛ . Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)
Λ ψ

Theoremselbergb 21182* Convert eventual boundedness in selberg 21181 to boundedness on . (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ

Theoremselberg2lem 21183* Lemma for selberg2 21184. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Λ ψ

Theoremselberg2 21184* Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
ψ Λ ψ

Theoremselberg2b 21185* Convert eventual boundedness in selberg2 21184 to boundedness on any interval . (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ

Theoremchpdifbndlem1 21186* Lemma for chpdifbnd 21188. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ                             ψ ψ

Theoremchpdifbndlem2 21187* Lemma for chpdifbnd 21188. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ               ψ ψ

Theoremchpdifbnd 21188* A bound on the difference of nearby ψ values. Theorem 10.5.2 of [Shapiro], p. 427. (Contributed by Mario Carneiro, 25-May-2016.)
ψ ψ

Theoremlogdivbnd 21189* A bound on a sum of logs, used in pntlemk 21239. This is not as precise as logdivsum 21166 in its asymptotic behavior, but it is valid for all and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremselberg3lem1 21190* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 21184 (written here as Λψ ). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ ψ Λ ψ

Theoremselberg3lem2 21191* Lemma for selberg3 21192. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ Λ ψ

Theoremselberg3 21192* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 21184 (written here as Λψ ). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ ψ

Theoremselberg4lem1 21193* Lemma for selberg4 21194. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ Λ ψ

Theoremselberg4 21194* The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form ΛΛΛ; we eliminate one of the nested sums by using the definition of ψ Λ. This statement can thus equivalently be written ψ ΛΛΛ . Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ Λ ψ

Theorempntrval 21195* Define the residual of the second Chebyshev function. The goal is to have , or . (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ        ψ

Theorempntrf 21196 Functionality of the residual. Lemma for pnt 21247. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theorempntrmax 21197* There is a bound on the residual valid for all . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theorempntrsumo1 21198* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd 21199* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd2 21200* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ

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