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Type | Label | Description |
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Statement | ||
Theorem | lgseisenlem2 24901* | Lemma for lgseisen 24904. The function 𝑀 is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃) & ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) & ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃) ⇒ ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2))) | ||
Theorem | lgseisenlem3 24902* | Lemma for lgseisen 24904. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-2019.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃) & ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) & ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃) & ⊢ 𝑌 = (ℤ/nℤ‘𝑃) & ⊢ 𝐺 = (mulGrp‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1r‘𝑌)) | ||
Theorem | lgseisenlem4 24903* | Lemma for lgseisen 24904. The function 𝑀 is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 18-Jun-2015.) (Proof shortened by AV, 15-Jun-2019.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃) & ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) & ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃) & ⊢ 𝑌 = (ℤ/nℤ‘𝑃) & ⊢ 𝐺 = (mulGrp‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃)) | ||
Theorem | lgseisen 24904* | Eisenstein's lemma, an expression for (𝑃 /L 𝑄) when 𝑃, 𝑄 are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → (𝑄 /L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) | ||
Theorem | lgsquadlem1 24905* | Lemma for lgsquad 24908. Count the members of 𝑆 with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ 𝑀 = ((𝑃 − 1) / 2) & ⊢ 𝑁 = ((𝑄 − 1) / 2) & ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⇒ ⊢ (𝜑 → (-1↑Σ𝑢 ∈ (((⌊‘(𝑀 / 2)) + 1)...𝑀)(⌊‘((𝑄 / 𝑃) · (2 · 𝑢)))) = (-1↑(#‘{𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1st ‘𝑧)}))) | ||
Theorem | lgsquadlem2 24906* | Lemma for lgsquad 24908. Count the members of 𝑆 with even coordinates, and combine with lgsquadlem1 24905 to get the total count of lattice points in 𝑆 (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ 𝑀 = ((𝑃 − 1) / 2) & ⊢ 𝑁 = ((𝑄 − 1) / 2) & ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⇒ ⊢ (𝜑 → (𝑄 /L 𝑃) = (-1↑(#‘𝑆))) | ||
Theorem | lgsquadlem3 24907* | Lemma for lgsquad 24908. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ 𝑀 = ((𝑃 − 1) / 2) & ⊢ 𝑁 = ((𝑄 − 1) / 2) & ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⇒ ⊢ (𝜑 → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(𝑀 · 𝑁))) | ||
Theorem | lgsquad 24908 | The Law of Quadratic Reciprocity, see also theorem 9.8 in [ApostolNT] p. 185. If 𝑃 and 𝑄 are distinct odd primes, then the product of the Legendre symbols (𝑃 /L 𝑄) and (𝑄 /L 𝑃) is the parity of ((𝑃 − 1) / 2) · ((𝑄 − 1) / 2). This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. This is Metamath 100 proof #7. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(((𝑃 − 1) / 2) · ((𝑄 − 1) / 2)))) | ||
Theorem | lgsquad2lem1 24909 | Lemma for lgsquad2 24911. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑀) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) & ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (𝐴 · 𝐵) = 𝑀) & ⊢ (𝜑 → ((𝐴 /L 𝑁) · (𝑁 /L 𝐴)) = (-1↑(((𝐴 − 1) / 2) · ((𝑁 − 1) / 2)))) & ⊢ (𝜑 → ((𝐵 /L 𝑁) · (𝑁 /L 𝐵)) = (-1↑(((𝐵 − 1) / 2) · ((𝑁 − 1) / 2)))) ⇒ ⊢ (𝜑 → ((𝑀 /L 𝑁) · (𝑁 /L 𝑀)) = (-1↑(((𝑀 − 1) / 2) · ((𝑁 − 1) / 2)))) | ||
Theorem | lgsquad2lem2 24910* | Lemma for lgsquad2 24911. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑀) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) & ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) & ⊢ ((𝜑 ∧ (𝑚 ∈ (ℙ ∖ {2}) ∧ (𝑚 gcd 𝑁) = 1)) → ((𝑚 /L 𝑁) · (𝑁 /L 𝑚)) = (-1↑(((𝑚 − 1) / 2) · ((𝑁 − 1) / 2)))) & ⊢ (𝜓 ↔ ∀𝑥 ∈ (1...𝑘)((𝑥 gcd (2 · 𝑁)) = 1 → ((𝑥 /L 𝑁) · (𝑁 /L 𝑥)) = (-1↑(((𝑥 − 1) / 2) · ((𝑁 − 1) / 2))))) ⇒ ⊢ (𝜑 → ((𝑀 /L 𝑁) · (𝑁 /L 𝑀)) = (-1↑(((𝑀 − 1) / 2) · ((𝑁 − 1) / 2)))) | ||
