P. 266 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2sqreultblem 26501*	Lemma for 2sqreultb 26512. (Contributed by AV, 10-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 18-Jun-2023.)
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
Theorem	2sqreunnlem1 26502*	Lemma 1 for 2sqreunn 26510. (Contributed by AV, 11-Jun-2023.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
Theorem	2sqreunnltlem 26503*	Lemma for 2sqreunnlt 26513. (Contributed by AV, 4-Jun-2023.) Specialization to different integers, proposed by GL. (Revised by AV, 11-Jun-2023.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
Theorem	2sqreunnltblem 26504*	Lemma for 2sqreunnltb 26514. (Contributed by AV, 11-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 18-Jun-2023.)
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
Theorem	2sqreulem2 26505	Lemma 2 for 2sqreu 26509 etc. (Contributed by AV, 25-Jun-2023.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((𝐴↑2) + (𝐵↑2)) = ((𝐴↑2) + (𝐶↑2)) → 𝐵 = 𝐶))
Theorem	2sqreulem3 26506	Lemma 3 for 2sqreu 26509 etc. (Contributed by AV, 25-Jun-2023.)
⊢ ((𝐴 ∈ ℕ0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0)) → (((𝜑 ∧ ((𝐴↑2) + (𝐵↑2)) = 𝑃) ∧ (𝜓 ∧ ((𝐴↑2) + (𝐶↑2)) = 𝑃)) → 𝐵 = 𝐶))
Theorem	2sqreulem4 26507*	Lemma 4 for 2sqreu 26509 et. (Contributed by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑
Theorem	2sqreunnlem2 26508*	Lemma 2 for 2sqreunn 26510. (Contributed by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ ∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑
Theorem	2sqreu 26509*	There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two nonnegative integers. See 2sqnn0 26491 for the existence of such a decomposition. (Contributed by AV, 4-Jun-2023.) (Revised by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑))
Theorem	2sqreunn 26510*	There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two positive integers. See 2sqnn 26492 for the existence of such a decomposition. (Contributed by AV, 11-Jun-2023.) (Revised by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))
Theorem	2sqreult 26511*	There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers. (Contributed by AV, 8-Jun-2023.) (Proposed by GL, 8-Jun-2023.) (Revised by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑))
Theorem	2sqreultb 26512*	There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers iff 𝑃≡1 (mod 4). (Contributed by AV, 10-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑)))
Theorem	2sqreunnlt 26513*	There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two different positive integers. (Contributed by AV, 4-Jun-2023.) Specialization to different integers, proposed by GL. (Revised by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))
Theorem	2sqreunnltb 26514*	There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. (Contributed by AV, 11-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 25-Jun-2023.)
⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))    ⇒   ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑)))
Theorem	2sqreuop 26515*	There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two nonnegative integers. Ordered pair variant of 2sqreu 26509. (Contributed by AV, 2-Jul-2023.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st ‘𝑝) ≤ (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃))
Theorem	2sqreuopnn 26516*	There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two positive integers. Ordered pair variant of 2sqreunn 26510. (Contributed by AV, 2-Jul-2023.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ × ℕ)((1st ‘𝑝) ≤ (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃))
Theorem	2sqreuoplt 26517*	There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers. Ordered pair variant of 2sqreult 26511. (Contributed by AV, 2-Jul-2023.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃))
Theorem	2sqreuopltb 26518*	There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers iff 𝑃≡1 (mod 4). Ordered pair variant of 2sqreultb 26512. (Contributed by AV, 3-Jul-2023.)
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃)))
Theorem	2sqreuopnnlt 26519*	There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two different positive integers. Ordered pair variant of 2sqreunnlt 26513. (Contributed by AV, 3-Jul-2023.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ × ℕ)((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃))
Theorem	2sqreuopnnltb 26520*	There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Ordered pair variant of 2sqreunnltb 26514. (Contributed by AV, 3-Jul-2023.)
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃)))
Theorem	2sqreuopb 26521*	There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Alternate ordered pair variant of 2sqreunnltb 26514. (Contributed by AV, 3-Jul-2023.)
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))))
14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem
Theorem	chebbnd1lem1 26522	Lemma for chebbnd1 26525: show a lower bound on π(𝑥) at even integers using similar techniques to those used to prove bpos 26346. (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 26337, 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 26333 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 26523	Lemma for chebbnd1 26525: 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 26524	Lemma for chebbnd1 26525: 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 26525	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 26526	Lemma for chtppilim 26528. (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 26527*	Lemma for chtppilim 26528. (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 26528	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 26529	The θ function is upper bounded by a linear term. Corollary of chtub 26265. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)
Theorem	chebbnd2 26530	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 26531	The θ function is lower bounded by a linear term. Corollary of chebbnd1 26525. (Contributed by Mario Carneiro, 8-Apr-2016.)
