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Type | Label | Description |
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Statement | ||
Theorem | 2sqlem6 26001* | Lemma for 2sq 26008. If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆)) & ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑆) | ||
Theorem | 2sqlem7 26002* | Lemma for 2sq 26008. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⇒ ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ) | ||
Theorem | 2sqlem8a 26003* | Lemma for 2sqlem8 26004. (Contributed by Mario Carneiro, 4-Jun-2016.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} & ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) & ⊢ (𝜑 → 𝑁 = ((𝐴↑2) + (𝐵↑2))) & ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) & ⊢ 𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) ⇒ ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ) | ||
Theorem | 2sqlem8 26004* | Lemma for 2sq 26008. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} & ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) & ⊢ (𝜑 → 𝑁 = ((𝐴↑2) + (𝐵↑2))) & ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) & ⊢ 𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) & ⊢ 𝐸 = (𝐶 / (𝐶 gcd 𝐷)) & ⊢ 𝐹 = (𝐷 / (𝐶 gcd 𝐷)) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑆) | ||
Theorem | 2sqlem9 26005* | Lemma for 2sq 26008. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} & ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ 𝑌) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑆) | ||
Theorem | 2sqlem10 26006* | Lemma for 2sq 26008. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⇒ ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆) | ||
Theorem | 2sqlem11 26007* | Lemma for 2sq 26008. (Contributed by Mario Carneiro, 19-Jun-2015.) |
⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) & ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⇒ ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → 𝑃 ∈ 𝑆) | ||
Theorem | 2sq 26008* | All primes of the form 4𝑘 + 1 are sums of two squares. This is Metamath 100 proof #20. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑃 = ((𝑥↑2) + (𝑦↑2))) | ||
Theorem | 2sqblem 26009 | Lemma for 2sqb 26010. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) & ⊢ (𝜑 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) & ⊢ (𝜑 → 𝑃 = ((𝑋↑2) + (𝑌↑2))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → (𝑃 gcd 𝑌) = ((𝑃 · 𝐴) + (𝑌 · 𝐵))) ⇒ ⊢ (𝜑 → (𝑃 mod 4) = 1) | ||
Theorem | 2sqb 26010* | The converse to 2sq 26008. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝑃 ∈ ℙ → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1))) | ||
Theorem | 2sq2 26011 | 2 is the sum of squares of two nonnegative integers iff the two integers are 1. (Contributed by AV, 19-Jun-2023.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (((𝐴↑2) + (𝐵↑2)) = 2 ↔ (𝐴 = 1 ∧ 𝐵 = 1))) | ||
Theorem | 2sqn0 26012 | If the sum of two squares is prime, none of the original number is zero. (Contributed by Thierry Arnoux, 4-Feb-2020.) |
⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | 2sqcoprm 26013 | If the sum of two squares is prime, the two original numbers are coprime. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃) ⇒ ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) | ||
Theorem | 2sqmod 26014 | Given two decompositions of a prime as a sum of two squares, show that they are equal. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ≤ 𝐷) & ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃) & ⊢ (𝜑 → ((𝐶↑2) + (𝐷↑2)) = 𝑃) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | 2sqmo 26015* | There exists at most one decomposition of a prime as a sum of two squares. See 2sqb 26010 for the existence of such a decomposition. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | ||
Theorem | 2sqnn0 26016* | All primes of the form 4𝑘 + 1 are sums of squares of two nonnegative integers. (Contributed by AV, 3-Jun-2023.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2))) | ||
Theorem | 2sqnn 26017* | All primes of the form 4𝑘 + 1 are sums of squares of two positive integers. (Contributed by AV, 11-Jun-2023.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))) | ||
Theorem | addsq2reu 26018* |
For each complex number 𝐶, there exists a unique complex
number
𝑎 added to the square of a unique
another complex number 𝑏
resulting in the given complex number 𝐶. The unique complex number
𝑎 is 𝐶, and the unique another complex
number 𝑏 is 0.
Remark: This, together with addsqnreup 26021, is an example showing that the pattern ∃!𝑎 ∈ 𝐴∃!𝑏 ∈ 𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑). See also comments for df-eu 2654 and 2eu4 2739. For more details see comment for addsqnreup 26021. (Contributed by AV, 21-Jun-2023.) |
⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) | ||
Theorem | addsqn2reu 26019* |
For each complex number 𝐶, there does not exist a unique
complex
number 𝑏, squared and added to a unique
another complex number
𝑎 resulting in the given complex number
𝐶.
Actually, for each
complex number 𝑏, 𝑎 = (𝐶 − (𝑏↑2)) is unique.