Theorem | lgsquad2 24911 | Extend lgsquad 24908 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑀) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) & ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) ⇒ ⊢ (𝜑 → ((𝑀 /L 𝑁) · (𝑁 /L 𝑀)) = (-1↑(((𝑀 − 1) / 2) · ((𝑁 − 1) / 2)))) | ||
Theorem | lgsquad3 24912 | Extend lgsquad2 24911 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ (((𝑀 ∈ ℕ ∧ ¬ 2 ∥ 𝑀) ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → (𝑀 /L 𝑁) = ((-1↑(((𝑀 − 1) / 2) · ((𝑁 − 1) / 2))) · (𝑁 /L 𝑀))) | ||
Theorem | m1lgs 24913 | The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime 𝑃 iff 𝑃≡1 (mod 4). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /L 𝑃) = 1 ↔ (𝑃 mod 4) = 1)) | ||
Theorem | 2lgslem1a1 24914* | Lemma 1 for 2lgslem1a 24916. (Contributed by AV, 16-Jun-2021.) |
⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ∀𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑖 · 2) = ((𝑖 · 2) mod 𝑃)) | ||
Theorem | 2lgslem1a2 24915 | Lemma 2 for 2lgslem1a 24916. (Contributed by AV, 18-Jun-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2))) | ||
Theorem | 2lgslem1a 24916* | Lemma 1 for 2lgslem1 24919. (Contributed by AV, 18-Jun-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))} = {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}) | ||
Theorem | 2lgslem1b 24917* | Lemma 2 for 2lgslem1 24919. (Contributed by AV, 18-Jun-2021.) |
⊢ 𝐼 = (𝐴...𝐵) & ⊢ 𝐹 = (𝑗 ∈ 𝐼 ↦ (𝑗 · 2)) ⇒ ⊢ 𝐹:𝐼–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} | ||
Theorem | 2lgslem1c 24918 | Lemma 3 for 2lgslem1 24919. (Contributed by AV, 19-Jun-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)) | ||
Theorem | 2lgslem1 24919* | Lemma 1 for 2lgs 24932. (Contributed by AV, 19-Jun-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (#‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))}) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) | ||
Theorem | 2lgslem2 24920 | Lemma 2 for 2lgs 24932. (Contributed by AV, 20-Jun-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ) | ||
Theorem | 2lgslem3a 24921 | Lemma for 2lgslem3a1 24925. (Contributed by AV, 14-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 1)) → 𝑁 = (2 · 𝐾)) | ||
Theorem | 2lgslem3b 24922 | Lemma for 2lgslem3b1 24926. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 3)) → 𝑁 = ((2 · 𝐾) + 1)) | ||
Theorem | 2lgslem3c 24923 | Lemma for 2lgslem3c1 24927. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 5)) → 𝑁 = ((2 · 𝐾) + 1)) | ||
Theorem | 2lgslem3d 24924 | Lemma for 2lgslem3d1 24928. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 7)) → 𝑁 = ((2 · 𝐾) + 2)) | ||
Theorem | 2lgslem3a1 24925 | Lemma 1 for 2lgslem3 24929. (Contributed by AV, 15-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) | ||
Theorem | 2lgslem3b1 24926 | Lemma 2 for 2lgslem3 24929. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 3) → (𝑁 mod 2) = 1) | ||
Theorem | 2lgslem3c1 24927 | Lemma 3 for 2lgslem3 24929. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 5) → (𝑁 mod 2) = 1) | ||
Theorem | 2lgslem3d1 24928 | Lemma 4 for 2lgslem3 24929. (Contributed by AV, 15-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 7) → (𝑁 mod 2) = 0) | ||
Theorem | 2lgslem3 24929 | Lemma 3 for 2lgs 24932. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)) | ||
Theorem | 2lgs2 24930 | The Legendre symbol for 2 at 2 is 0. (Contributed by AV, 20-Jun-2021.) |
⊢ (2 /L 2) = 0 | ||
Theorem | 2lgslem4 24931 | Lemma 4 for 2lgs 24932: special case of 2lgs 24932 for 𝑃 = 2. (Contributed by AV, 20-Jun-2021.) |
⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) | ||
Theorem | 2lgs 24932 | The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime 𝑃 iff 𝑃≡±1 (mod 8), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of (𝑁 /L 2) (lgs2 24839) to some degree, by demanding that reciprocity extend to the case 𝑄 = 2. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.) |
⊢ (𝑃 ∈ ℙ → ((2 /L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7})) | ||