⊢ (𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∈ 𝑂(1)
Theorem	chpchtlim 26532	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 26533	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 26534*	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 26535*	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 26536*	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 26537*	Lemma for rplogsum 26580. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)
Theorem	rplogsumlem2 26538*	Lemma for rplogsum 26580. 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 26539	Lemma for dchrisum0lem1 26569. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋))))
Theorem	rpvmasumlem 26540*	Lemma for rpvmasum 26579. 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 26541*	Lemma for dchrisum 26545. 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 26542*	Lemma for dchrisum 26545. 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 26543*	Lemma for dchrisum 26545. 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 26544*	Lemma for dchrisum 26545. 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 26545*	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 26546*	Lemma for dchrmusum 26577 and dchrisumn0 26574. Apply dchrisum 26545 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 26547*	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 26548*	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 26549*	Give an expression for log𝑥 remarkably similar to Σ𝑛 ≤ 𝑥(𝑋(𝑛)Λ(𝑛) / 𝑛) given in dchrvmasumlem1 26548. 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 26550*	Combine the results of dchrvmasumlem1 26548 and dchrvmasum2lem 26549 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 26551*	Lemma for dchrvmasum 26578. (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 26552*	Lemma for dchrvmasum 26578. (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 26553*	Lemma for dchrvmasum 26578 and dchrvmasumif 26556. Apply dchrisum 26545 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 26554*	Lemma for dchrvmasumif 26556. (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 26555*	Lemma for dchrvmasum 26578. (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 26556*	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 26578.) (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 26557*	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 26558*	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 26559*	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 26560*	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 26561*	Lemma for dchrisum0flb 26563. 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 26562*	Lemma for dchrisum0flb 26563. 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 26563*	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 26564*	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
Theorem	rpvmasum2 26565*	A partial result along the lines of rpvmasum 26579. The sum of the von Mangoldt function over those integers 𝑛≡𝐴 (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)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ 𝑈 = (Unit‘𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ 𝑇 = (◡𝐿 “ {𝐴})    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → 𝐴 = (1r‘𝑍))    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈ 𝑂(1))
Theorem	dchrisum0re 26566*	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)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
Theorem	dchrisum0lema 26567*	Lemma for dchrisum0 26573. Apply dchrisum 26545 for the function 1 / √𝑦. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))    ⇒   ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))
Theorem	dchrisum0lem1b 26568*	Lemma for dchrisum0lem1 26569. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ 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+) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))
Theorem	dchrisum0lem1 26569*	Lemma for dchrisum0 26573. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ 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+ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
Theorem	dchrisum0lem2a 26570*	Lemma for dchrisum0 26573. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))    &   ⊢ (𝜑 → 𝐶 ∈ (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))))    &   ⊢ (𝜑 → 𝐻 ⇝𝑟 𝑈)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))
Theorem	dchrisum0lem2 26571*	Lemma for dchrisum0 26573. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))    &   ⊢ (𝜑 → 𝐶 ∈ (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))))    &   ⊢ (𝜑 → 𝐻 ⇝𝑟 𝑈)    &   ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))    &   ⊢ (𝜑 → 𝐸 ∈ (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‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
Theorem	dchrisum0lem3 26572*	Lemma for dchrisum0 26573. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ 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+ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1))
Theorem	dchrisum0 26573*	The sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is nonzero for all non-principal Dirichlet characters (i.e. the assumption 𝑋 ∈ 𝑊 is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 26547 and dchrvmasumif 26556. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    ⇒   ⊢ ¬ 𝜑
Theorem	dchrisumn0 26574*	The sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is nonzero for all non-principal Dirichlet characters (i.e. the assumption 𝑋 ∈ 𝑊 is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 26547 and dchrvmasumif 26556. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-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 → 𝑇 ≠ 0)
Theorem	dchrmusumlem 26575*	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)    &   ⊢ (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+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))
