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Theorem | ftalem3 24801* | Lemma for fta 24806. There exists a global minimum of the function abs ∘ 𝐹. The proof uses a circle of radius 𝑟 where 𝑟 is the value coming from ftalem1 24799; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅} & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) | ||
Theorem | ftalem4 24802* | Lemma for fta 24806: Closure of the auxiliary variables for ftalem5 24803. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐹‘0) ≠ 0) & ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) & ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) & ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) & ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) ⇒ ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+))) | ||
Theorem | ftalem5 24803* | Lemma for fta 24806: Main proof. We have already shifted the minimum found in ftalem3 24801 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let 𝐾 be the lowest term in the polynomial that is nonzero, and let 𝑇 be a 𝐾-th root of -𝐹(0) / 𝐴(𝐾). Then an evaluation of 𝐹(𝑇𝑋) where 𝑋 is a sufficiently small positive number yields 𝐹(0) for the first term and -𝐹(0) · 𝑋↑𝐾 for the 𝐾-th term, and all higher terms are bounded because 𝑋 is small. Thus, abs(𝐹(𝑇𝑋)) ≤ abs(𝐹(0))(1 − 𝑋↑𝐾) < abs(𝐹(0)), in contradiction to our choice of 𝐹(0) as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.) (Revised by AV, 28-Sep-2020.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐹‘0) ≠ 0) & ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) & ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) & ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) & ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0))) | ||
Theorem | ftalem6 24804* | Lemma for fta 24806: Discharge the auxiliary variables in ftalem5 24803. (Contributed by Mario Carneiro, 20-Sep-2014.) (Proof shortened by AV, 28-Sep-2020.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐹‘0) ≠ 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0))) | ||
Theorem | ftalem7 24805* | Lemma for fta 24806. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → (𝐹‘𝑋) ≠ 0) ⇒ ⊢ (𝜑 → ¬ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑋)) ≤ (abs‘(𝐹‘𝑥))) | ||
Theorem | fta 24806* | The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. This is Metamath 100 proof #2. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) ∈ ℕ) → ∃𝑧 ∈ ℂ (𝐹‘𝑧) = 0) | ||
Theorem | basellem1 24807 | Lemma for basel 24816. 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))) | ||
Theorem | basellem2 24808* | Lemma for basel 24816. 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↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ⇒ ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))))) | ||
Theorem | basellem3 24809* | Lemma for basel 24816. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁 − 𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀 − 𝑗) (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↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ⇒ ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))) | ||
Theorem | basellem4 24810* | Lemma for basel 24816. By basellem3 24809, 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 24063. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ 𝑁 = ((2 · 𝑀) + 1) & ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2)) ⇒ ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(◡𝑃 “ {0})) | ||
Theorem | basellem5 24811* | Lemma for basel 24816. Using vieta1 24067, 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↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) & ⊢ 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)) | ||
Theorem | basellem6 24812 | Lemma for basel 24816. 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 | ||
Theorem | basellem7 24813 | Lemma for basel 24816. 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 | ||
Theorem | basellem8 24814* | Lemma for basel 24816. 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 24811. (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}) ∘𝑓 − 𝐺)) & ⊢ 𝐽 = (𝐻 ∘𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺))) & ⊢ 𝐾 = (𝐻 ∘𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺)) & ⊢ 𝑁 = ((2 · 𝑀) + 1) ⇒ ⊢ (𝑀 ∈ ℕ → ((𝐽‘𝑀) ≤ (𝐹‘𝑀) ∧ (𝐹‘𝑀) ≤ (𝐾‘𝑀))) | ||
Theorem | basellem9 24815* | Lemma for basel 24816. Since by basellem8 24814 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 14371. 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}) ∘𝑓 − 𝐺)) & ⊢ 𝐽 = (𝐻 ∘𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺))) & ⊢ 𝐾 = (𝐻 ∘𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺)) ⇒ ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6) | ||
Theorem | basel 24816 | 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) | ||
Syntax | ccht 24817 | Extend class notation with the first Chebyshev function. |
class θ | ||
Syntax | cvma 24818 | Extend class notation with the von Mangoldt function. |
class Λ | ||
Syntax | cchp 24819 | Extend class notation with the second Chebyshev function. |
class ψ | ||
Syntax | cppi 24820 | Extend class notation with the prime-counting function pi. |
class π | ||
Syntax | cmu 24821 | Extend class notation with the Möbius function. |
class μ | ||
Syntax | csgm 24822 | Extend class notation with the divisor function. |
class σ | ||
Definition | df-cht 24823* | 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 24825. 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)) | ||
Definition | df-vma 24824* | 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 ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0)) | ||
Definition | df-chp 24825* | 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)) | ||
Definition | df-ppi 24826 | 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))) | ||
Definition | df-mu 24827* | 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 ∣ 𝑝 ∥ 𝑥})))) | ||
Definition | df-sgm 24828* | 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 ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥)) | ||
Theorem | efnnfsumcl 24829* | Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (exp‘𝐵) ∈ ℕ) ⇒ ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ) | ||
Theorem | ppisval 24830 | 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)) | ||
Theorem | ppisval2 24831 | The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ)) | ||
Theorem | ppifi 24832 | The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin) | ||
Theorem | prmdvdsfi 24833* | The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) | ||
Theorem | chtf 24834 | Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ θ:ℝ⟶ℝ | ||
Theorem | chtcl 24835 | Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ) | ||