Theorem | 2lgsoddprmlem1 24933 | Lemma 1 for 2lgsoddprm 24941. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (((𝑁↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) | ||
Theorem | 2lgsoddprmlem2 24934 | Lemma 2 for 2lgsoddprm 24941. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ 2 ∥ (((𝑅↑2) − 1) / 8))) | ||
Theorem | 2lgsoddprmlem3a 24935 | Lemma 1 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((1↑2) − 1) / 8) = 0 | ||
Theorem | 2lgsoddprmlem3b 24936 | Lemma 2 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((3↑2) − 1) / 8) = 1 | ||
Theorem | 2lgsoddprmlem3c 24937 | Lemma 3 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((5↑2) − 1) / 8) = 3 | ||
Theorem | 2lgsoddprmlem3d 24938 | Lemma 4 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((7↑2) − 1) / 8) = (2 · 3) | ||
Theorem | 2lgsoddprmlem3 24939 | Lemma 3 for 2lgsoddprm 24941. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑅↑2) − 1) / 8) ↔ 𝑅 ∈ {1, 7})) | ||
Theorem | 2lgsoddprmlem4 24940 | Lemma 4 for 2lgsoddprm 24941. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ (𝑁 mod 8) ∈ {1, 7})) | ||
Theorem | 2lgsoddprm 24941 | The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for 2 at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight ((2 /L 𝑃) = -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 /L 𝑃) = (-1↑(((𝑃↑2) − 1) / 8))) | ||
Theorem | 2sqlem1 24942* | Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ⇒ ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) | ||
Theorem | 2sqlem2 24943* | Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ⇒ ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))) | ||
Theorem | mul2sq 24944 | 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.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆) | ||
Theorem | 2sqlem3 24945 | Lemma for 2sqlem5 24947. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℤ) & ⊢ (𝜑 → (𝑁 · 𝑃) = ((𝐴↑2) + (𝐵↑2))) & ⊢ (𝜑 → 𝑃 = ((𝐶↑2) + (𝐷↑2))) & ⊢ (𝜑 → 𝑃 ∥ ((𝐶 · 𝐵) + (𝐴 · 𝐷))) ⇒ ⊢ (𝜑 → 𝑁 ∈ 𝑆) | ||
Theorem | 2sqlem4 24946 | Lemma for 2sqlem5 24947. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℤ) & ⊢ (𝜑 → (𝑁 · 𝑃) = ((𝐴↑2) + (𝐵↑2))) & ⊢ (𝜑 → 𝑃 = ((𝐶↑2) + (𝐷↑2))) ⇒ ⊢ (𝜑 → 𝑁 ∈ 𝑆) | ||
Theorem | 2sqlem5 24947 | Lemma for 2sq 24955. 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.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → (𝑁 · 𝑃) ∈ 𝑆) & ⊢ (𝜑 → 𝑃 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑁 ∈ 𝑆) | ||
Theorem | 2sqlem6 24948* | Lemma for 2sq 24955. 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.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆)) & ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑆) | ||
Theorem | 2sqlem7 24949* | Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⇒ ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ) | ||
Theorem | 2sqlem8a 24950* | Lemma for 2sqlem8 24951. (Contributed by Mario Carneiro, 4-Jun-2016.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} & ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) & ⊢ (𝜑 → 𝑁 = ((𝐴↑2) + (𝐵↑2))) & ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) & ⊢ 𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) ⇒ ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ) | ||
Theorem | 2sqlem8 24951* | Lemma for 2sq 24955. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} & ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) & ⊢ (𝜑 → 𝑁 = ((𝐴↑2) + (𝐵↑2))) & ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) & ⊢ 𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) & ⊢ 𝐸 = (𝐶 / (𝐶 gcd 𝐷)) & ⊢ 𝐹 = (𝐷 / (𝐶 gcd 𝐷)) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑆) | ||
Theorem | 2sqlem9 24952* | Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} & ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ 𝑌) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑆) | ||
Theorem | 2sqlem10 24953* | Lemma for 2sq 24955. 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.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⇒ ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆) | ||
