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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrpvmasum2 25001* A partial result along the lines of rpvmasum 25015. 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 25002* 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 25003* Lemma for dchrisum0 25009. Apply dchrisum 24981 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 25004* Lemma for dchrisum0lem1 25005. (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 25005* Lemma for dchrisum0 25009. (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 25006* Lemma for dchrisum0 25009. (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 25007* Lemma for dchrisum0 25009. (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 25008* Lemma for dchrisum0 25009. (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 25009* 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 24983 and dchrvmasumif 24992. 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 25010* 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 24983 and dchrvmasumif 24992. 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 25011* 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))
 
Theoremdchrvmasumlem 25012* 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)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
 
Theoremdchrmusum 25013* 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 )       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))
 
Theoremdchrvmasum 25014* 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
 
Theoremrpvmasum 25015* The sum of the von Mangoldt function over those integers 𝑛𝐴 (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)    &   (𝜑𝑁 ∈ ℕ)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝐴𝑈)    &   𝑇 = (𝐿 “ {𝐴})       (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
 
Theoremrplogsum 25016* The sum of log𝑝 / 𝑝 over the primes 𝑝𝐴 (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)    &   (𝜑𝑁 ∈ ℕ)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝐴𝑈)    &   𝑇 = (𝐿 “ {𝐴})       (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑝 ∈ ((1...(⌊‘𝑥)) ∩ (ℙ ∩ 𝑇))((log‘𝑝) / 𝑝)) − (log‘𝑥))) ∈ 𝑂(1))
 
Theoremdirith2 25017 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)    &   (𝜑𝑁 ∈ ℕ)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝐴𝑈)    &   𝑇 = (𝐿 “ {𝐴})       (𝜑 → (ℙ ∩ 𝑇) ≈ ℕ)
 
Theoremdirith 25018* Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to 𝑁. Theorem 9.4.1 of [Shapiro], p. 375. See http://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 ∥ (𝑝𝐴)} ≈ ℕ)
 
14.4.13  The Prime Number Theorem
 
Theoremmudivsum 25019* Asymptotic formula for Σ𝑛𝑥, 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)
 
Theoremmulogsumlem 25020* Lemma for mulogsum 25021. (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)
 
Theoremmulogsum 25021* Asymptotic formula for Σ𝑛𝑥, (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)
 
Theoremlogdivsum 25022* Asymptotic analysis of Σ𝑛𝑥, 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 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
 
Theoremmulog2sumlem1 25023* Asymptotic formula for Σ𝑛𝑥, log(𝑥 / 𝑛) / 𝑛 = (1 / 2)log↑2(𝑥) + γ · log𝑥𝐿 + 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)))    &   (𝜑𝐹𝑟 𝐿)    &   (𝜑𝐴 ∈ 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)))
 
Theoremmulog2sumlem2 25024* Lemma for mulog2sum 25026. (Contributed by Mario Carneiro, 19-May-2016.)
𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))    &   (𝜑𝐹𝑟 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))
 
Theoremmulog2sumlem3 25025* Lemma for mulog2sum 25026. (Contributed by Mario Carneiro, 13-May-2016.)
𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))    &   (𝜑𝐹𝑟 𝐿)       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
 
Theoremmulog2sum 25026* Asymptotic formula for Σ𝑛𝑥, (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)
 
Theoremvmalogdivsum2 25027* The sum Σ𝑛𝑥, 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)
 
Theoremvmalogdivsum 25028* The sum Σ𝑛𝑥, 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)
 
Theorem2vmadivsumlem 25029* Lemma for 2vmadivsum 25030. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ 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))
 
Theorem2vmadivsum 25030* The sum Σ𝑚𝑛𝑥, 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)
 
Theoremlogsqvma 25031* 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 ∣ 𝑥𝑁} (Σ𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑑} ((Λ‘𝑢) · (Λ‘(𝑑 / 𝑢))) + ((Λ‘𝑑) · (log‘𝑑))) = ((log‘𝑁)↑2))
 
