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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlgsquad3 24801 Extend lgsquad2 24800 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 24802 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 24803* Lemma 1 for 2lgslem1a 24805. (Contributed by AV, 16-Jun-2021.)
((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ∀𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑖 · 2) = ((𝑖 · 2) mod 𝑃))
 
Theorem2lgslem1a2 24804 Lemma 2 for 2lgslem1a 24805. (Contributed by AV, 18-Jun-2021.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2)))
 
Theorem2lgslem1a 24805* Lemma 1 for 2lgslem1 24808. (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 24806* Lemma 2 for 2lgslem1 24808. (Contributed by AV, 18-Jun-2021.)
𝐼 = (𝐴...𝐵)    &   𝐹 = (𝑗𝐼 ↦ (𝑗 · 2))       𝐹:𝐼1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖𝐼 𝑥 = (𝑖 · 2)}
 
Theorem2lgslem1c 24807 Lemma 3 for 2lgslem1 24808. (Contributed by AV, 19-Jun-2021.)
((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
 
Theorem2lgslem1 24808* Lemma 1 for 2lgs 24821. (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 24809 Lemma 2 for 2lgs 24821. (Contributed by AV, 20-Jun-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ)
 
Theorem2lgslem3a 24810 Lemma for 2lgslem3a1 24814. (Contributed by AV, 14-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 1)) → 𝑁 = (2 · 𝐾))
 
Theorem2lgslem3b 24811 Lemma for 2lgslem3b1 24815. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 3)) → 𝑁 = ((2 · 𝐾) + 1))
 
Theorem2lgslem3c 24812 Lemma for 2lgslem3c1 24816. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 5)) → 𝑁 = ((2 · 𝐾) + 1))
 
Theorem2lgslem3d 24813 Lemma for 2lgslem3d1 24817. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝐾 ∈ ℕ0𝑃 = ((8 · 𝐾) + 7)) → 𝑁 = ((2 · 𝐾) + 2))
 
Theorem2lgslem3a1 24814 Lemma 1 for 2lgslem3 24818. (Contributed by AV, 15-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0)
 
Theorem2lgslem3b1 24815 Lemma 2 for 2lgslem3 24818. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 3) → (𝑁 mod 2) = 1)
 
Theorem2lgslem3c1 24816 Lemma 3 for 2lgslem3 24818. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 5) → (𝑁 mod 2) = 1)
 
Theorem2lgslem3d1 24817 Lemma 4 for 2lgslem3 24818. (Contributed by AV, 15-Jul-2021.)
𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))       ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 7) → (𝑁 mod 2) = 0)
 
Theorem2lgslem3 24818 Lemma 3 for 2lgs 24821. (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 24819 The Legendre symbol for 2 at 2 is 0. (Contributed by AV, 20-Jun-2021.)
(2 /L 2) = 0
 
Theorem2lgslem4 24820 Lemma 4 for 2lgs 24821: special case of 2lgs 24821 for 𝑃 = 2. (Contributed by AV, 20-Jun-2021.)
((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7})
 
Theorem2lgs 24821 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 24728) 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 24822 Lemma 1 for 2lgsoddprm 24830. (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 24823 Lemma 2 for 2lgsoddprm 24830. (Contributed by AV, 19-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ 2 ∥ (((𝑅↑2) − 1) / 8)))
 
Theorem2lgsoddprmlem3a 24824 Lemma 1 for 2lgsoddprmlem3 24828. (Contributed by AV, 20-Jul-2021.)
(((1↑2) − 1) / 8) = 0
 
Theorem2lgsoddprmlem3b 24825 Lemma 2 for 2lgsoddprmlem3 24828. (Contributed by AV, 20-Jul-2021.)
(((3↑2) − 1) / 8) = 1
 
Theorem2lgsoddprmlem3c 24826 Lemma 3 for 2lgsoddprmlem3 24828. (Contributed by AV, 20-Jul-2021.)
(((5↑2) − 1) / 8) = 3
 
Theorem2lgsoddprmlem3d 24827 Lemma 4 for 2lgsoddprmlem3 24828. (Contributed by AV, 20-Jul-2021.)
(((7↑2) − 1) / 8) = (2 · 3)
 
Theorem2lgsoddprmlem3 24828 Lemma 3 for 2lgsoddprm 24830. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑅↑2) − 1) / 8) ↔ 𝑅 ∈ {1, 7}))
 
Theorem2lgsoddprmlem4 24829 Lemma 4 for 2lgsoddprm 24830. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ (𝑁 mod 8) ∈ {1, 7}))
 
Theorem2lgsoddprm 24830 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 24831* Lemma for 2sq 24844. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))       (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
 
Theorem2sqlem2 24832* Lemma for 2sq 24844. (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 24833 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 24834 Lemma for 2sqlem5 24836. (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 24835 Lemma for 2sqlem5 24836. (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 24836 Lemma for 2sq 24844. 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 24837* Lemma for 2sq 24844. 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 24838* Lemma for 2sq 24844. (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 24839* Lemma for 2sqlem8 24840. (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 24840* Lemma for 2sq 24844. (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 24841* Lemma for 2sq 24844. (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 24842* Lemma for 2sq 24844. 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 24843* Lemma for 2sq 24844. (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 24844* 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 24845 The converse to 2sq 24844. (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 24846* The converse to 2sq 24844. (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 24847 Lemma for chebbnd1 24850: show a lower bound on π(𝑥) at even integers using similar techniques to those used to prove bpos 24707. (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 24698, 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 24694 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 24848 Lemma for chebbnd1 24850: 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 24849 Lemma for chebbnd1 24850: 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 24850 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 24851 Lemma for chtppilim 24853. (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 24852* Lemma for chtppilim 24853. (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 24853 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 24854 The θ function is upper bounded by a linear term. Corollary of chtub 24626. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)
 
