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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlgseisenlem2 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.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)    &   𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)    &   𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))    &   𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)       (𝜑𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2)))
 
Theoremlgseisenlem3 24902* Lemma for lgseisen 24904. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-2019.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)    &   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)))) = (1r𝑌))
 
Theoremlgseisenlem4 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.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)    &   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))
 
Theoremlgseisen 24904* Eisenstein's lemma, an expression for (𝑃 /L 𝑄) when 𝑃, 𝑄 are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)       (𝜑 → (𝑄 /L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
 
Theoremlgsquadlem1 24905* Lemma for lgsquad 24908. Count the members of 𝑆 with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)    &   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↑(#‘{𝑧𝑆 ∣ ¬ 2 ∥ (1st𝑧)})))
 
Theoremlgsquadlem2 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.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)    &   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)))
 
Theoremlgsquadlem3 24907* Lemma for lgsquad 24908. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝜑𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑𝑃𝑄)    &   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)))
 
Theoremlgsquad 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}) ∧ 𝑃𝑄) → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(((𝑃 − 1) / 2) · ((𝑄 − 1) / 2))))
 
Theoremlgsquad2lem1 24909 Lemma for lgsquad2 24911. (Contributed by Mario Carneiro, 19-Jun-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑀)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑁)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ 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))))
 
Theoremlgsquad2lem2 24910* Lemma for lgsquad2 24911. (Contributed by Mario Carneiro, 19-Jun-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑀)    &   (𝜑𝑁 ∈ 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))))
 
Theoremlgsquad2 24911 Extend lgsquad 24908 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑀)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑁)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)       (𝜑 → ((𝑀 /L 𝑁) · (𝑁 /L 𝑀)) = (-1↑(((𝑀 − 1) / 2) · ((𝑁 − 1) / 2))))
 
Theoremlgsquad3 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)))
 
Theoremm1lgs 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))
 
Theorem2lgslem1a1 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))
 
Theorem2lgslem1a2 24915 Lemma 2 for 2lgslem1a 24916. (Contributed by AV, 18-Jun-2021.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2)))
 
Theorem2lgslem1a 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)})
 
Theorem2lgslem1b 24917* Lemma 2 for 2lgslem1 24919. (Contributed by AV, 18-Jun-2021.)
𝐼 = (𝐴...𝐵)    &   𝐹 = (𝑗𝐼 ↦ (𝑗 · 2))       𝐹:𝐼1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖𝐼 𝑥 = (𝑖 · 2)}
 
Theorem2lgslem1c 24918 Lemma 3 for 2lgslem1 24919. (Contributed by AV, 19-Jun-2021.)
((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
 
Theorem2lgslem1 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))))
 
Theorem2lgslem2 24920 Lemma 2 for 2lgs 24932. (Contributed by AV, 20-Jun-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ)
 
Theorem2lgslem3a 24921 Lemma for 2lgslem3a1 24925. (Contributed by AV, 14-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 1)) → 𝑁 = (2 · 𝐾))
 
Theorem2lgslem3b 24922 Lemma for 2lgslem3b1 24926. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 3)) → 𝑁 = ((2 · 𝐾) + 1))
 
Theorem2lgslem3c 24923 Lemma for 2lgslem3c1 24927. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 5)) → 𝑁 = ((2 · 𝐾) + 1))
 
Theorem2lgslem3d 24924 Lemma for 2lgslem3d1 24928. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 7)) → 𝑁 = ((2 · 𝐾) + 2))
 
Theorem2lgslem3a1 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)
 
Theorem2lgslem3b1 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)
 
Theorem2lgslem3c1 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)
 
Theorem2lgslem3d1 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)
 
Theorem2lgslem3 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))
 
Theorem2lgs2 24930 The Legendre symbol for 2 at 2 is 0. (Contributed by AV, 20-Jun-2021.)
(2 /L 2) = 0
 
Theorem2lgslem4 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})
 
Theorem2lgs 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}))
 
Theorem2lgsoddprmlem1 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)))
 
Theorem2lgsoddprmlem2 24934 Lemma 2 for 2lgsoddprm 24941. (Contributed by AV, 19-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ 2 ∥ (((𝑅↑2) − 1) / 8)))
 
Theorem2lgsoddprmlem3a 24935 Lemma 1 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.)
(((1↑2) − 1) / 8) = 0
 
Theorem2lgsoddprmlem3b 24936 Lemma 2 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.)
(((3↑2) − 1) / 8) = 1
 
Theorem2lgsoddprmlem3c 24937 Lemma 3 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.)
(((5↑2) − 1) / 8) = 3
 
Theorem2lgsoddprmlem3d 24938 Lemma 4 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.)
(((7↑2) − 1) / 8) = (2 · 3)
 
