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

Theoremlgsqr 20601* The Legendre symbol for odd primes is iff the number is not a multiple of the prime (in which case it is , see lgsne0 20588) and the number is a quadratic residue (it is for nonresidues by the process of elimination from lgsabs1 20589). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsdchrval 20602* The Legendre symbol function , where is an odd positive number, is a Dirichlet character modulo . (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                     RHom

Theoremlgsdchr 20603* The Legendre symbol function , where is an odd positive number, is a real Dirichlet character modulo . (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                     RHom

13.4.9  Quadratic reciprocity

Theoremlgseisenlem1 20604* Lemma for lgseisen 20608. If and , then for any even , is also an even integer . To simplify these statements, we divide all the even numbers by , so that it becomes the statement that is an integer between and . (Contributed by Mario Carneiro, 17-Jun-2015.)

Theoremlgseisenlem2 20605* Lemma for lgseisen 20608. The function is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)

Theoremlgseisenlem3 20606* Lemma for lgseisen 20608. (Contributed by Mario Carneiro, 17-Jun-2015.)
ℤ/n       mulGrp       RHom       g

Theoremlgseisenlem4 20607* Lemma for lgseisen 20608. The function is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 18-Jun-2015.)
ℤ/n       mulGrp       RHom

Theoremlgseisen 20608* Eisenstein's lemma, an expression for when are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquadlem1 20609* Lemma for lgsquad 20612. Count the members of with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquadlem2 20610* Lemma for lgsquad 20612. Count the members of with even coordinates, and combine with lgsquadlem1 20609 to get the total count of lattice points in (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquadlem3 20611* Lemma for lgsquad 20612. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquad 20612 The Law of Quadratic Reciprocity. If and are distinct odd primes, then the product of the Legendre symbols and is the parity of . This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad2lem1 20613 Lemma for lgsquad2 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad2lem2 20614* Lemma for lgsquad2 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad2 20615 Extend lgsquad 20612 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad3 20616 Extend lgsquad2 20615 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremm1lgs 20617 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 20618* Lemma for 2sq 20631. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem2 20619* Lemma for 2sq 20631. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremmul2sq 20620 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 20621 Lemma for 2sqlem5 20623. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem4 20622 Lemma for 2sqlem5 20623. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem5 20623 Lemma for 2sq 20631. 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 20624* Lemma for 2sq 20631. 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 20625* Lemma for 2sq 20631. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem8a 20626* Lemma for 2sqlem8 20627. (Contributed by Mario Carneiro, 4-Jun-2016.)

Theorem2sqlem8 20627* Lemma for 2sq 20631. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem9 20628* Lemma for 2sq 20631. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem10 20629* Lemma for 2sq 20631. 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 20630* Lemma for 2sq 20631. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sq 20631* All primes of the form are sums of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqblem 20632 The converse to 2sq 20631. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqb 20633* The converse to 2sq 20631. (Contributed by Mario Carneiro, 20-Jun-2015.)

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

Theoremchebbnd1lem1 20634 Lemma for chebbnd1 20637: show a lower bound on π at even integers using similar techniques to those used to prove bpos 20548. (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 20539, 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 20535 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
π

Theoremchebbnd1lem2 20635 Lemma for chebbnd1 20637: Show that does not change too much between and . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd1lem3 20636 Lemma for chebbnd1 20637: get a lower bound on π that is independent of . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchebbnd1 20637 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 20638 Lemma for chtppilim 20640. (Contributed by Mario Carneiro, 22-Sep-2014.)
π        π

Theoremchtppilimlem2 20639* Lemma for chtppilim 20640. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilim 20640 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 20641 The function is upper bounded by a linear term. Corollary of chtub 20467. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd2 20642 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 20643 The function is lower bounded by a linear term. Corollary of chebbnd1 20637. (Contributed by Mario Carneiro, 8-Apr-2016.)

Theoremchpchtlim 20644 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 20645 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ

Theoremchpo1ubb 20646* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
ψ

Theoremvmadivsum 20647* 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 20648* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremrplogsumlem1 20649* Lemma for rplogsum 20692. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremrplogsumlem2 20650* Lemma for rplogsum 20692. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Λ

Theoremdchrisum0lem1a 20651 Lemma for dchrisum0lem1 20681. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremrpvmasumlem 20652* Lemma for rpvmasum 20691. 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 20653* Lemma for dchrisum 20657. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumlem1 20654* Lemma for dchrisum 20657. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^        ..^

Theoremdchrisumlem2 20655* Lemma for dchrisum 20657. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisumlem3 20656* Lemma for dchrisum 20657. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisum 20657* 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 20658* Lemma for dchrmusum 20689 and dchrisumn0 20686. Apply dchrisum 20657 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 20659* 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 20660* 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 20661* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 20660. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 20662* Combine the results of dchrvmasumlem1 20660 and dchrvmasum2lem 20661 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 20663* Lemma for dchrvmasum 20690. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 20664* Lemma for dchrvmasum 20690. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 20665* Lemma for dchrvmasum 20690 and dchrvmasumif 20668. Apply dchrisum 20657 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 20666* Lemma for dchrvmasumif 20668. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 20667* Lemma for dchrvmasum 20690. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 20668* 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 20690.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrvmaeq0 20669* 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 20670* Value of the function , the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fmul 20671* 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 20672* The function is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem1 20673* Lemma for dchrisum0flb 20675. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem2 20674* Lemma for dchrisum0flb 20675. 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 20675* 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 20676* 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 20677* A partial result along the lines of rpvmasum 20691. 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 20678* 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 20679* Lemma for dchrisum0 20685. Apply dchrisum 20657 for the function . (Contributed by Mario Carneiro, 10-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1b 20680* Lemma for dchrisum0lem1 20681. (Contributed by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1 20681* Lemma for dchrisum0 20685. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2a 20682* Lemma for dchrisum0 20685. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2 20683* Lemma for dchrisum0 20685. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem3 20684* Lemma for dchrisum0 20685. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0 20685* 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 20659 and dchrvmasumif 20668. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumn0 20686* 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 20659 and dchrvmasumif 20668. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlem 20687* 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 20688* 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 20689* 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 20690* 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 20691* 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 20692* 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 20693 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 20694* 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 20695* Asymptotic formula for . Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsumlem 20696* Lemma for mulogsum 20697. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsum 20697* Asymptotic formula for . Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremlogdivsum 20698* Asymptotic analysis of . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem1 20699* Asymptotic formula for , with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem2 20700* Lemma for mulog2sum 20702. (Contributed by Mario Carneiro, 19-May-2016.)

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