Theoremlogsqvma2 25032* The Möbius inverse of logsqvma 25031. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
(𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ((μ‘𝑑) · ((log‘(𝑁 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ((Λ‘𝑑) · (Λ‘(𝑁 / 𝑑))) + ((Λ‘𝑁) · (log‘𝑁))))
 
Theoremlog2sumbnd 25033* Bound on the difference between Σ𝑛𝐴, 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))
 
Theoremselberglem1 25034* Lemma for selberg 25037. Estimation of the asymptotic part of selberglem3 25036. (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)
 
Theoremselberglem2 25035* Lemma for selberg 25037. (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)
 
Theoremselberglem3 25036* Lemma for selberg 25037. Estimation of the left-hand side of logsqvma2 25032. (Contributed by Mario Carneiro, 23-May-2016.)
(𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
 
Theoremselberg 25037* Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Σ𝑛𝑥, Λ(𝑛)log𝑛 + Σ𝑚 · 𝑛𝑥, 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)
 
Theoremselbergb 25038* Convert eventual boundedness in selberg 25037 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
 
Theoremselberg2lem 25039* Lemma for selberg2 25040. 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)
 
Theoremselberg2 25040* 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)
 
Theoremselberg2b 25041* Convert eventual boundedness in selberg2 25040 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
 
Theoremchpdifbndlem1 25042* Lemma for chpdifbnd 25044. (Contributed by Mario Carneiro, 25-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝐵 ∈ 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)))    &   (𝜑𝑋 ∈ (1(,)+∞))    &   (𝜑𝑌 ∈ (𝑋[,](𝐴 · 𝑋)))       (𝜑 → ((ψ‘𝑌) − (ψ‘𝑋)) ≤ ((2 · (𝑌𝑋)) + (𝐶 · (𝑋 / (log‘𝑋)))))
 
Theoremchpdifbndlem2 25043* Lemma for chpdifbnd 25044. (Contributed by Mario Carneiro, 25-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝐵 ∈ 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 · (𝑦𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))))
 
Theoremchpdifbnd 25044* 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 · (𝑦𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))))
 
Theoremlogdivbnd 25045* A bound on a sum of logs, used in pntlemk 25095. This is not as precise as logdivsum 25022 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))
 
Theoremselberg3lem1 25046* Introduce a log weighting on the summands of Σ𝑚 · 𝑛𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 25040 (written here as Σ𝑛𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ 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))
 
Theoremselberg3lem2 25047* Lemma for selberg3 25048. 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)
 
Theoremselberg3 25048* Introduce a log weighting on the summands of Σ𝑚 · 𝑛𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 25040 (written here as Σ𝑛𝑥, 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)
 
Theoremselberg4lem1 25049* Lemma for selberg4 25050. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ 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))
 
Theoremselberg4 25050* The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form Σ𝑖𝑗𝑘𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘); we eliminate one of the nested sums by using the definition of ψ(𝑥) = Σ𝑘𝑥, Λ(𝑘). This statement can thus equivalently be written ψ(𝑥)log↑2(𝑥) = 𝑖𝑗𝑘𝑥, 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)
 
Theorempntrval 25051* Define the residual of the second Chebyshev function. The goal is to have 𝑅(𝑥) ∈ 𝑜(𝑥), or 𝑅(𝑥) / 𝑥𝑟 0. (Contributed by Mario Carneiro, 8-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((ψ‘𝐴) − 𝐴))
 
Theorempntrf 25052 Functionality of the residual. Lemma for pnt 25103. (Contributed by Mario Carneiro, 8-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       𝑅:ℝ+⟶ℝ
 
Theorempntrmax 25053* 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‘((𝑅𝑥) / 𝑥)) ≤ 𝑐
 
Theorempntrsumo1 25054* 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))((𝑅𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)
 
Theorempntrsumbnd 25055* 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)((𝑅𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
 
Theorempntrsumbnd2 25056* 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)((𝑅𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
 
Theoremselbergr 25057* 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+ ↦ ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1)
 