Theorem | 2sqlem11 24954* | Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⇒ ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → 𝑃 ∈ 𝑆) | ||
Theorem | 2sq 24955* | All primes of the form 4𝑘 + 1 are sums of two squares. This is Metamath 100 proof #20. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑃 = ((𝑥↑2) + (𝑦↑2))) | ||
Theorem | 2sqblem 24956 | The converse to 2sq 24955. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) & ⊢ (𝜑 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) & ⊢ (𝜑 → 𝑃 = ((𝑋↑2) + (𝑌↑2))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → (𝑃 gcd 𝑌) = ((𝑃 · 𝐴) + (𝑌 · 𝐵))) ⇒ ⊢ (𝜑 → (𝑃 mod 4) = 1) | ||
Theorem | 2sqb 24957* | The converse to 2sq 24955. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝑃 ∈ ℙ → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1))) | ||
Theorem | chebbnd1lem1 24958 | Lemma for chebbnd1 24961: show a lower bound on π(𝑥) at even integers using similar techniques to those used to prove bpos 24818. (Note that the expression 𝐾 is actually equal to 2 · 𝑁, but proving that is not necessary for the proof, and it's too much work.) The key to the proof is bposlem1 24809, which shows that each term in the expansion ((2 · 𝑁)C𝑁) = ∏𝑝 ∈ ℙ (𝑝↑(𝑝 pCnt ((2 · 𝑁)C𝑁))) is at most 2 · 𝑁, so that the sum really only has nonzero elements up to 2 · 𝑁, and since each term is at most 2 · 𝑁, after taking logs we get the inequality π(2 · 𝑁) · log(2 · 𝑁) ≤ log((2 · 𝑁)C𝑁), and bclbnd 24805 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.) |
⊢ 𝐾 = if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘4) → (log‘((4↑𝑁) / 𝑁)) < ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁)))) | ||
Theorem | chebbnd1lem2 24959 | Lemma for chebbnd1 24961: Show that log(𝑁) / 𝑁 does not change too much between 𝑁 and 𝑀 = ⌊(𝑁 / 2). (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ 𝑀 = (⌊‘(𝑁 / 2)) ⇒ ⊢ ((𝑁 ∈ ℝ ∧ 8 ≤ 𝑁) → ((log‘(2 · 𝑀)) / (2 · 𝑀)) < (2 · ((log‘𝑁) / 𝑁))) | ||
Theorem | chebbnd1lem3 24960 | Lemma for chebbnd1 24961: get a lower bound on π(𝑁) / (𝑁 / log(𝑁)) that is independent of 𝑁. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ 𝑀 = (⌊‘(𝑁 / 2)) ⇒ ⊢ ((𝑁 ∈ ℝ ∧ 8 ≤ 𝑁) → (((log‘2) − (1 / (2 · e))) / 2) < ((π‘𝑁) · ((log‘𝑁) / 𝑁))) | ||
Theorem | chebbnd1 24961 | The Chebyshev bound: The function π(𝑥) is eventually lower bounded by a positive constant times 𝑥 / log(𝑥). Alternatively stated, the function (𝑥 / log(𝑥)) / π(𝑥) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∈ 𝑂(1) | ||
Theorem | chtppilimlem1 24962 | Lemma for chtppilim 24964. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 𝑁 ∈ (2[,)+∞)) & ⊢ (𝜑 → ((𝑁↑𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴)) ⇒ ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁)) | ||
Theorem | chtppilimlem2 24963* | Lemma for chtppilim 24964. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 1) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))) | ||
Theorem | chtppilim 24964 | The θ function is asymptotic to π(𝑥)log(𝑥), so it is sufficient to prove θ(𝑥) / 𝑥 ⇝𝑟 1 to establish the PNT. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / ((π‘𝑥) · (log‘𝑥)))) ⇝𝑟 1 | ||
Theorem | chto1ub 24965 | The θ function is upper bounded by a linear term. Corollary of chtub 24737. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | chebbnd2 24966 | The Chebyshev bound, part 2: The function π(𝑥) is eventually upper bounded by a positive constant times 𝑥 / log(𝑥). Alternatively stated, the function π(𝑥) / (𝑥 / log(𝑥)) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝑥 ∈ (2[,)+∞) ↦ ((π‘𝑥) / (𝑥 / (log‘𝑥)))) ∈ 𝑂(1) | ||
Theorem | chto1lb 24967 | The θ function is lower bounded by a linear term. Corollary of chebbnd1 24961. (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ (𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∈ 𝑂(1) | ||
