P. 276 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	chebbnd1lem3 27501	Lemma for chebbnd1 27502: 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 27502	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 27503	Lemma for chtppilim 27505. (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 27504*	Lemma for chtppilim 27505. (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 27505	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 27506	The θ function is upper bounded by a linear term. Corollary of chtub 27242. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)
Theorem	chebbnd2 27507	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 27508	The θ function is lower bounded by a linear term. Corollary of chebbnd1 27502. (Contributed by Mario Carneiro, 8-Apr-2016.)
⊢ (𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∈ 𝑂(1)
Theorem	chpchtlim 27509	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 27510	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 27511*	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 27512*	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 27513*	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 27514*	Lemma for rplogsum 27557. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)
Theorem	rplogsumlem2 27515*	Lemma for rplogsum 27557. 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 27516	Lemma for dchrisum0lem1 27546. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋))))
Theorem	rpvmasumlem 27517*	Lemma for rpvmasum 27556. 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 27518*	Lemma for dchrisum 27522. 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 27519*	Lemma for dchrisum 27522. 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 27520*	Lemma for dchrisum 27522. 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 27521*	Lemma for dchrisum 27522. 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 27522*	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 27523*	Lemma for dchrmusum 27554 and dchrisumn0 27551. Apply dchrisum 27522 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 27524*	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 27525*	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 27526*	Give an expression for log𝑥 remarkably similar to Σ𝑛 ≤ 𝑥(𝑋(𝑛)Λ(𝑛) / 𝑛) given in dchrvmasumlem1 27525. 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 27527*	Combine the results of dchrvmasumlem1 27525 and dchrvmasum2lem 27526 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 27528*	Lemma for dchrvmasum 27555. (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 27529*	Lemma for dchrvmasum 27555. (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 27530*	Lemma for dchrvmasum 27555 and dchrvmasumif 27533. Apply dchrisum 27522 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 27531*	Lemma for dchrvmasumif 27533. (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 27532*	Lemma for dchrvmasum 27555. (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 27533*	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 27555.) (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 27534*	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 27535*	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 27536*	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 27537*	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 27538*	Lemma for dchrisum0flb 27540. 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 27539*	Lemma for dchrisum0flb 27540. 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 27540*	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 27541*	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 27542*	A partial result along the lines of rpvmasum 27556. 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 27543*	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 27544*	Lemma for dchrisum0 27550. Apply dchrisum 27522 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 27545*	Lemma for dchrisum0lem1 27546. (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 27546*	Lemma for dchrisum0 27550. (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 27547*	Lemma for dchrisum0 27550. (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 27548*	Lemma for dchrisum0 27550. (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 27549*	Lemma for dchrisum0 27550. (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 27550*	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 to eliminate the conditionals from dchrmusum2 27524 and dchrvmasumif 27533. 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 27551*	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 to eliminate the conditionals from dchrmusum2 27524 and dchrvmasumif 27533. 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 27552*	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 27553*	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 27554*	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 27555*	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 27556*	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 27557*	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 27558	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 27559*	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 27560*	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 27561*	Lemma for mulogsum 27562. (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 27562*	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 27563*	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 27564*	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 27565*	Lemma for mulog2sum 27567. (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 27566*	Lemma for mulog2sum 27567. (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 27567*	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 27568*	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 27569*	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 27570*	Lemma for 2vmadivsum 27571. (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 27571*	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 27572*	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 27573*	The Möbius inverse of logsqvma 27572. 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 27574*	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 27575*	Lemma for selberg 27578. Estimation of the asymptotic part of selberglem3 27577. (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 27576*	Lemma for selberg 27578. (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 27577*	Lemma for selberg 27578. Estimation of the left-hand side of logsqvma2 27573. (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)
Theorem	selberg 27578*	Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Σ𝑛 ≤ 𝑥, Λ(𝑛)log𝑛 + Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛) = 2𝑥log𝑥 + 𝑂(𝑥). Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Theorem	selbergb 27579*	Convert eventual boundedness in selberg 27578 to boundedness on [1, +∞). (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 30-May-2016.)
⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐
Theorem	selberg2lem 27580*	Lemma for selberg2 27581. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
Theorem	selberg2 27581*	Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
⊢ (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Theorem	selberg2b 27582*	Convert eventual boundedness in selberg2 27581 to boundedness on any interval [𝐴, +∞). (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 25-May-2016.)
⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐
Theorem	chpdifbndlem1 27583*	Lemma for chpdifbnd 27585. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 1 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵)    &   ⊢ 𝐶 = ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴)))    &   ⊢ (𝜑 → 𝑋 ∈ (1(,)+∞))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑋[,](𝐴 · 𝑋)))    ⇒   ⊢ (𝜑 → ((ψ‘𝑌) − (ψ‘𝑋)) ≤ ((2 · (𝑌 − 𝑋)) + (𝐶 · (𝑋 / (log‘𝑋)))))
Theorem	chpdifbndlem2 27584*	Lemma for chpdifbnd 27585. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 1 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵)    &   ⊢ 𝐶 = ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴)))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))))
Theorem	chpdifbnd 27585*	A bound on the difference of nearby ψ values. Theorem 10.5.2 of [Shapiro], p. 427. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))))
Theorem	logdivbnd 27586*	A bound on a sum of logs, used in pntlemk 27636. This is not as precise as logdivsum 27563 in its asymptotic behavior, but it is valid for all 𝑁 and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.)
⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2))
Theorem	selberg3lem1 27587*	Introduce a log weighting on the summands of Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 27581 (written here as Σ𝑛 ≤ 𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))
Theorem	selberg3lem2 27588*	Lemma for selberg3 27589. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1)
Theorem	selberg3 27589*	Introduce a log weighting on the summands of Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 27581 (written here as Σ𝑛 ≤ 𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝑥 ∈ (1(,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Theorem	selberg4lem1 27590*	Lemma for selberg4 27591. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
Theorem	selberg4 27591*	The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form Σ𝑖𝑗𝑘 ≤ 𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘); we eliminate one of the nested sums by using the definition of ψ(𝑥) = Σ𝑘 ≤ 𝑥, Λ(𝑘). This statement can thus equivalently be written ψ(𝑥)log↑2(𝑥) = 2Σ𝑖𝑗𝑘 ≤ 𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘) + 𝑂(𝑥log𝑥). Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)
Theorem	pntrval 27592*	Define the residual of the second Chebyshev function. The goal is to have 𝑅(𝑥) ∈ 𝑜(𝑥), or 𝑅(𝑥) / 𝑥 ⇝𝑟 0. (Contributed by Mario Carneiro, 8-Apr-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴))
Theorem	pntrf 27593	Functionality of the residual. Lemma for pnt 27644. (Contributed by Mario Carneiro, 8-Apr-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ 𝑅:ℝ+⟶ℝ
Theorem	pntrmax 27594*	There is a bound on the residual valid for all 𝑥. (Contributed by Mario Carneiro, 9-Apr-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐
Theorem	pntrsumo1 27595*	A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)
Theorem	pntrsumbnd 27596*	A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ ∃𝑐 ∈ ℝ+ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
Theorem	pntrsumbnd2 27597*	A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ ∃𝑐 ∈ ℝ+ ∀𝑘 ∈ ℕ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑘...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
Theorem	selbergr 27598*	Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ (𝑥 ∈ ℝ+ ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1)
Theorem	selberg3r 27599*	Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ 𝑂(1)
Theorem	selberg4r 27600*	Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    ⇒   ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)