Theoremselberg3r 25058* 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(,)+∞) ↦ ((((𝑅𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ 𝑂(1)
 
Theoremselberg4r 25059* 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(,)+∞) ↦ ((((𝑅𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)
 
Theoremselberg34r 25060* The sum of selberg3r 25058 and selberg4r 25059. (Contributed by Mario Carneiro, 31-May-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
 
Theorempntsval 25061* Define the "Selberg function", whose asymptotic behavior is the content of selberg 25037. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))       (𝐴 ∈ ℝ → (𝑆𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛)))))
 
Theorempntsf 25062* Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))       𝑆:ℝ⟶ℝ
 
Theoremselbergs 25063* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))       (𝑥 ∈ ℝ+ ↦ (((𝑆𝑥) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
 
Theoremselbergsb 25064* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))       𝑐 ∈ ℝ+𝑥 ∈ (1[,)+∞)(abs‘(((𝑆𝑥) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐
 
Theorempntsval2 25065* The Selberg function can be expressed using the convolution product of the von Mangoldt function with itself. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))       (𝐴 ∈ ℝ → (𝑆𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))))
 
Theorempntrlog2bndlem1 25066* The sum of selberg3r 25058 and selberg4r 25059. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1)
 
Theorempntrlog2bndlem2 25067* Lemma for pntrlog2bnd 25073. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑦 ∈ ℝ+ (ψ‘𝑦) ≤ (𝐴 · 𝑦))       (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
 
Theorempntrlog2bndlem3 25068* Lemma for pntrlog2bnd 25073. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘(((𝑆𝑦) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴)       (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) − (abs‘(𝑅‘(𝑥 / (𝑛 + 1))))) · ((𝑆𝑛) − (2 · (𝑛 · (log‘𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
 
Theorempntrlog2bndlem4 25069* Lemma for pntrlog2bnd 25073. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))       (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1)
 
Theorempntrlog2bndlem5 25070* Lemma for pntrlog2bnd 25073. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)       (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
 
Theorempntrlog2bndlem6a 25071* Lemma for pntrlog2bndlem6 25072. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)       ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))))
 
Theorempntrlog2bndlem6 25072* Lemma for pntrlog2bnd 25073. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))    &   𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
 
Theorempntrlog2bnd 25073* A bound on 𝑅(𝑥)log↑2(𝑥). Equation 10.6.15 of [Shapiro], p. 431. (Contributed by Mario Carneiro, 1-Jun-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑐 ∈ ℝ+𝑥 ∈ (1(,)+∞)((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ≤ 𝑐)
 
Theorempntpbnd1a 25074* Lemma for pntpbnd 25077. (Contributed by Mario Carneiro, 11-Apr-2016.) Replace reference to OLD theorem. (Revised by Wolf Lammen, 8-Sep-2020.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐸 ∈ (0(,)1))    &   𝑋 = (exp‘(2 / 𝐸))    &   (𝜑𝑌 ∈ (𝑋(,)+∞))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑌 < 𝑁𝑁 ≤ (𝐾 · 𝑌)))    &   (𝜑 → (abs‘(𝑅𝑁)) ≤ (abs‘((𝑅‘(𝑁 + 1)) − (𝑅𝑁))))       (𝜑 → (abs‘((𝑅𝑁) / 𝑁)) ≤ 𝐸)
 
Theorempntpbnd1 25075* Lemma for pntpbnd 25077. (Contributed by Mario Carneiro, 11-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐸 ∈ (0(,)1))    &   𝑋 = (exp‘(2 / 𝐸))    &   (𝜑𝑌 ∈ (𝑋(,)+∞))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑦 ∈ (𝑖...𝑗)((𝑅𝑦) / (𝑦 · (𝑦 + 1)))) ≤ 𝐴)    &   𝐶 = (𝐴 + 2)    &   (𝜑𝐾 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞))    &   (𝜑 → ¬ ∃𝑦 ∈ ℕ ((𝑌 < 𝑦𝑦 ≤ (𝐾 · 𝑌)) ∧ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐸))       (𝜑 → Σ𝑛 ∈ (((⌊‘𝑌) + 1)...(⌊‘(𝐾 · 𝑌)))(abs‘((𝑅𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝐴)
 