Theoremchebbnd2 24855 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 24856 The θ function is lower bounded by a linear term. Corollary of chebbnd1 24850. (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∈ 𝑂(1)
 
Theoremchpchtlim 24857 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 24858 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 24859* 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 24860* 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 24861* 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 24862* Lemma for rplogsum 24905. (Contributed by Mario Carneiro, 2-May-2016.)
(𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)
 
Theoremrplogsumlem2 24863* Lemma for rplogsum 24905. 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 24864 Lemma for dchrisum0lem1 24894. (Contributed by Mario Carneiro, 7-Jun-2016.)
(((𝜑𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ‘(⌊‘𝑋))))
 
Theoremrpvmasumlem 24865* Lemma for rpvmasum 24904. 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 24866* Lemma for dchrisum 24870. 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 24867* Lemma for dchrisum 24870. 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 24868* Lemma for dchrisum 24870. 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 24869* Lemma for dchrisum 24870. 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 24870* 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 24871* Lemma for dchrmusum 24902 and dchrisumn0 24899. Apply dchrisum 24870 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 24872* 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 24873* 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 24874* Give an expression for log𝑥 remarkably similar to Σ𝑛𝑥(𝑋(𝑛)Λ(𝑛) / 𝑛) given in dchrvmasumlem1 24873. 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 24875* Combine the results of dchrvmasumlem1 24873 and dchrvmasum2lem 24874 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 24876* Lemma for dchrvmasum 24903. (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 24877* Lemma for dchrvmasum 24903. (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 24878* Lemma for dchrvmasum 24903 and dchrvmasumif 24881. Apply dchrisum 24870 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 24879* Lemma for dchrvmasumif 24881. (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 24880* Lemma for dchrvmasum 24903. (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 24881* 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 24903.) (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 24882* 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 24883* 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 24884* 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 24885* 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 24886* Lemma for dchrisum0flb 24888. 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 24887* Lemma for dchrisum0flb 24888. 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 24888* 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 24889* 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
 
Theoremrpvmasum2 24890* A partial result along the lines of rpvmasum 24904. The sum of the von Mangoldt function over those integers 𝑛𝐴 (mod 𝑁) is asymptotic to (1 − 𝑀)(log𝑥 / ϕ(𝑥)) + 𝑂(1), where 𝑀 is the number of non-principal Dirichlet characters with Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 = 0. Our goal is to show this set is empty. Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝐴𝑈)    &   𝑇 = (𝐿 “ {𝐴})    &   ((𝜑𝑓𝑊) → 𝐴 = (1r𝑍))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (#‘𝑊))))) ∈ 𝑂(1))
 
Theoremdchrisum0re 24891* Suppose 𝑋 is a non-principal Dirichlet character with Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 = 0. Then 𝑋 is a real character. Part of Lemma 9.4.4 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)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)       (𝜑𝑋:(Base‘𝑍)⟶ℝ)
 
Theoremdchrisum0lema 24892* Lemma for dchrisum0 24898. Apply dchrisum 24870 for the function 1 / √𝑦. (Contributed by Mario Carneiro, 10-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / (√‘𝑎)))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))
 
Theoremdchrisum0lem1b 24893* Lemma for dchrisum0lem1 24894. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / (√‘𝑎)))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))       (((𝜑𝑥 ∈ ℝ+) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))
 
Theoremdchrisum0lem1 24894* Lemma for dchrisum0 24898. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)    &   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+ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
 
Theoremdchrisum0lem2a 24895* Lemma for dchrisum0 24898. (Contributed by Mario Carneiro, 12-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / (√‘𝑎)))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))    &   𝐻 = (𝑦 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))    &   (𝜑𝐻𝑟 𝑈)       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))
 
Theoremdchrisum0lem2 24896* Lemma for dchrisum0 24898. (Contributed by Mario Carneiro, 12-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / (√‘𝑎)))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))    &   𝐻 = (𝑦 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))    &   (𝜑𝐻𝑟 𝑈)    &   𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐸 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦))       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
 
Theoremdchrisum0lem3 24897* Lemma for dchrisum0 24898. (Contributed by Mario Carneiro, 12-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)    &   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+ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1))
 
Theoremdchrisum0 24898* The sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is nonzero for all non-principal Dirichlet characters (i.e. the assumption 𝑋𝑊 is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 24872 and dchrvmasumif 24881. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}    &   (𝜑𝑋𝑊)        ¬ 𝜑
 
Theoremdchrisumn0 24899* The sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is nonzero for all non-principal Dirichlet characters (i.e. the assumption 𝑋𝑊 is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 24872 and dchrvmasumif 24881. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-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))       (𝜑𝑇 ≠ 0)
 
Theoremdchrmusumlem 24900* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-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+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))
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