Theorem | chpchtlim 24968 | The ψ and θ functions are asymptotic to each other, so is sufficient to prove either θ(𝑥) / 𝑥 ⇝𝑟 1 or ψ(𝑥) / 𝑥 ⇝𝑟 1 to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟 1 | ||
Theorem | chpo1ub 24969 | The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.) |
⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | chpo1ubb 24970* | The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ ℝ+ (ψ‘𝑥) ≤ (𝑐 · 𝑥) | ||
Theorem | vmadivsum 24971* | The sum of the von Mangoldt function over 𝑛 is asymptotic to log𝑥 + 𝑂(1). Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.) |
⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) | ||
Theorem | vmadivsumb 24972* | Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ≤ 𝑐 | ||
Theorem | rplogsumlem1 24973* | Lemma for rplogsum 25016. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2) | ||
Theorem | rplogsumlem2 24974* | Lemma for rplogsum 25016. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2) | ||
Theorem | dchrisum0lem1a 24975 | Lemma for dchrisum0lem1 25005. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋)))) | ||
Theorem | rpvmasumlem 24976* | Lemma for rpvmasum 25015. Calculate the "trivial case" estimate Σ𝑛 ≤ 𝑥( 1 (𝑛)Λ(𝑛) / 𝑛) = log𝑥 + 𝑂(1), where 1 (𝑥) is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1)) | ||
Theorem | dchrisumlema 24977* | Lemma for dchrisum 24981. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ 𝐴) ⇝𝑟 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴)) ⇒ ⊢ (𝜑 → ((𝐼 ∈ ℝ+ → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ) ∧ (𝐼 ∈ (𝑀[,)+∞) → 0 ≤ ⦋𝐼 / 𝑛⦌𝐴))) | ||
Theorem | dchrisumlem1 24978* | Lemma for dchrisum 24981. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ 𝐴) ⇝𝑟 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛))) ≤ 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝑈 ∈ ℕ0) → (abs‘Σ𝑛 ∈ (0..^𝑈)(𝑋‘(𝐿‘𝑛))) ≤ 𝑅) | ||
Theorem | dchrisumlem2 24979* | Lemma for dchrisum 24981. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ 𝐴) ⇝𝑟 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛))) ≤ 𝑅) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑀 ≤ 𝑈) & ⊢ (𝜑 → 𝑈 ≤ (𝐼 + 1)) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐽 ∈ (ℤ≥‘𝐼)) ⇒ ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝐽) − (seq1( + , 𝐹)‘𝐼))) ≤ ((2 · 𝑅) · ⦋𝑈 / 𝑛⦌𝐴)) | ||
Theorem | dchrisumlem3 24980* | Lemma for dchrisum 24981. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ 𝐴) ⇝𝑟 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛))) ≤ 𝑅) ⇒ ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵))) | ||
Theorem | dchrisum 24981* | If 𝑛 ∈ [𝑀, +∞) ↦ 𝐴(𝑛) is a positive decreasing function approaching zero, then the infinite sum Σ𝑛, 𝑋(𝑛)𝐴(𝑛) is convergent, with the partial sum Σ𝑛 ≤ 𝑥, 𝑋(𝑛)𝐴(𝑛) within 𝑂(𝐴(𝑀)) of the limit 𝑇. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ 𝐴) ⇝𝑟 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴)) ⇒ ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵))) | ||
Theorem | dchrmusumlema 24982* | Lemma for dchrmusum 25013 and dchrisumn0 25010. Apply dchrisum 24981 for the function 1 / 𝑦. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) ⇒ ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))) | ||
Theorem | dchrmusum2 24983* | The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded, provided that 𝑇 ≠ 0. Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1)) | ||
Theorem | dchrvmasumlem1 24984* | 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 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)))) | ||
Theorem | dchrvmasum2lem 24985* | Give an expression for log𝑥 remarkably similar to Σ𝑛 ≤ 𝑥(𝑋(𝑛)Λ(𝑛) / 𝑛) given in dchrvmasumlem1 24984. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 1 ≤ 𝐴) ⇒ ⊢ (𝜑 → (log‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))) | ||
Theorem | dchrvmasum2if 24986* | Combine the results of dchrvmasumlem1 24984 and dchrvmasum2lem 24985 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 1 ≤ 𝐴) ⇒ ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)))) | ||