Theorempntpbnd2 25076* Lemma for pntpbnd 25077. (Contributed by Mario Carneiro, 11-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐸 ∈ (0(,)1))    &   𝑋 = (exp‘(2 / 𝐸))    &   (𝜑𝑌 ∈ (𝑋(,)+∞))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑦 ∈ (𝑖...𝑗)((𝑅𝑦) / (𝑦 · (𝑦 + 1)))) ≤ 𝐴)    &   𝐶 = (𝐴 + 2)    &   (𝜑𝐾 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞))    &   (𝜑 → ¬ ∃𝑦 ∈ ℕ ((𝑌 < 𝑦𝑦 ≤ (𝐾 · 𝑌)) ∧ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐸))        ¬ 𝜑
 
Theorempntpbnd 25077* Lemma for pnt 25103. Establish smallness of 𝑅 at a point. Lemma 10.6.1 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       𝑐 ∈ ℝ+𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅𝑛) / 𝑛)) ≤ 𝑒)
 
Theorempntibndlem1 25078 Lemma for pntibnd 25082. (Contributed by Mario Carneiro, 10-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   𝐿 = ((1 / 4) / (𝐴 + 3))       (𝜑𝐿 ∈ (0(,)1))
 
Theorempntibndlem2a 25079* Lemma for pntibndlem2 25080. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   𝐿 = ((1 / 4) / (𝐴 + 3))    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   𝐾 = (exp‘(𝐵 / (𝐸 / 2)))    &   𝐶 = ((2 · 𝐵) + (log‘2))    &   (𝜑𝐸 ∈ (0(,)1))    &   (𝜑𝑍 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)       ((𝜑𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁))) → (𝑢 ∈ ℝ ∧ 𝑁𝑢𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁)))
 
Theorempntibndlem2 25080* Lemma for pntibnd 25082. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   𝐿 = ((1 / 4) / (𝐴 + 3))    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   𝐾 = (exp‘(𝐵 / (𝐸 / 2)))    &   𝐶 = ((2 · 𝐵) + (log‘2))    &   (𝜑𝐸 ∈ (0(,)1))    &   (𝜑𝑍 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](2 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦𝑥)) + (𝑇 · (𝑥 / (log‘𝑥)))))    &   𝑋 = ((exp‘(𝑇 / (𝐸 / 4))) + 𝑍)    &   (𝜑𝑀 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞))    &   (𝜑𝑌 ∈ (𝑋(,)+∞))    &   (𝜑 → ((𝑌 < 𝑁𝑁 ≤ ((𝑀 / 2) · 𝑌)) ∧ (abs‘((𝑅𝑁) / 𝑁)) ≤ (𝐸 / 2)))       (𝜑 → ∃𝑧 ∈ ℝ+ ((𝑌 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑀 · 𝑌)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))
 
Theorempntibndlem3 25081* Lemma for pntibnd 25082. Package up pntibndlem2 25080 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   𝐿 = ((1 / 4) / (𝐴 + 3))    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   𝐾 = (exp‘(𝐵 / (𝐸 / 2)))    &   𝐶 = ((2 · 𝐵) + (log‘2))    &   (𝜑𝐸 ∈ (0(,)1))    &   (𝜑𝑍 ∈ ℝ+)    &   (𝜑 → ∀𝑚 ∈ (𝐾[,)+∞)∀𝑣 ∈ (𝑍(,)+∞)∃𝑖 ∈ ℕ ((𝑣 < 𝑖𝑖 ≤ (𝑚 · 𝑣)) ∧ (abs‘((𝑅𝑖) / 𝑖)) ≤ (𝐸 / 2)))       (𝜑 → ∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))
 
Theorempntibnd 25082* Lemma for pnt 25103. Establish smallness of 𝑅 on an interval. Lemma 10.6.2 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))       𝑐 ∈ ℝ+𝑙 ∈ (0(,)1)∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝑙 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝑙 · 𝑒)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝑒)
 