Theorem2lgsoddprmlem3 24939 Lemma 3 for 2lgsoddprm 24941. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑅↑2) − 1) / 8) ↔ 𝑅 ∈ {1, 7}))
 
Theorem2lgsoddprmlem4 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}))
 
Theorem2lgsoddprm 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)))
 
14.4.11  All primes 4n+1 are the sum of two squares
 
Theorem2sqlem1 24942* Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))       (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
 
Theorem2sqlem2 24943* Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))       (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)))
 
Theoremmul2sq 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))       ((𝐴𝑆𝐵𝑆) → (𝐴 · 𝐵) ∈ 𝑆)
 
Theorem2sqlem3 24945 Lemma for 2sqlem5 24947. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑 → (𝑁 · 𝑃) = ((𝐴↑2) + (𝐵↑2)))    &   (𝜑𝑃 = ((𝐶↑2) + (𝐷↑2)))    &   (𝜑𝑃 ∥ ((𝐶 · 𝐵) + (𝐴 · 𝐷)))       (𝜑𝑁𝑆)
 
Theorem2sqlem4 24946 Lemma for 2sqlem5 24947. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑 → (𝑁 · 𝑃) = ((𝐴↑2) + (𝐵↑2)))    &   (𝜑𝑃 = ((𝐶↑2) + (𝐷↑2)))       (𝜑𝑁𝑆)
 
Theorem2sqlem5 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))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (𝑁 · 𝑃) ∈ 𝑆)    &   (𝜑𝑃𝑆)       (𝜑𝑁𝑆)
 
Theorem2sqlem6 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))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → ∀𝑝 ∈ ℙ (𝑝𝐵𝑝𝑆))    &   (𝜑 → (𝐴 · 𝐵) ∈ 𝑆)       (𝜑𝐴𝑆)
 
Theorem2sqlem7 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)
 
Theorem2sqlem8a 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))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))    &   (𝜑𝑀𝑁)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑𝑁 = ((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)
 
Theorem2sqlem8 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))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))    &   (𝜑𝑀𝑁)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑𝑁 = ((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))       (𝜑𝑀𝑆)
 
Theorem2sqlem9 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))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))    &   (𝜑𝑀𝑁)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁𝑌)       (𝜑𝑀𝑆)
 
Theorem2sqlem10 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)))}       ((𝐴𝑌𝐵 ∈ ℕ ∧ 𝐵𝐴) → 𝐵𝑆)
 
Theorem2sqlem11 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) → 𝑃𝑆)
 
Theorem2sq 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)))
 
Theorem2sqblem 24956 The converse to 2sq 24955. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑 → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))    &   (𝜑 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))    &   (𝜑𝑃 = ((𝑋↑2) + (𝑌↑2)))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → (𝑃 gcd 𝑌) = ((𝑃 · 𝐴) + (𝑌 · 𝐵)))       (𝜑 → (𝑃 mod 4) = 1)
 
Theorem2sqb 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)))
 
14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem
 
Theoremchebbnd1lem1 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))))
 
Theoremchebbnd1lem2 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)))
 
Theoremchebbnd1lem3 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) < ((π𝑁) · ((log‘𝑁) / 𝑁)))
 
Theoremchebbnd1 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)) / (π𝑥))) ∈ 𝑂(1)
 
Theoremchtppilimlem1 24962 Lemma for chtppilim 24964. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐴 < 1)    &   (𝜑𝑁 ∈ (2[,)+∞))    &   (𝜑 → ((𝑁𝑐𝐴) / (π𝑁)) < (1 − 𝐴))       (𝜑 → ((𝐴↑2) · ((π𝑁) · (log‘𝑁))) < (θ‘𝑁))
 
Theoremchtppilimlem2 24963* Lemma for chtppilim 24964. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐴 < 1)       (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧𝑥 → ((𝐴↑2) · ((π𝑥) · (log‘𝑥))) < (θ‘𝑥)))
 
Theoremchtppilim 24964 The θ function is asymptotic to π(𝑥)log(𝑥), so it is sufficient to prove θ(𝑥) / 𝑥𝑟 1 to establish the PNT. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / ((π𝑥) · (log‘𝑥)))) ⇝𝑟 1
 
Theoremchto1ub 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)
 
Theoremchebbnd2 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[,)+∞) ↦ ((π𝑥) / (𝑥 / (log‘𝑥)))) ∈ 𝑂(1)
 
Theoremchto1lb 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)
 
Theoremchpchtlim 24968 The ψ and θ functions are asymptotic to each other, so is sufficient to prove either θ(𝑥) / 𝑥𝑟 1 or ψ(𝑥) / 𝑥𝑟 1 to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟 1
 