Theorem	dchrvmasumlem 26576*	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)    &   ⊢ (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+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
Theorem	dchrmusum 26577*	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)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 1 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ≠ 1 )    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))
Theorem	dchrvmasum 26578*	The sum of the von Mangoldt function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded. Equation 9.4.8 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 12-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 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
Theorem	rpvmasum 26579*	The sum of the von Mangoldt function over those integers 𝑛≡𝐴 (mod 𝑁) is asymptotic to log𝑥 / ϕ(𝑥) + 𝑂(1). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑈 = (Unit‘𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ 𝑇 = (◡𝐿 “ {𝐴})    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
Theorem	rplogsum 26580*	The sum of log𝑝 / 𝑝 over the primes 𝑝≡𝐴 (mod 𝑁) is asymptotic to log𝑥 / ϕ(𝑥) + 𝑂(1). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 16-Apr-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑈 = (Unit‘𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ 𝑇 = (◡𝐿 “ {𝐴})    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑝 ∈ ((1...(⌊‘𝑥)) ∩ (ℙ ∩ 𝑇))((log‘𝑝) / 𝑝)) − (log‘𝑥))) ∈ 𝑂(1))
Theorem	dirith2 26581	Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to 𝑁. Theorem 9.4.1 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑈 = (Unit‘𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ 𝑇 = (◡𝐿 “ {𝐴})    ⇒   ⊢ (𝜑 → (ℙ ∩ 𝑇) ≈ ℕ)
Theorem	dirith 26582*	Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to 𝑁. Theorem 9.4.1 of [Shapiro], p. 375. See https://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html for an informal exposition. This is Metamath 100 proof #48. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → {𝑝 ∈ ℙ ∣ 𝑁 ∥ (𝑝 − 𝐴)} ≈ ℕ)
14.4.13  The Prime Number Theorem
Theorem	mudivsum 26583*	Asymptotic formula for Σ𝑛 ≤ 𝑥, μ(𝑛) / 𝑛 = 𝑂(1). Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)
Theorem	mulogsumlem 26584*	Lemma for mulogsum 26585. (Contributed by Mario Carneiro, 14-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)
Theorem	mulogsum 26585*	Asymptotic formula for Σ𝑛 ≤ 𝑥, (μ(𝑛) / 𝑛)log(𝑥 / 𝑛) = 𝑂(1). Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)
Theorem	logdivsum 26586*	Asymptotic analysis of Σ𝑛 ≤ 𝑥, log𝑛 / 𝑛 = (log𝑥)↑2 / 2 + 𝐿 + 𝑂(log𝑥 / 𝑥). (Contributed by Mario Carneiro, 18-May-2016.)
⊢ 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))    ⇒   ⊢ (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
Theorem	mulog2sumlem1 26587*	Asymptotic formula for Σ𝑛 ≤ 𝑥, log(𝑥 / 𝑛) / 𝑛 = (1 / 2)log↑2(𝑥) + γ · log𝑥 − 𝐿 + 𝑂(log𝑥 / 𝑥), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)
⊢ 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))    &   ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → e ≤ 𝐴)    ⇒   ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))
Theorem	mulog2sumlem2 26588*	Lemma for mulog2sum 26590. (Contributed by Mario Carneiro, 19-May-2016.)
⊢ 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))    &   ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)    &   ⊢ 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))    &   ⊢ 𝑅 = (((1 / 2) + (γ + (abs‘𝐿))) + Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ∈ 𝑂(1))
Theorem	mulog2sumlem3 26589*	Lemma for mulog2sum 26590. (Contributed by Mario Carneiro, 13-May-2016.)
⊢ 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))    &   ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
Theorem	mulog2sum 26590*	Asymptotic formula for Σ𝑛 ≤ 𝑥, (μ(𝑛) / 𝑛)log↑2(𝑥 / 𝑛) = 2log𝑥 + 𝑂(1). Equation 10.2.8 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 19-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Theorem	vmalogdivsum2 26591*	The sum Σ𝑛 ≤ 𝑥, Λ(𝑛)log(𝑥 / 𝑛) / 𝑛 is asymptotic to log↑2(𝑥) / 2 + 𝑂(log𝑥). Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Theorem	vmalogdivsum 26592*	The sum Σ𝑛 ≤ 𝑥, Λ(𝑛)log𝑛 / 𝑛 is asymptotic to log↑2(𝑥) / 2 + 𝑂(log𝑥). Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Theorem	2vmadivsumlem 26593*	Lemma for 2vmadivsum 26594. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
Theorem	2vmadivsum 26594*	The sum Σ𝑚𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛) / 𝑚𝑛 is asymptotic to log↑2(𝑥) / 2 + 𝑂(log𝑥). (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Theorem	logsqvma 26595*	A formula for log↑2(𝑁) in terms of the primes. Equation 10.4.6 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (Σ𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑑} ((Λ‘𝑢) · (Λ‘(𝑑 / 𝑢))) + ((Λ‘𝑑) · (log‘𝑑))) = ((log‘𝑁)↑2))
Theorem	logsqvma2 26596*	The Möbius inverse of logsqvma 26595. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ((μ‘𝑑) · ((log‘(𝑁 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ((Λ‘𝑑) · (Λ‘(𝑁 / 𝑑))) + ((Λ‘𝑁) · (log‘𝑁))))
Theorem	log2sumbnd 26597*	Bound on the difference between Σ𝑛 ≤ 𝐴, log↑2(𝑛) and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)
⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2))
Theorem	selberglem1 26598*	Lemma for selberg 26601. Estimation of the asymptotic part of selberglem3 26600. (Contributed by Mario Carneiro, 20-May-2016.)
⊢ 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)    ⇒   ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Theorem	selberglem2 26599*	Lemma for selberg 26601. (Contributed by Mario Carneiro, 23-May-2016.)
⊢ 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)    ⇒   ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Theorem	selberglem3 26600*	Lemma for selberg 26601. Estimation of the left-hand side of logsqvma2 26596. (Contributed by Mario Carneiro, 23-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)