Home Metamath Proof ExplorerTheorem List (p. 262 of 458) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28805) Hilbert Space Explorer (28806-30328) Users' Mathboxes (30329-45797)

Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2lgsoddprmlem3d 26101 Lemma 4 for 2lgsoddprmlem3 26102. (Contributed by AV, 20-Jul-2021.)
(((7↑2) − 1) / 8) = (2 · 3)

Theorem2lgsoddprmlem3 26102 Lemma 3 for 2lgsoddprm 26104. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑅↑2) − 1) / 8) ↔ 𝑅 ∈ {1, 7}))

Theorem2lgsoddprmlem4 26103 Lemma 4 for 2lgsoddprm 26104. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ (𝑁 mod 8) ∈ {1, 7}))

Theorem2lgsoddprm 26104 The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for 2 at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight ((2 /L 𝑃) = -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.)
(𝑃 ∈ (ℙ ∖ {2}) → (2 /L 𝑃) = (-1↑(((𝑃↑2) − 1) / 8)))

14.4.11  All primes 4n+1 are the sum of two squares

Theorem2sqlem1 26105* Lemma for 2sq 26118. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))       (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))

Theorem2sqlem2 26106* Lemma for 2sq 26118. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))       (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)))

Theoremmul2sq 26107 Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))       ((𝐴𝑆𝐵𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

Theorem2sqlem3 26108 Lemma for 2sqlem5 26110. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑 → (𝑁 · 𝑃) = ((𝐴↑2) + (𝐵↑2)))    &   (𝜑𝑃 = ((𝐶↑2) + (𝐷↑2)))    &   (𝜑𝑃 ∥ ((𝐶 · 𝐵) + (𝐴 · 𝐷)))       (𝜑𝑁𝑆)

Theorem2sqlem4 26109 Lemma for 2sqlem5 26110. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑 → (𝑁 · 𝑃) = ((𝐴↑2) + (𝐵↑2)))    &   (𝜑𝑃 = ((𝐶↑2) + (𝐷↑2)))       (𝜑𝑁𝑆)

Theorem2sqlem5 26110 Lemma for 2sq 26118. If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (𝑁 · 𝑃) ∈ 𝑆)    &   (𝜑𝑃𝑆)       (𝜑𝑁𝑆)

Theorem2sqlem6 26111* Lemma for 2sq 26118. If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → ∀𝑝 ∈ ℙ (𝑝𝐵𝑝𝑆))    &   (𝜑 → (𝐴 · 𝐵) ∈ 𝑆)       (𝜑𝐴𝑆)

Theorem2sqlem7 26112* Lemma for 2sq 26118. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}       𝑌 ⊆ (𝑆 ∩ ℕ)

Theorem2sqlem8a 26113* Lemma for 2sqlem8 26114. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}    &   (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))    &   (𝜑𝑀𝑁)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑𝑁 = ((𝐴↑2) + (𝐵↑2)))    &   𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))       (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ)

Theorem2sqlem8 26114* Lemma for 2sq 26118. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}    &   (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))    &   (𝜑𝑀𝑁)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑𝑁 = ((𝐴↑2) + (𝐵↑2)))    &   𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐸 = (𝐶 / (𝐶 gcd 𝐷))    &   𝐹 = (𝐷 / (𝐶 gcd 𝐷))       (𝜑𝑀𝑆)

Theorem2sqlem9 26115* Lemma for 2sq 26118. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}    &   (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))    &   (𝜑𝑀𝑁)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁𝑌)       (𝜑𝑀𝑆)

Theorem2sqlem10 26116* Lemma for 2sq 26118. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}       ((𝐴𝑌𝐵 ∈ ℕ ∧ 𝐵𝐴) → 𝐵𝑆)

Theorem2sqlem11 26117* Lemma for 2sq 26118. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))    &   𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}       ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → 𝑃𝑆)

Theorem2sq 26118* All primes of the form 4𝑘 + 1 are sums of two squares. This is Metamath 100 proof #20. (Contributed by Mario Carneiro, 20-Jun-2015.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑃 = ((𝑥↑2) + (𝑦↑2)))