Theoremchpo1ub 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)
 
Theoremchpo1ubb 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)
 
Theoremvmadivsum 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)
 
Theoremvmadivsumb 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
 
Theoremrplogsumlem1 24973* Lemma for rplogsum 25016. (Contributed by Mario Carneiro, 2-May-2016.)
(𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)
 
Theoremrplogsumlem2 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)
 
Theoremdchrisum0lem1a 24975 Lemma for dchrisum0lem1 25005. (Contributed by Mario Carneiro, 7-Jun-2016.)
(((𝜑𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ‘(⌊‘𝑋))))
 
Theoremrpvmasumlem 24976* Lemma for rpvmasum 25015. Calculate the "trivial case" estimate Σ𝑛𝑥( 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
 
Theoremdchrisumlema 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))       (𝜑 → ((𝐼 ∈ ℝ+𝐼 / 𝑛𝐴 ∈ ℝ) ∧ (𝐼 ∈ (𝑀[,)+∞) → 0 ≤ 𝐼 / 𝑛𝐴)))
 
Theoremdchrisumlem1 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿𝑛))) ≤ 𝑅)       ((𝜑𝑈 ∈ ℕ0) → (abs‘Σ𝑛 ∈ (0..^𝑈)(𝑋‘(𝐿𝑛))) ≤ 𝑅)
 
Theoremdchrisumlem2 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿𝑛))) ≤ 𝑅)    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑀𝑈)    &   (𝜑𝑈 ≤ (𝐼 + 1))    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐽 ∈ (ℤ𝐼))       (𝜑 → (abs‘((seq1( + , 𝐹)‘𝐽) − (seq1( + , 𝐹)‘𝐼))) ≤ ((2 · 𝑅) · 𝑈 / 𝑛𝐴))
 
Theoremdchrisumlem3 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿𝑛))) ≤ 𝑅)       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))
 
Theoremdchrisum 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 Σ𝑛𝑥, 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))
 
Theoremdchrmusumlema 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))
 
Theoremdchrmusum2 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1))
 
Theoremdchrvmasumlem1 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿𝑚)) · ((log‘𝑚) / 𝑚))))
 
Theoremdchrvmasum2lem 24985* Give an expression for log𝑥 remarkably similar to Σ𝑛𝑥(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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → (log‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
 
Theoremdchrvmasum2if 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))))
 
Theoremdchrvmasumlem2 24987* Lemma for dchrvmasum 25014. (Contributed by Mario Carneiro, 4-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   ((𝜑𝑚 ∈ ℝ+) → 𝐹 ∈ ℂ)    &   (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑𝑇 ∈ ℂ)    &   ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹𝑇)) ≤ 𝑅)       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾𝑇)) / 𝑑)) ∈ 𝑂(1))
 
Theoremdchrvmasumlem3 24988* Lemma for dchrvmasum 25014. (Contributed by Mario Carneiro, 3-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   ((𝜑𝑚 ∈ ℝ+) → 𝐹 ∈ ℂ)    &   (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑𝑇 ∈ ℂ)    &   ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹𝑇)) ≤ 𝑅)       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾𝑇))) ∈ 𝑂(1))
 
Theoremdchrvmasumlema 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))
 
Theoremdchrvmasumiflem1 24990* Lemma for dchrvmasumif 24992. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))    &   𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))    &   (𝜑𝐸 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) ∈ 𝑂(1))
 
Theoremdchrvmasumiflem2 24991* Lemma for dchrvmasum 25014. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))    &   𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))    &   (𝜑𝐸 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))
 
Theoremdchrvmasumif 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))
 
Theoremdchrvmaeq0 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}       (𝜑 → (𝑋𝑊𝑆 = 0))
 
Theoremdchrisum0fval 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))       (𝐴 ∈ ℕ → (𝐹𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))
 
Theoremdchrisum0fmul 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)       (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) · (𝐹𝐵)))
 
Theoremdchrisum0ff 24996* The function 𝐹 is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)       (𝜑𝐹:ℕ⟶ℝ)
 
Theoremdchrisum0flblem1 24997* Lemma for dchrisum0flb 24999. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℕ0)       (𝜑 → if((√‘(𝑃𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃𝐴)))
 
Theoremdchrisum0flblem2 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝐴)    &   (𝜑 → ∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹𝑦))       (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹𝐴))
 
Theoremdchrisum0flb 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑𝐴 ∈ ℕ)       (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹𝐴))
 
Theoremdchrisum0fno1 25000* The sum Σ𝑘𝑥, 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))) ∈ 𝑂(1))        ¬ 𝜑
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