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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremm1lgs 20601 The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime iff mod 4. (Contributed by Mario Carneiro, 19-Jun-2015.)

13.4.10  All primes 4n+1 are the sum of two squares

Theorem2sqlem1 20602* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem2 20603* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremmul2sq 20604 Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem3 20605 Lemma for 2sqlem5 20607. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem4 20606 Lemma for 2sqlem5 20607. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem5 20607 Lemma for 2sq 20615. If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem6 20608* Lemma for 2sq 20615. If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem7 20609* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem8a 20610* Lemma for 2sqlem8 20611. (Contributed by Mario Carneiro, 4-Jun-2016.)

Theorem2sqlem8 20611* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem9 20612* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem10 20613* Lemma for 2sq 20615. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem11 20614* Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sq 20615* All primes of the form are sums of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqblem 20616 The converse to 2sq 20615. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqb 20617* The converse to 2sq 20615. (Contributed by Mario Carneiro, 20-Jun-2015.)

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

Theoremchebbnd1lem1 20618 Lemma for chebbnd1 20621: show a lower bound on π at even integers using similar techniques to those used to prove bpos 20532. (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 20523, 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 20519 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
π

Theoremchebbnd1lem2 20619 Lemma for chebbnd1 20621: Show that does not change too much between and . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd1lem3 20620 Lemma for chebbnd1 20621: get a lower bound on π that is independent of . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchebbnd1 20621 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 20622 Lemma for chtppilim 20624. (Contributed by Mario Carneiro, 22-Sep-2014.)
π        π

Theoremchtppilimlem2 20623* Lemma for chtppilim 20624. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilim 20624 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 20625 The function is upper bounded by a linear term. Corollary of chtub 20451. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd2 20626 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 20627 The function is lower bounded by a linear term. Corollary of chebbnd1 20621. (Contributed by Mario Carneiro, 8-Apr-2016.)

Theoremchpchtlim 20628 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 20629 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ

Theoremchpo1ubb 20630* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
ψ

Theoremvmadivsum 20631* 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 20632* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremrplogsumlem1 20633* Lemma for rplogsum 20676. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremrplogsumlem2 20634* Lemma for rplogsum 20676. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Λ

Theoremdchrisum0lem1a 20635 Lemma for dchrisum0lem1 20665. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremrpvmasumlem 20636* Lemma for rpvmasum 20675. 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 20637* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumlem1 20638* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^        ..^

Theoremdchrisumlem2 20639* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisumlem3 20640* Lemma for dchrisum 20641. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisum 20641* 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 20642* Lemma for dchrmusum 20673 and dchrisumn0 20670. Apply dchrisum 20641 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 20643* 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 20644* 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 20645* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 20644. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 20646* Combine the results of dchrvmasumlem1 20644 and dchrvmasum2lem 20645 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 20647* Lemma for dchrvmasum 20674. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 20648* Lemma for dchrvmasum 20674. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 20649* Lemma for dchrvmasum 20674 and dchrvmasumif 20652. Apply dchrisum 20641 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 20650* Lemma for dchrvmasumif 20652. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 20651* Lemma for dchrvmasum 20674. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 20652* 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 20674.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrvmaeq0 20653* 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 20654* Value of the function , the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fmul 20655* 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 20656* The function is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem1 20657* Lemma for dchrisum0flb 20659. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem2 20658* Lemma for dchrisum0flb 20659. 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 20659* 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 20660* 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 20661* A partial result along the lines of rpvmasum 20675. 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 20662* 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 20663* Lemma for dchrisum0 20669. Apply dchrisum 20641 for the function . (Contributed by Mario Carneiro, 10-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1b 20664* Lemma for dchrisum0lem1 20665. (Contributed by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1 20665* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2a 20666* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2 20667* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem3 20668* Lemma for dchrisum0 20669. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0 20669* 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 20643 and dchrvmasumif 20652. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumn0 20670* 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 20643 and dchrvmasumif 20652. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlem 20671* 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 20672* 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 20673* 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 20674* 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 20675* 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 20676* 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 20677 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 20678* 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 20679* Asymptotic formula for . Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsumlem 20680* Lemma for mulogsum 20681. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsum 20681* Asymptotic formula for . Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremlogdivsum 20682* Asymptotic analysis of . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem1 20683* Asymptotic formula for , with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem2 20684* Lemma for mulog2sum 20686. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremmulog2sumlem3 20685* Lemma for mulog2sum 20686. (Contributed by Mario Carneiro, 13-May-2016.)

Theoremmulog2sum 20686* Asymptotic formula for . Equation 10.2.8 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremvmalogdivsum2 20687* The sum Λ is asymptotic to . Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremvmalogdivsum 20688* 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 20690* The sum ΛΛ is asymptotic to . (Contributed by Mario Carneiro, 30-May-2016.)
Λ Λ

Theoremlogsqvma 20691* 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 20692* The Möbius inverse of logsqvma 20691. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Λ Λ Λ

Theoremlog2sumbnd 20693* Bound on the difference between and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem1 20694* Lemma for selberg 20697. Estimation of the asymptotic part of selberglem3 20696. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem2 20695* Lemma for selberg 20697. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberglem3 20696* Lemma for selberg 20697. Estimation of the left hand side of logsqvma2 20692. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberg 20697* 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 20698* Convert eventual boundedness in selberg 20697 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 20699* Lemma for selberg2 20700. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Λ ψ

Theoremselberg2 20700* 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

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