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Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcsgm 25601 Extend class notation with the divisor function.
class σ
 
Definitiondf-cht 25602* Define the first Chebyshev function, which adds up the logarithms of all primes less than 𝑥, see definition in [ApostolNT] p. 75. 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 df-chp 25604. See https://en.wikipedia.org/wiki/Chebyshev_function 25604 for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
 
Definitiondf-vma 25603* Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere, see definition in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0))
 
Definitiondf-chp 25604* Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than 𝑥, see definition in [ApostolNT] p. 75. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
 
Definitiondf-ppi 25605 Define the prime π function, which counts the number of primes less than or equal to 𝑥, see definition in [ApostolNT] p. 8. (Contributed by Mario Carneiro, 15-Sep-2014.)
π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
 
Definitiondf-mu 25606* Define the Möbius function, which is zero for non-squarefree numbers and is -1 or 1 for squarefree numbers according as to the number of prime divisors of the number is even or odd, see definition in [ApostolNT] p. 24. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
 
Definitiondf-sgm 25607* Define the sum of positive divisors function (𝑥 σ 𝑛), which is the sum of the xth powers of the positive integer divisors of n, see definition in [ApostolNT] p. 38. For 𝑥 = 0, (𝑥 σ 𝑛) counts the number of divisors of 𝑛, i.e. (0 σ 𝑛) is the divisor function, see remark in [ApostolNT] p. 38. (Contributed by Mario Carneiro, 22-Sep-2014.)
σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝑛} (𝑘𝑐𝑥))
 
Theoremefnnfsumcl 25608* Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → (exp‘𝐵) ∈ ℕ)       (𝜑 → (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ)
 
Theoremppisval 25609 The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ))
 
Theoremppisval2 25610 The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ))
 
Theoremppifi 25611 The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
 
Theoremprmdvdsfi 25612* The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝𝐴} ∈ Fin)
 
Theoremchtf 25613 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ:ℝ⟶ℝ
 
Theoremchtcl 25614 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)
 
Theoremchtval 25615* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
 
Theoremefchtcl 25616 The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(θ‘𝐴)) ∈ ℕ)
 
Theoremchtge0 25617 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴))
 
Theoremvmaval 25618* Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}       (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
 
Theoremisppw 25619* Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝𝐴))
 
Theoremisppw2 25620* Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝐴 = (𝑝𝑘)))
 
Theoremvmappw 25621 Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃𝐾)) = (log‘𝑃))
 
Theoremvmaprm 25622 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃))
 
Theoremvmacl 25623 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ)
 
Theoremvmaf 25624 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ:ℕ⟶ℝ
 
Theoremefvmacl 25625 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)
 
Theoremvmage0 25626 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))
 
Theoremchpval 25627* Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
 
Theoremchpf 25628 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ:ℝ⟶ℝ
 
Theoremchpcl 25629 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
 
Theoremefchpcl 25630 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)
 
Theoremchpge0 25631 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
 
Theoremppival 25632 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
 
Theoremppival2 25633 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℤ → (π𝐴) = (♯‘((2...𝐴) ∩ ℙ)))
 
Theoremppival2g 25634 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ𝑀)) → (π𝐴) = (♯‘((𝑀...𝐴) ∩ ℙ)))
 
Theoremppif 25635 Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
π:ℝ⟶ℕ0
 
Theoremppicl 25636 Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) ∈ ℕ0)
 
Theoremmuval 25637* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))))
 
Theoremmuval1 25638 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℕ ∧ 𝑃 ∈ (ℤ‘2) ∧ (𝑃↑2) ∥ 𝐴) → (μ‘𝐴) = 0)
 
Theoremmuval2 25639* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴})))
 
Theoremisnsqf 25640* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴))
 
Theoremissqf 25641* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1))
 
Theoremsqfpc 25642 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)
((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt 𝐴) ≤ 1)
 
Theoremdvdssqf 25643 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵𝐴) → ((μ‘𝐴) ≠ 0 → (μ‘𝐵) ≠ 0))
 
Theoremsqf11 25644* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)
(((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝𝐴𝑝𝐵)))
 
Theoremmuf 25645 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ:ℕ⟶ℤ
 
Theoremmucl 25646 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ)
 
Theoremsgmval 25647* The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝐵} (𝑘𝑐𝐴))
 
Theoremsgmval2 25648* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝐵} (𝑘𝐴))
 
Theorem0sgm 25649* The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝𝐴}))
 
Theoremsgmf 25650 The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
σ :(ℂ × ℕ)⟶ℂ
 