Remark: This, together with addsq2reu 26018, shows that commutation of two unique quantifications need not be equivalent, and provides an evident justification of the fact that considering the pair of variables is necessary to obtain what we intuitively understand as "double unique existence". (Proposed by GL, 23-Jun-2023.). (Contributed by AV, 23-Jun-2023.) |
⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) | ||
Theorem | addsqrexnreu 26020* |
For each complex number, there exists a complex number to which the
square of more than one (or no) other complex numbers can be added to
result in the given complex number.
Remark: This theorem, together with addsq2reu 26018, shows that there are cases in which there is a set together with a not unique other set fulfilling a wff, although there is a unique set fulfilling the wff together with another unique set (see addsq2reu 26018). For more details see comment for addsqnreup 26021. (Contributed by AV, 20-Jun-2023.) |
⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ ¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) | ||
Theorem | addsqnreup 26021* |
There is no unique decomposition of a complex number as a sum of a
complex number and a square of a complex number.
Remark: This theorem, together with addsq2reu 26018, is a real life example (about a numerical property) showing that the pattern ∃!𝑎 ∈ 𝐴∃!𝑏 ∈ 𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑"). See also comments for df-eu 2654 and 2eu4 2739. In the case of decompositions of complex numbers as a sum of a complex number and a square of a complex number, the only/unique complex number to which the square of a unique complex number is added yields in the given complex number is the given number itself, and the unique complex number to be squared is 0 (see comment for addsq2reu 26018). There are, however, complex numbers to which the square of more than one other complex numbers can be added to yield the given complex number (see addsqrexnreu 26020). For example, 〈1, (√‘(𝐶 − 1))〉 and 〈1, -(√‘(𝐶 − 1))〉 are two different decompositions of 𝐶 (if 𝐶 ≠ 1). Therefore, there is no unique decomposition of any complex number as a sum of a complex number and a square of a complex number, as generally proved by this theorem. As a consequence, a theorem must claim the existence of a unique pair of sets to express "There are unique 𝑎 and 𝑏 so that .." (more formally ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 with 𝑝 = 〈𝑎, 𝑏〉), or by showing (∃!𝑥 ∈ 𝐴∃𝑦 ∈ 𝐵𝜑 ∧ ∃!𝑦 ∈ 𝐵∃𝑥 ∈ 𝐴𝜑) (see 2reu4 4468 resp. 2eu4 2739). These two representations are equivalent (see opreu2reurex 6147). An analogon of this theorem using the latter variant is given in addsqn2reurex2 26023. In some cases, however, the variant with (ordered!) pairs may be possible only for ordered sets (like ℝ or ℙ) and claiming that the first component is less than or equal to the second component (see, for example, 2sqreunnltb 26039 and 2sqreuopb 26046). Alternatively, (proper) unordered pairs can be used: ∃!𝑝𝑒𝒫 𝐴((♯‘𝑝) = 2 ∧ 𝜑), or, using the definition of proper pairs: ∃!𝑝 ∈ (Pairsproper‘𝐴)𝜑 (see, for example, inlinecirc02preu 44782). (Contributed by AV, 21-Jun-2023.) |
⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶) | ||
Theorem | addsq2nreurex 26022* | For each complex number 𝐶, there is no unique complex number 𝑎 added to the square of another complex number 𝑏 resulting in the given complex number 𝐶. (Contributed by AV, 2-Jul-2023.) |
⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑎 ∈ ℂ ∃𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) | ||
Theorem | addsqn2reurex2 26023* |
For each complex number 𝐶, there does not uniquely exist two
complex numbers 𝑎 and 𝑏, with 𝑏 squared
and added to 𝑎
resulting in the given complex number 𝐶.
