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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
14.4.3  The Basel problem (ζ(2) = π2/6)
 
Theorembasellem1 24801 Lemma for basel 24810. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.) Replace OLD theorem. (Revised ba Wolf Lammen, 18-Sep-2020.)
𝑁 = ((2 · 𝑀) + 1)       ((𝑀 ∈ ℕ ∧ 𝐾 ∈ (1...𝑀)) → ((𝐾 · π) / 𝑁) ∈ (0(,)(π / 2)))
 
Theorembasellem2 24802* Lemma for basel 24810. Show that 𝑃 is a polynomial of degree 𝑀, and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))       (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀𝑛))))))
 
Theorembasellem3 24803* Lemma for basel 24810. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀𝑗) (cot𝑥)↑(-2𝑗) = 𝑃((cot𝑥)↑2), where 𝑁 = 2𝑀 + 1. (Contributed by Mario Carneiro, 30-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))       ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))
 
Theorembasellem4 24804* Lemma for basel 24810. By basellem3 24803, the expression 𝑃((cot𝑥)↑2) = sin(𝑁𝑥) / (sin𝑥)↑𝑁 goes to zero whenever 𝑥 = 𝑛π / 𝑁 for some 𝑛 ∈ (1...𝑀), so this function enumerates 𝑀 distinct roots of a degree- 𝑀 polynomial, which must therefore be all the roots by fta1 24057. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))    &   𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))       (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(𝑃 “ {0}))
 
Theorembasellem5 24805* Lemma for basel 24810. Using vieta1 24061, we can calculate the sum of the roots of 𝑃 as the quotient of the top two coefficients, and since the function 𝑇 enumerates the roots, we are left with an equation that sums the cot↑2 function at the 𝑀 different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))    &   𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))       (𝑀 ∈ ℕ → Σ𝑘 ∈ (1...𝑀)((tan‘((𝑘 · π) / 𝑁))↑-2) = (((2 · 𝑀) · ((2 · 𝑀) − 1)) / 6))
 
Theorembasellem6 24806 Lemma for basel 24810. The function 𝐺 goes to zero because it is bounded by 1 / 𝑛. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))       𝐺 ⇝ 0
 
Theorembasellem7 24807 Lemma for basel 24810. The function 1 + 𝐴 · 𝐺 for any fixed 𝐴 goes to 1. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐴 ∈ ℂ       ((ℕ × {1}) ∘𝑓 + ((ℕ × {𝐴}) ∘𝑓 · 𝐺)) ⇝ 1
 
Theorembasellem8 24808* Lemma for basel 24810. The function 𝐹 of partial sums of the inverse squares is bounded below by 𝐽 and above by 𝐾, obtained by summing the inequality cot↑2𝑥 ≤ 1 / 𝑥↑2 ≤ csc↑2𝑥 = cot↑2𝑥 + 1 over the 𝑀 roots of the polynomial 𝑃, and applying the identity basellem5 24805. (Contributed by Mario Carneiro, 29-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))    &   𝐻 = ((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓𝐺))    &   𝐽 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺)))    &   𝐾 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺))    &   𝑁 = ((2 · 𝑀) + 1)       (𝑀 ∈ ℕ → ((𝐽𝑀) ≤ (𝐹𝑀) ∧ (𝐹𝑀) ≤ (𝐾𝑀)))
 
Theorembasellem9 24809* Lemma for basel 24810. Since by basellem8 24808 𝐹 is bounded by two expressions that tend to π↑2 / 6, 𝐹 must also go to π↑2 / 6 by the squeeze theorem climsqz 14365. But the series 𝐹 is exactly the partial sums of 𝑘↑-2, so it follows that this is also the value of the infinite sum Σ𝑘 ∈ ℕ(𝑘↑-2). (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))    &   𝐻 = ((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓𝐺))    &   𝐽 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺)))    &   𝐾 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺))       Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)
 
Theorembasel 24810 The sum of the inverse squares is π↑2 / 6. This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). This is Metamath 100 proof #14. (Contributed by Mario Carneiro, 30-Jul-2014.)
Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)
 