Theoremsgmcl 25651 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)
 
Theoremsgmnncl 25652 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)
 
Theoremmule1 25653 The Möbius function takes on values in magnitude at most 1. (Together with mucl 25646, this implies that it takes a value in {-1, 0, 1} for every positive integer.) (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (abs‘(μ‘𝐴)) ≤ 1)
 
Theoremchtfl 25654 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))
 
Theoremchpfl 25655 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
 
Theoremppiprm 25656 The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = ((π𝐴) + 1))
 
Theoremppinprm 25657 The prime-counting function π at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = (π𝐴))
 
Theoremchtprm 25658 The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = ((θ‘𝐴) + (log‘(𝐴 + 1))))
 
Theoremchtnprm 25659 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = (θ‘𝐴))
 
Theoremchpp1 25660 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
(𝐴 ∈ ℕ0 → (ψ‘(𝐴 + 1)) = ((ψ‘𝐴) + (Λ‘(𝐴 + 1))))
 
Theoremchtwordi 25661 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (θ‘𝐴) ≤ (θ‘𝐵))
 
Theoremchpwordi 25662 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (ψ‘𝐴) ≤ (ψ‘𝐵))
 
Theoremchtdif 25663* The difference of the Chebyshev function at two points sums the logarithms of the primes in an interval. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑁 ∈ (ℤ𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))
 
Theoremefchtdvds 25664 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (exp‘(θ‘𝐴)) ∥ (exp‘(θ‘𝐵)))
 
Theoremppifl 25665 The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π𝐴))
 
Theoremppip1le 25666 The prime-counting function π cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(𝐴 + 1)) ≤ ((π𝐴) + 1))
 
Theoremppiwordi 25667 The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (π𝐴) ≤ (π𝐵))
 
Theoremppidif 25668 The difference of the prime-counting function π at two points counts the number of primes in an interval. (Contributed by Mario Carneiro, 21-Sep-2014.)
(𝑁 ∈ (ℤ𝑀) → ((π𝑁) − (π𝑀)) = (♯‘(((𝑀 + 1)...𝑁) ∩ ℙ)))
 
Theoremppi1 25669 The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘1) = 0
 
Theoremcht1 25670 The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘1) = 0
 
Theoremvma1 25671 The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(Λ‘1) = 0
 
Theoremchp1 25672 The second Chebyshev function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(ψ‘1) = 0
 
Theoremppi1i 25673 Inference form of ppiprm 25656. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &   𝑁 ∈ ℙ       (π𝑁) = (𝐾 + 1)
 
Theoremppi2i 25674 Inference form of ppinprm 25657. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &    ¬ 𝑁 ∈ ℙ       (π𝑁) = 𝐾
 
Theoremppi2 25675 The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘2) = 1
 
Theoremppi3 25676 The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘3) = 2
 
Theoremcht2 25677 The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘2) = (log‘2)
 
Theoremcht3 25678 The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘3) = (log‘6)
 
Theoremppinncl 25679 Closure of the prime-counting function π in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π𝐴) ∈ ℕ)
 
Theoremchtrpcl 25680 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ+)
 
Theoremppieq0 25681 The prime-counting function π is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → ((π𝐴) = 0 ↔ 𝐴 < 2))
 
Theoremppiltx 25682 The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (π𝐴) < 𝐴)
 
Theoremprmorcht 25683 Relate the primorial (product of the first 𝑛 primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, 𝑛, 1))       (𝐴 ∈ ℕ → (exp‘(θ‘𝐴)) = (seq1( · , 𝐹)‘𝐴))
 
Theoremmumullem1 25684 Lemma for mumul 25686. A multiple of a non-squarefree number is non-squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (μ‘𝐴) = 0) → (μ‘(𝐴 · 𝐵)) = 0)
 
Theoremmumullem2 25685 Lemma for mumul 25686. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) ≠ 0)
 
Theoremmumul 25686 The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵)))
 
Theoremsqff1o 25687* There is a bijection from the squarefree divisors of a number 𝑁 to the powerset of the prime divisors of 𝑁. Among other things, this implies that a number has 2↑𝑘 squarefree divisors where 𝑘 is the number of prime divisors, and a squarefree number has 2↑𝑘 divisors (because all divisors of a squarefree number are squarefree). The inverse function to 𝐹 takes the product of all the primes in some subset of prime divisors of 𝑁. (Contributed by Mario Carneiro, 1-Jul-2015.)
𝑆 = {𝑥 ∈ ℕ ∣ ((μ‘𝑥) ≠ 0 ∧ 𝑥𝑁)}    &   𝐹 = (𝑛𝑆 ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑛})    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       (𝑁 ∈ ℕ → 𝐹:𝑆1-1-onto→𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁})
 