Remark: This, together with addsq2reu 26018, is an example showing that the pattern ∃!𝑎 ∈ 𝐴∃!𝑏 ∈ 𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑), as it is the case with the pattern (∃!𝑎 ∈ 𝐴∃𝑏 ∈ 𝐵𝜑 ∧ ∃!𝑏 ∈ 𝐵∃𝑎 ∈ 𝐴𝜑. See also comments for df-eu 2654 and 2eu4 2739. (Contributed by AV, 2-Jul-2023.) |
⊢ (𝐶 ∈ ℂ → ¬ (∃!𝑎 ∈ ℂ ∃𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∃!𝑏 ∈ ℂ ∃𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)) | ||
Theorem | 2sqreulem1 26024* | Lemma 1 for 2sqreu 26034. (Contributed by AV, 4-Jun-2023.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | ||
Theorem | 2sqreultlem 26025* | Lemma for 2sqreult 26036. (Contributed by AV, 8-Jun-2023.) (Proposed by GL, 8-Jun-2023.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | ||
Theorem | 2sqreultblem 26026* | Lemma for 2sqreultb 26037. (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 26027* | Lemma 1 for 2sqreunn 26035. (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 26028* | Lemma for 2sqreunnlt 26038. (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 26029* | Lemma for 2sqreunnltb 26039. (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 26030 | Lemma 2 for 2sqreu 26034 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 26031 | Lemma 3 for 2sqreu 26034 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 26032* | Lemma 4 for 2sqreu 26034 et. (Contributed by AV, 25-Jun-2023.) |
⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) ⇒ ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 | ||
Theorem | 2sqreunnlem2 26033* | Lemma 2 for 2sqreunn 26035. (Contributed by AV, 25-Jun-2023.) |
⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) ⇒ ⊢ ∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑 | ||
Theorem | 2sqreu 26034* | There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two nonnegative integers. See 2sqnn0 26016 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 26035* | There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two positive integers. See 2sqnn 26017 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 26036* | 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 26037* | 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 26038* | 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 26039* | 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 26040* | 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 26034. (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 26041* | 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 26035. (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 26042* | There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers. Ordered pair variant of 2sqreult 26036. (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 26043* | 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 26037. (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 26044* | 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 26038. (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 26045* | 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 26039. (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 26046* | 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 26039. (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)))) | ||
Theorem | chebbnd1lem1 26047 | Lemma for chebbnd1 26050: show a lower bound on π(𝑥) at even integers using similar techniques to those used to prove bpos 25871. (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 25862, 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 25858 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 26048 | Lemma for chebbnd1 26050: 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 26049 | Lemma for chebbnd1 26050: 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 26050 | 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 26051 | Lemma for chtppilim 26053. (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 26052* | Lemma for chtppilim 26053. (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 26053 | 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 26054 | The θ function is upper bounded by a linear term. Corollary of chtub 25790. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | chebbnd2 26055 | 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 26056 | The θ function is lower bounded by a linear term. Corollary of chebbnd1 26050. (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ (𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∈ 𝑂(1) | ||
Theorem | chpchtlim 26057 | 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 26058 | 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 26059* | 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 26060* | 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 26061* | 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 26062* | Lemma for rplogsum 26105. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2) | ||
Theorem | rplogsumlem2 26063* | Lemma for rplogsum 26105. 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 26064 | Lemma for dchrisum0lem1 26094. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋)))) | ||
Theorem | rpvmasumlem 26065* | Lemma for rpvmasum 26104. 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 26066* | Lemma for dchrisum 26070. 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 26067* | Lemma for dchrisum 26070. 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 26068* | Lemma for dchrisum 26070. 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 26069* | Lemma for dchrisum 26070. 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 26070* | 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 26071* | Lemma for dchrmusum 26102 and dchrisumn0 26099. Apply dchrisum 26070 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 26072* | 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 26073* | 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 26074* | Give an expression for log𝑥 remarkably similar to Σ𝑛 ≤ 𝑥(𝑋(𝑛)Λ(𝑛) / 𝑛) given in dchrvmasumlem1 26073. 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 26075* | Combine the results of dchrvmasumlem1 26073 and dchrvmasum2lem 26074 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 26076* | Lemma for dchrvmasum 26103. (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 26077* | Lemma for dchrvmasum 26103. (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 26078* | Lemma for dchrvmasum 26103 and dchrvmasumif 26081. Apply dchrisum 26070 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 26079* | Lemma for dchrvmasumif 26081. (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 26080* | Lemma for dchrvmasum 26103. (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 26081* | 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 26103.) (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 26082* | 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 26083* | 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 26084* | 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 26085* | 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 26086* | Lemma for dchrisum0flb 26088. 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 26087* | Lemma for dchrisum0flb 26088. 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 26088* | 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 26089* | 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 26090* | A partial result along the lines of rpvmasum 26104. 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 26091* | 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 26092* | Lemma for dchrisum0 26098. Apply dchrisum 26070 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 26093* | Lemma for dchrisum0lem1 26094. (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 26094* | Lemma for dchrisum0 26098. (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 26095* | Lemma for dchrisum0 26098. (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 26096* | Lemma for dchrisum0 26098. (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 26097* | Lemma for dchrisum0 26098. (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 26098* | 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 26072 and dchrvmasumif 26081. 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 26099* | 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 26072 and dchrvmasumif 26081. 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 26100* | 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)) |
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