14.4.4  Number-theoretical functions
 
Syntaxccht 24811 Extend class notation with the first Chebyshev function.
class θ
 
Syntaxcvma 24812 Extend class notation with the von Mangoldt function.
class Λ
 
Syntaxcchp 24813 Extend class notation with the second Chebyshev function.
class ψ
 
Syntaxcppi 24814 Extend class notation with the prime-counting function pi.
class π
 
Syntaxcmu 24815 Extend class notation with the Möbius function.
class μ
 
Syntaxcsgm 24816 Extend class notation with the divisor function.
class σ
 
Definitiondf-cht 24817* 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 24819. See https://en.wikipedia.org/wiki/Chebyshev_function 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 24818* 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 24819* 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 24820 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 24821* 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 24822* 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 24823* 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 24824 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 24825 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 24826 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 24827* 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 24828 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ:ℝ⟶ℝ
 
Theoremchtcl 24829 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)
 
Theoremchtval 24830* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
 
Theoremefchtcl 24831 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 24832 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴))
 
Theoremvmaval 24833* 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 24834* 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 24835* 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 24836 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 24837 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃))
 
Theoremvmacl 24838 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ)
 
Theoremvmaf 24839 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ:ℕ⟶ℝ
 
Theoremefvmacl 24840 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)
 
Theoremvmage0 24841 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))
 
Theoremchpval 24842* 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 24843 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ:ℝ⟶ℝ
 
Theoremchpcl 24844 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
 
Theoremefchpcl 24845 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)
 
Theoremchpge0 24846 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
 
Theoremppival 24847 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))
 
Theoremppival2 24848 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℤ → (π𝐴) = (#‘((2...𝐴) ∩ ℙ)))
 
Theoremppival2g 24849 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ𝑀)) → (π𝐴) = (#‘((𝑀...𝐴) ∩ ℙ)))
 
Theoremppif 24850 Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
π:ℝ⟶ℕ0
 
Theoremppicl 24851 Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) ∈ ℕ0)
 
Theoremmuval 24852* 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 24853 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 24854* 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 24855* 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 24856* 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 24857 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 24858 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 24859* 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 24860 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ:ℕ⟶ℤ
 
Theoremmucl 24861 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ)
 
Theoremsgmval 24862* 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 24863* 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 24864* 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 24865 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 24866 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)
 
Theoremsgmnncl 24867 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)
 
Theoremmule1 24868 The Möbius function takes on values in magnitude at most 1. (Together with mucl 24861, 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 24869 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))
 
Theoremchpfl 24870 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
 
Theoremppiprm 24871 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 24872 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 24873 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 24874 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 24875 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 24876 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (θ‘𝐴) ≤ (θ‘𝐵))
 
Theoremchpwordi 24877 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (ψ‘𝐴) ≤ (ψ‘𝐵))
 
Theoremchtdif 24878* 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 24879 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 24880 The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π𝐴))
 
Theoremppip1le 24881 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 24882 The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (π𝐴) ≤ (π𝐵))
 
Theoremppidif 24883 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 24884 The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘1) = 0
 
Theoremcht1 24885 The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘1) = 0
 
Theoremvma1 24886 The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(Λ‘1) = 0
 
Theoremchp1 24887 The second Chebyshev function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(ψ‘1) = 0
 
Theoremppi1i 24888 Inference form of ppiprm 24871. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &   𝑁 ∈ ℙ       (π𝑁) = (𝐾 + 1)
 
Theoremppi2i 24889 Inference form of ppinprm 24872. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &    ¬ 𝑁 ∈ ℙ       (π𝑁) = 𝐾
 
Theoremppi2 24890 The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘2) = 1
 
Theoremppi3 24891 The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘3) = 2
 
Theoremcht2 24892 The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘2) = (log‘2)
 
Theoremcht3 24893 The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘3) = (log‘6)
 
Theoremppinncl 24894 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 24895 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ+)
 
Theoremppieq0 24896 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 24897 The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (π𝐴) < 𝐴)
 
Theoremprmorcht 24898 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 24899 Lemma for mumul 24901. 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 24900 Lemma for mumul 24901. 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)
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