Theorempntlemd 25083 Lemma for pnt 25103. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝐴 is C^*, 𝐵 is c1, 𝐿 is λ, 𝐷 is c2, and 𝐹 is c3. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))       (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
 
Theorempntlemc 25084* Lemma for pnt 25103. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑈 is α, 𝐸 is ε, and 𝐾 is K. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))       (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
 
Theorempntlema 25085* Lemma for pnt 25103. Closure for the constants used in the proof. The mammoth expression 𝑊 is a number large enough to satisfy all the lower bounds needed for 𝑍. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑌 is x2, 𝑋 is x1, 𝐶 is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and 𝑊 is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))       (𝜑𝑊 ∈ ℝ+)
 
Theorempntlemb 25086* Lemma for pnt 25103. Unpack all the lower bounds contained in 𝑊, in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑍 is x. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))       (𝜑 → (𝑍 ∈ ℝ+ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
 
Theorempntlemg 25087* Lemma for pnt 25103. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑀 is j^* and 𝑁 is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))       (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
 
Theorempntlemh 25088* Lemma for pnt 25103. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))       ((𝜑𝐽 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝐽) ∧ (𝐾𝐽) ≤ (√‘𝑍)))
 
Theorempntlemn 25089* Lemma for pnt 25103. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)       ((𝜑 ∧ (𝐽 ∈ ℕ ∧ 𝐽 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝐽) − (abs‘((𝑅‘(𝑍 / 𝐽)) / 𝑍))) · (log‘𝐽)))
 
Theorempntlemq 25090* Lemma for pntlemj 25092. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))    &   (𝜑𝑉 ∈ ℝ+)    &   (𝜑 → (((𝐾𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑𝐽 ∈ (𝑀..^𝑁))    &   𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))       (𝜑𝐼𝑂)
 
Theorempntlemr 25091* Lemma for pntlemj 25092. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))    &   (𝜑𝑉 ∈ ℝ+)    &   (𝜑 → (((𝐾𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑𝐽 ∈ (𝑀..^𝑁))    &   𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))       (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ ((#‘𝐼) · ((𝑈𝐸) · ((log‘(𝑍 / 𝑉)) / (𝑍 / 𝑉)))))
 
Theorempntlemj 25092* Lemma for pnt 25103. The induction step. Using pntibnd 25082, we find an interval in 𝐾𝐽...𝐾↑(𝐽 + 1) which is sufficiently large and has a much smaller value, 𝑅(𝑧) / 𝑧𝐸 (instead of our original bound 𝑅(𝑧) / 𝑧𝑈). (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))    &   (𝜑𝑉 ∈ ℝ+)    &   (𝜑 → (((𝐾𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑𝐽 ∈ (𝑀..^𝑁))    &   𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))       (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛𝑂 (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
 
Theorempntlemi 25093* Lemma for pnt 25103. Eliminate some assumptions from pntlemj 25092. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))       ((𝜑𝐽 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛𝑂 (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
 
Theorempntlemf 25094* Lemma for pnt 25103. Add up the pieces in pntlemi 25093 to get an estimate slightly better than the naive lower bound 0. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))       (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
 
Theorempntlemk 25095* Lemma for pnt 25103. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))       (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))
 
Theorempntlemo 25096* Lemma for pnt 25103. Combine all the estimates to establish a smaller eventual bound on 𝑅(𝑍) / 𝑍. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)       (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
 
Theorempntleme 25097* Lemma for pnt 25103. Package up pntlemo 25096 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑘 ∈ (𝐾[,)+∞)∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)       (𝜑 → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
 
Theorempntlem3 25098* Lemma for pnt 25103. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.) (Proof shortened by AV, 27-Sep-2020.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   𝑇 = {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ+𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑡}    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑢𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇)       (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1)
 
Theorempntlemp 25099* Lemma for pnt 25103. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝑒))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)       (𝜑 → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
 
Theorempntleml 25100* Lemma for pnt 25103. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝑒))       (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1)
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