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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvmaf 25301 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ:ℕ⟶ℝ

Theoremefvmacl 25302 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)

Theoremvmage0 25303 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))

Theoremchpval 25304* 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 25305 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ:ℝ⟶ℝ

Theoremchpcl 25306 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)

Theoremefchpcl 25307 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)

Theoremchpge0 25308 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))

Theoremppival 25309 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))

Theoremppival2 25310 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℤ → (π𝐴) = (♯‘((2...𝐴) ∩ ℙ)))

Theoremppival2g 25311 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ𝑀)) → (π𝐴) = (♯‘((𝑀...𝐴) ∩ ℙ)))

Theoremppif 25312 Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
π:ℝ⟶ℕ0

Theoremppicl 25313 Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) ∈ ℕ0)

Theoremmuval 25314* 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 25315 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 25316* 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 25317* 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 25318* 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 25319 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 25320 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 25321* 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 25322 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ:ℕ⟶ℤ

Theoremmucl 25323 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ)

Theoremsgmval 25324* 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 25325* 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 25326* 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 25327 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 25328 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)

Theoremsgmnncl 25329 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)

Theoremmule1 25330 The Möbius function takes on values in magnitude at most 1. (Together with mucl 25323, 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 25331 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))

Theoremchpfl 25332 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))

Theoremppiprm 25333 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 25334 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 25335 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 25336 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 25337 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 25338 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (θ‘𝐴) ≤ (θ‘𝐵))

Theoremchpwordi 25339 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (ψ‘𝐴) ≤ (ψ‘𝐵))

Theoremchtdif 25340* 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 25341 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 25342 The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π𝐴))

Theoremppip1le 25343 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 25344 The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (π𝐴) ≤ (π𝐵))

Theoremppidif 25345 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 25346 The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘1) = 0

Theoremcht1 25347 The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘1) = 0

Theoremvma1 25348 The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(Λ‘1) = 0

Theoremchp1 25349 The second Chebyshev function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(ψ‘1) = 0

Theoremppi1i 25350 Inference form of ppiprm 25333. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &   𝑁 ∈ ℙ       (π𝑁) = (𝐾 + 1)

Theoremppi2i 25351 Inference form of ppinprm 25334. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &    ¬ 𝑁 ∈ ℙ       (π𝑁) = 𝐾

Theoremppi2 25352 The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘2) = 1

Theoremppi3 25353 The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘3) = 2

Theoremcht2 25354 The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘2) = (log‘2)

Theoremcht3 25355 The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘3) = (log‘6)

Theoremppinncl 25356 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 25357 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ+)

Theoremppieq0 25358 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 25359 The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (π𝐴) < 𝐴)

Theoremprmorcht 25360 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 25361 Lemma for mumul 25363. 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 25362 Lemma for mumul 25363. 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 25363 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 25364* 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 25365* A "diagonal commutation" of divisor sums analogous to fsum0diag 14917. (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 25366* A "diagonal commutation" of divisor sums analogous to fsum0diag 14917. (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 25367* A double commutation of divisor sums based on fsumdvdsdiag 25366. 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 25368* 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 25369* 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 25370* 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 25371* Lemma for fsumfldivdiag 25372. (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 25372* The right-hand side of dvdsflsumcom 25370 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 25373* 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 25375. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))

Theoremmusumsum 25374* 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 25375* 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 25376* 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 25377* 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)

Theoremsgmppw 25378* The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)
((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝐴 σ (𝑃𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃𝑐𝐴)↑𝑘))

Theorem0sgmppw 25379 A prime power 𝑃𝐾 has 𝐾 + 1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ0) → (0 σ (𝑃𝐾)) = (𝐾 + 1))

Theorem1sgmprm 25380 The sum of divisors for a prime is 𝑃 + 1 because the only divisors are 1 and 𝑃. (Contributed by Mario Carneiro, 17-May-2016.)
(𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Theorem1sgm2ppw 25381 The sum of the divisors of 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 17-May-2016.)
(𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Theoremsgmmul 25382 The divisor function for fixed parameter 𝐴 is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)
((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)))

Theoremppiublem1 25383 Lemma for ppiub 25385. (Contributed by Mario Carneiro, 12-Mar-2014.)
(𝑁 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5})))    &   𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (2 ∥ 𝑀 ∨ 3 ∥ 𝑀𝑀 ∈ {1, 5})       (𝑀 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5})))

Theoremppiublem2 25384 A prime greater than 3 does not divide 2 or 3, so its residue mod 6 is 1 or 5. (Contributed by Mario Carneiro, 12-Mar-2014.)
((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ {1, 5})

Theoremppiub 25385 An upper bound on the prime-counting function π, which counts the number of primes less than 𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.)
((𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) → (π𝑁) ≤ ((𝑁 / 3) + 2))

Theoremvmalelog 25386 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ≤ (log‘𝐴))

Theoremchtlepsi 25387 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (θ‘𝐴) ≤ (ψ‘𝐴))

Theoremchprpcl 25388 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ+)

Theoremchpeq0 25389 The second Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

Theoremchteq0 25390 The first Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → ((θ‘𝐴) = 0 ↔ 𝐴 < 2))

Theoremchtleppi 25391 Upper bound on the θ function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (θ‘𝐴) ≤ ((π𝐴) · (log‘𝐴)))

Theoremchtublem 25392 Lemma for chtub 25393. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ → (θ‘((2 · 𝑁) − 1)) ≤ ((θ‘𝑁) + ((log‘4) · (𝑁 − 1))))

Theoremchtub 25393 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)
((𝑁 ∈ ℝ ∧ 2 < 𝑁) → (θ‘𝑁) < ((log‘2) · ((2 · 𝑁) − 3)))

Theoremfsumvma 25394* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝑥 = (𝑝𝑘) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑 → ((𝑝𝑃𝑘𝐾) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝𝑘) ∈ 𝐴)))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝐴 ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)       (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑝𝑃 Σ𝑘𝐾 𝐶)

Theoremfsumvma2 25395* Apply fsumvma 25394 for the common case of all numbers less than a real number 𝐴. (Contributed by Mario Carneiro, 30-Apr-2016.)
(𝑥 = (𝑝𝑘) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ ℝ)    &   ((𝜑𝑥 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)       (𝜑 → Σ𝑥 ∈ (1...(⌊‘𝐴))𝐵 = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))𝐶)

Theorempclogsum 25396* The logarithmic analogue of pcprod 16007. The sum of the logarithms of the primes dividing 𝐴 multiplied by their powers yields the logarithm of 𝐴. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)) = (log‘𝐴))

Theoremvmasum 25397* The sum of the von Mangoldt function over the divisors of 𝑛. Equation 9.2.4 of [Shapiro], p. 328 and theorem 2.10 in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝐴 ∈ ℕ → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝐴} (Λ‘𝑛) = (log‘𝐴))

Theoremlogfac2 25398* Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))

Theoremchpval2 25399* Express the second Chebyshev function directly as a sum over the primes less than 𝐴 (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))

Theoremchpchtsum 25400* The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))

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