Theoremfsumdvdsdiaglem 25688* A "diagonal commutation" of divisor sums analogous to fsum0diag 15122. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑗)}) → (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)})))
 
Theoremfsumdvdsdiag 25689* A "diagonal commutation" of divisor sums analogous to fsum0diag 15122. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)
(𝜑𝑁 ∈ ℕ)    &   ((𝜑 ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑗)})) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑗)}𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐴)
 
Theoremfsumdvdscom 25690* A double commutation of divisor sums based on fsumdvdsdiag 25689. Note that 𝐴 depends on both 𝑗 and 𝑘. (Contributed by Mario Carneiro, 13-May-2016.)
(𝜑𝑁 ∈ ℕ)    &   (𝑗 = (𝑘 · 𝑚) → 𝐴 = 𝐵)    &   ((𝜑 ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑗})) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑗}𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵)
 
Theoremdvdsppwf1o 25691* A bijection from the divisors of a prime power to the integers less than the prime count. (Contributed by Mario Carneiro, 5-May-2016.)
𝐹 = (𝑛 ∈ (0...𝐴) ↦ (𝑃𝑛))       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
 
Theoremdvdsflf1o 25692* A bijection from the numbers less than 𝑁 / 𝐴 to the multiples of 𝐴 less than 𝑁. Useful for some sum manipulations. (Contributed by Mario Carneiro, 3-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   𝐹 = (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↦ (𝑁 · 𝑛))       (𝜑𝐹:(1...(⌊‘(𝐴 / 𝑁)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
 
Theoremdvdsflsumcom 25693* A sum commutation from Σ𝑛𝐴, Σ𝑑𝑛, 𝐵(𝑛, 𝑑) to Σ𝑑𝐴, Σ𝑚𝐴 / 𝑑, 𝐵(𝑛, 𝑑𝑚). (Contributed by Mario Carneiro, 4-May-2016.)
(𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
 
Theoremfsumfldivdiaglem 25694* Lemma for fsumfldivdiag 25695. (Contributed by Mario Carneiro, 10-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))))
 
Theoremfsumfldivdiag 25695* The right-hand side of dvdsflsumcom 25693 is commutative in the variables, because it can be written as the manifestly symmetric sum over those 𝑚, 𝑛 such that 𝑚 · 𝑛𝐴. (Contributed by Mario Carneiro, 4-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))𝐵 = Σ𝑚 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚)))𝐵)
 
Theoremmusum 25696* The sum of the Möbius function over the divisors of 𝑁 gives one if 𝑁 = 1, but otherwise always sums to zero. Theorem 2.1 in [ApostolNT] p. 25. This makes the Möbius function useful for inverting divisor sums; see also muinv 25698. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))
 
Theoremmusumsum 25697* Evaluate a collapsing sum over the Möbius function. (Contributed by Mario Carneiro, 4-May-2016.)
(𝑚 = 1 → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑 → 1 ∈ 𝐴)    &   ((𝜑𝑚𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑚𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑚} ((μ‘𝑘) · 𝐵) = 𝐶)
 
Theoremmuinv 25698* The Möbius inversion formula. If 𝐺(𝑛) = Σ𝑘𝑛𝐹(𝑘) for every 𝑛 ∈ ℕ, then 𝐹(𝑛) = Σ𝑘𝑛 μ(𝑘)𝐺(𝑛 / 𝑘) = Σ𝑘𝑛μ(𝑛 / 𝑘)𝐺(𝑘), i.e. the Möbius function is the Dirichlet convolution inverse of the constant function 1. Theorem 2.9 in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝐹:ℕ⟶ℂ)    &   (𝜑𝐺 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (𝐹𝑘)))       (𝜑𝐹 = (𝑚 ∈ ℕ ↦ Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑚} ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗)))))
 
Theoremdvdsmulf1o 25699* If 𝑀 and 𝑁 are two coprime integers, multiplication forms a bijection from the set of pairs 𝑗, 𝑘 where 𝑗𝑀 and 𝑘𝑁, to the set of divisors of 𝑀 · 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}       (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto𝑍)
 
Theoremfsumdvdsmul 25700* Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    &   ((𝜑𝑗𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑌) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) = 𝐷)    &   (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)       (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑖𝑍 𝐶)
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