Theorem | dchrvmasumlem2 24987* | Lemma for dchrvmasum 25014. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) → 𝐹 ∈ ℂ) & ⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → 𝑇 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚))) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1)) | ||
Theorem | dchrvmasumlem3 24988* | Lemma for dchrvmasum 25014. (Contributed by Mario Carneiro, 3-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) → 𝐹 ∈ ℂ) & ⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → 𝑇 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚))) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ∈ 𝑂(1)) | ||
Theorem | dchrvmasumlema 24989* | Lemma for dchrvmasum 25014 and dchrvmasumif 24992. Apply dchrisum 24981 for the function log(𝑦) / 𝑦, which is decreasing above e (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) ⇒ ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) | ||
Theorem | dchrvmasumiflem1 24990* | Lemma for dchrvmasumif 24992. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦)) & ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) & ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇) & ⊢ (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦))) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) ∈ 𝑂(1)) | ||
Theorem | dchrvmasumiflem2 24991* | Lemma for dchrvmasum 25014. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦)) & ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) & ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇) & ⊢ (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦))) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1)) | ||
Theorem | dchrvmasumif 24992* | An asymptotic approximation for the sum of 𝑋(𝑛)Λ(𝑛) / 𝑛 conditional on the value of the infinite sum 𝑆. (We will later show that the case 𝑆 = 0 is impossible, and hence establish dchrvmasum 25014.) (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦)) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1)) | ||
Theorem | dchrvmaeq0 24993* | The set 𝑊 is the collection of all non-principal Dirichlet characters such that the sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ≠ 1 ) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) & ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦)) & ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0)) | ||
Theorem | dchrisum0fval 24994* | Value of the function 𝐹, the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) ⇒ ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑡))) | ||
Theorem | dchrisum0fmul 24995* | The function 𝐹, 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 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) ⇒ ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵))) | ||
Theorem | dchrisum0ff 24996* | The function 𝐹 is a real function. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) ⇒ ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | ||
Theorem | dchrisum0flblem1 24997* | Lemma for dchrisum0flb 24999. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑𝐴))) | ||
Theorem | dchrisum0flblem2 24998* | Lemma for dchrisum0flb 24999. Induction over relatively prime factors, with the prime power case handled in dchrisum0flblem1 . (Contributed by Mario Carneiro, 5-May-2016.) Replace reference to OLD theorem. (Revised by Wolf Lammen, 8-Sep-2020.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝐴) & ⊢ (𝜑 → ∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) ⇒ ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴)) | ||
Theorem | dchrisum0flb 24999* | 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 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴)) | ||
Theorem | dchrisum0fno1 25000* | The sum Σ𝑘 ≤ 𝑥, 𝐹(𝑥) / √𝑘 is divergent (i.e. not eventually bounded). Equation 9.4.30 of [Shapiro], p. 383. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) & ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ 𝑂(1)) ⇒ ⊢ ¬ 𝜑 |
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