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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdchrmusumlema 23401* Lemma for dchrmusum 23432 and dchrisumn0 23429. Apply dchrisum 23400 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 23402* 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 23403* 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 23404* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 23403. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 23405* Combine the results of dchrvmasumlem1 23403 and dchrvmasum2lem 23404 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 23406* Lemma for dchrvmasum 23433. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 23407* Lemma for dchrvmasum 23433. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 23408* Lemma for dchrvmasum 23433 and dchrvmasumif 23411. Apply dchrisum 23400 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 23409* Lemma for dchrvmasumif 23411. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 23410* Lemma for dchrvmasum 23433. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 23411* 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 23433.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrvmaeq0 23412* 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 23413* Value of the function , the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fmul 23414* 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 23415* The function is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem1 23416* Lemma for dchrisum0flb 23418. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem2 23417* Lemma for dchrisum0flb 23418. 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 23418* 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 23419* 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 23420* A partial result along the lines of rpvmasum 23434. 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 23421* 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 23422* Lemma for dchrisum0 23428. Apply dchrisum 23400 for the function . (Contributed by Mario Carneiro, 10-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1b 23423* Lemma for dchrisum0lem1 23424. (Contributed by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1 23424* Lemma for dchrisum0 23428. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2a 23425* Lemma for dchrisum0 23428. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2 23426* Lemma for dchrisum0 23428. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem3 23427* Lemma for dchrisum0 23428. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0 23428* 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 23402 and dchrvmasumif 23411. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumn0 23429* 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 23402 and dchrvmasumif 23411. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlem 23430* 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 23431* 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 23432* 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 23433* 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 23434* 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 23435* 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 23436 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 23437* 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. This is Metamath 100 proof #48. (Contributed by Mario Carneiro, 12-May-2016.)

14.4.12  The Prime Number Theorem

Theoremmudivsum 23438* Asymptotic formula for . Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsumlem 23439* Lemma for mulogsum 23440. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsum 23440* Asymptotic formula for . Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremlogdivsum 23441* Asymptotic analysis of . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem1 23442* Asymptotic formula for , with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem2 23443* Lemma for mulog2sum 23445. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremmulog2sumlem3 23444* Lemma for mulog2sum 23445. (Contributed by Mario Carneiro, 13-May-2016.)

Theoremmulog2sum 23445* Asymptotic formula for . Equation 10.2.8 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremvmalogdivsum2 23446* The sum Λ is asymptotic to . Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremvmalogdivsum 23447* 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 23449* The sum ΛΛ is asymptotic to . (Contributed by Mario Carneiro, 30-May-2016.)
Λ Λ

Theoremlogsqvma 23450* 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 23451* The Möbius inverse of logsqvma 23450. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Λ Λ Λ

Theoremlog2sumbnd 23452* Bound on the difference between and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem1 23453* Lemma for selberg 23456. Estimation of the asymptotic part of selberglem3 23455. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem2 23454* Lemma for selberg 23456. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberglem3 23455* Lemma for selberg 23456. Estimation of the left-hand side of logsqvma2 23451. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberg 23456* 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 23457* Convert eventual boundedness in selberg 23456 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 23458* Lemma for selberg2 23459. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Λ ψ

Theoremselberg2 23459* 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 23460* Convert eventual boundedness in selberg2 23459 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 23461* Lemma for chpdifbnd 23463. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ                             ψ ψ

Theoremchpdifbndlem2 23462* Lemma for chpdifbnd 23463. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ               ψ ψ

Theoremchpdifbnd 23463* 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 23464* A bound on a sum of logs, used in pntlemk 23514. This is not as precise as logdivsum 23441 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 23465* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 23459 (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 23466* Lemma for selberg3 23467. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ Λ ψ

Theoremselberg3 23467* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 23459 (written here as Λψ ). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ ψ

Theoremselberg4lem1 23468* Lemma for selberg4 23469. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ Λ ψ

Theoremselberg4 23469* 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 23470* Define the residual of the second Chebyshev function. The goal is to have , or . (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ        ψ

Theorempntrf 23471 Functionality of the residual. Lemma for pnt 23522. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theorempntrmax 23472* There is a bound on the residual valid for all . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theorempntrsumo1 23473* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd 23474* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd2 23475* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ

Theoremselbergr 23476* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ        Λ

Theoremselberg3r 23477* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.)
ψ        Λ

Theoremselberg4r 23478* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.)
ψ        Λ Λ

Theoremselberg34r 23479* The sum of selberg3r 23477 and selberg4r 23478. (Contributed by Mario Carneiro, 31-May-2016.)
ψ        Λ Λ Λ

Theorempntsval 23480* Define the "Selberg function", whose asymptotic behavior is the content of selberg 23456. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        Λ ψ

Theorempntsf 23481* Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theoremselbergs 23482* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theoremselbergsb 23483* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theorempntsval2 23484* The Selberg function can be expressed using the convolution product of the von Mangoldt function with itself. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        Λ Λ Λ

Theorempntrlog2bndlem1 23485* The sum of selberg3r 23477 and selberg4r 23478. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem2 23486* Lemma for pntrlog2bnd 23492. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ               ψ

Theorempntrlog2bndlem3 23487* Lemma for pntrlog2bnd 23492. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem4 23488* Lemma for pntrlog2bnd 23492. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem5 23489* Lemma for pntrlog2bnd 23492. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem6a 23490* Lemma for pntrlog2bndlem6 23491. (Contributed by Mario Carneiro, 7-Jun-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem6 23491* Lemma for pntrlog2bnd 23492. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bnd 23492* A bound on . Equation 10.6.15 of [Shapiro], p. 431. (Contributed by Mario Carneiro, 1-Jun-2016.)
ψ

Theorempntpbnd1a 23493* Lemma for pntpbnd 23496. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ

Theorempntpbnd1 23494* Lemma for pntpbnd 23496. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ

Theorempntpbnd2 23495* Lemma for pntpbnd 23496. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ

Theorempntpbnd 23496* Lemma for pnt 23522. Establish smallness of at a point. Lemma 10.6.1 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntibndlem1 23497 Lemma for pntibnd 23501. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntibndlem2a 23498* Lemma for pntibndlem2 23499. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ

Theorempntibndlem2 23499* Lemma for pntibnd 23501. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ                                                                              ψ ψ

Theorempntibndlem3 23500* Lemma for pntibnd 23501. Package up pntibndlem2 23499 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

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