Theorem | chtval 24836* | Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) | ||
Theorem | efchtcl 24837 | 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) | ||
Theorem | chtge0 24838 | The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴)) | ||
Theorem | vmaval 24839* | Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ⇒ ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) | ||
Theorem | isppw 24840* | Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴)) | ||
Theorem | isppw2 24841* | 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 = (𝑝↑𝑘))) | ||
Theorem | vmappw 24842 | Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃↑𝐾)) = (log‘𝑃)) | ||
Theorem | vmaprm 24843 | Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃)) | ||
Theorem | vmacl 24844 | Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ) | ||
Theorem | vmaf 24845 | Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ Λ:ℕ⟶ℝ | ||
Theorem | efvmacl 24846 | The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ) | ||
Theorem | vmage0 24847 | The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴)) | ||
Theorem | chpval 24848* | 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)) | ||
Theorem | chpf 24849 | Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ ψ:ℝ⟶ℝ | ||
Theorem | chpcl 24850 | Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ) | ||
Theorem | efchpcl 24851 | The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ) | ||
Theorem | chpge0 24852 | The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴)) | ||
Theorem | ppival 24853 | Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (#‘((0[,]𝐴) ∩ ℙ))) | ||
Theorem | ppival2 24854 | Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.) |
⊢ (𝐴 ∈ ℤ → (π‘𝐴) = (#‘((2...𝐴) ∩ ℙ))) | ||
Theorem | ppival2g 24855 | Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ≥‘𝑀)) → (π‘𝐴) = (#‘((𝑀...𝐴) ∩ ℙ))) | ||
Theorem | ppif 24856 | Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ π:ℝ⟶ℕ0 | ||
Theorem | ppicl 24857 | Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0) | ||
Theorem | muval 24858* | 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 ∣ 𝑝 ∥ 𝐴})))) | ||
Theorem | muval1 24859 | 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) | ||
Theorem | muval2 24860* | 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 ∣ 𝑝 ∥ 𝐴}))) | ||
Theorem | isnsqf 24861* | 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)) | ||
Theorem | issqf 24862* | 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)) | ||
Theorem | sqfpc 24863 | 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) | ||
Theorem | dvdssqf 24864 | A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → ((μ‘𝐴) ≠ 0 → (μ‘𝐵) ≠ 0)) | ||
Theorem | sqf11 24865* | 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 (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵))) | ||
Theorem | muf 24866 | The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ μ:ℕ⟶ℤ | ||
Theorem | mucl 24867 | Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ) | ||
Theorem | sgmval 24868* | 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 ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴)) | ||
Theorem | sgmval2 24869* | The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴)) | ||
Theorem | 0sgm 24870* | 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 ∣ 𝑝 ∥ 𝐴})) | ||
Theorem | sgmf 24871 | 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 | ||
Theorem | sgmcl 24872 | Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ) | ||
Theorem | sgmnncl 24873 | Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ) | ||
Theorem | mule1 24874 | The Möbius function takes on values in magnitude at most 1. (Together with mucl 24867, 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) | ||
Theorem | chtfl 24875 | The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴)) | ||
Theorem | chpfl 24876 | The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.) |
⊢ (𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴)) | ||
Theorem | ppiprm 24877 | The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = ((π‘𝐴) + 1)) | ||
Theorem | ppinprm 24878 | The prime-counting function π at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = (π‘𝐴)) | ||
Theorem | chtprm 24879 | 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)))) | ||
Theorem | chtnprm 24880 | The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = (θ‘𝐴)) | ||
Theorem | chpp1 24881 | The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.) |
⊢ (𝐴 ∈ ℕ0 → (ψ‘(𝐴 + 1)) = ((ψ‘𝐴) + (Λ‘(𝐴 + 1)))) | ||
Theorem | chtwordi 24882 | The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (θ‘𝐴) ≤ (θ‘𝐵)) | ||
Theorem | chpwordi 24883 | The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (ψ‘𝐴) ≤ (ψ‘𝐵)) | ||
Theorem | chtdif 24884* | 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 ∈ (ℤ≥‘𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝)) | ||
Theorem | efchtdvds 24885 | The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (exp‘(θ‘𝐴)) ∥ (exp‘(θ‘𝐵))) | ||
Theorem | ppifl 24886 | The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴)) | ||
Theorem | ppip1le 24887 | 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)) ≤ ((π‘𝐴) + 1)) | ||
Theorem | ppiwordi 24888 | The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (π‘𝐴) ≤ (π‘𝐵)) | ||
Theorem | ppidif 24889 | 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 ∈ (ℤ≥‘𝑀) → ((π‘𝑁) − (π‘𝑀)) = (#‘(((𝑀 + 1)...𝑁) ∩ ℙ))) | ||
Theorem | ppi1 24890 | The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ (π‘1) = 0 | ||
Theorem | cht1 24891 | The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (θ‘1) = 0 | ||
Theorem | vma1 24892 | The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.) |
⊢ (Λ‘1) = 0 | ||
Theorem | chp1 24893 | The second Chebyshev function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.) |
⊢ (ψ‘1) = 0 | ||
Theorem | ppi1i 24894 | Inference form of ppiprm 24877. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 = (𝑀 + 1) & ⊢ (π‘𝑀) = 𝐾 & ⊢ 𝑁 ∈ ℙ ⇒ ⊢ (π‘𝑁) = (𝐾 + 1) | ||
Theorem | ppi2i 24895 | Inference form of ppinprm 24878. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 = (𝑀 + 1) & ⊢ (π‘𝑀) = 𝐾 & ⊢ ¬ 𝑁 ∈ ℙ ⇒ ⊢ (π‘𝑁) = 𝐾 | ||
Theorem | ppi2 24896 | The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ (π‘2) = 1 | ||
Theorem | ppi3 24897 | The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ (π‘3) = 2 | ||
Theorem | cht2 24898 | The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (θ‘2) = (log‘2) | ||
Theorem | cht3 24899 | The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (θ‘3) = (log‘6) | ||
Theorem | ppinncl 24900 | Closure of the prime-counting function π in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) |
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