Theorem2sqblem 26119 Lemma for 2sqb 26120. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑 → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))    &   (𝜑 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))    &   (𝜑𝑃 = ((𝑋↑2) + (𝑌↑2)))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → (𝑃 gcd 𝑌) = ((𝑃 · 𝐴) + (𝑌 · 𝐵)))       (𝜑 → (𝑃 mod 4) = 1)

Theorem2sqb 26120* The converse to 2sq 26118. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝑃 ∈ ℙ → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1)))

Theorem2sq2 26121 2 is the sum of squares of two nonnegative integers iff the two integers are 1. (Contributed by AV, 19-Jun-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (((𝐴↑2) + (𝐵↑2)) = 2 ↔ (𝐴 = 1 ∧ 𝐵 = 1)))

Theorem2sqn0 26122 If the sum of two squares is prime, none of the original number is zero. (Contributed by Thierry Arnoux, 4-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)       (𝜑𝐴 ≠ 0)

Theorem2sqcoprm 26123 If the sum of two squares is prime, the two original numbers are coprime. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)       (𝜑 → (𝐴 gcd 𝐵) = 1)

Theorem2sqmod 26124 Given two decompositions of a prime as a sum of two squares, show that they are equal. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)    &   (𝜑 → ((𝐶↑2) + (𝐷↑2)) = 𝑃)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))

Theorem2sqmo 26125* There exists at most one decomposition of a prime as a sum of two squares. See 2sqb 26120 for the existence of such a decomposition. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Theorem2sqnn0 26126* 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 ∈ ℕ0𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2)))

Theorem2sqnn 26127* 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)))

Theoremaddsq2reu 26128* 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 26131, is an example showing that the pattern ∃!𝑎𝐴∃!𝑏𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑). See also comments for df-eu 2588 and 2eu4 2675. For more details see comment for addsqnreup 26131. (Contributed by AV, 21-Jun-2023.)

(𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Theoremaddsqn2reu 26129* 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 26128, 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)

Theoremaddsqrexnreu 26130* 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 26128, 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 26128). For more details see comment for addsqnreup 26131. (Contributed by AV, 20-Jun-2023.)

(𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ ¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Theoremaddsqnreup 26131* 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 26128, is a real life example (about a numerical property) showing that the pattern ∃!𝑎𝐴∃!𝑏𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑"). See also comments for df-eu 2588 and 2eu4 2675.

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 26128). 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 26130). 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 (∃!𝑥𝐴𝑦𝐵𝜑 ∧ ∃!𝑦𝐵𝑥𝐴𝜑) (see 2reu4 4422 resp. 2eu4 2675). These two representations are equivalent (see opreu2reurex 6127). An analogon of this theorem using the latter variant is given in addsqn2reurex2 26133. In some cases, however, the variant with (ordered!) pairs may be possible only for ordered sets (like or ) and claiming that the first component is less than or equal to the second component (see, for example, 2sqreunnltb 26149 and 2sqreuopb 26156). Alternatively, (proper) unordered pairs can be used: ∃!𝑝𝑒𝒫 𝐴((♯‘𝑝) = 2 ∧ 𝜑), or, using the definition of proper pairs: ∃!𝑝 ∈ (Pairsproper𝐴)𝜑 (see, for example, inlinecirc02preu 45595). (Contributed by AV, 21-Jun-2023.)

(𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)

Theoremaddsq2nreurex 26132* 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)

Theoremaddsqn2reurex2 26133* 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 26128, is an example showing that the pattern ∃!𝑎𝐴∃!𝑏𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑), as it is the case with the pattern (∃!𝑎𝐴𝑏𝐵𝜑 ∧ ∃!𝑏𝐵𝑎𝐴𝜑. See also comments for df-eu 2588 and 2eu4 2675. (Contributed by AV, 2-Jul-2023.)

(𝐶 ∈ ℂ → ¬ (∃!𝑎 ∈ ℂ ∃𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∃!𝑏 ∈ ℂ ∃𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶))

Theorem2sqreulem1 26134* Lemma 1 for 2sqreu 26144. (Contributed by AV, 4-Jun-2023.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Theorem2sqreultlem 26135* Lemma for 2sqreult 26146. (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))

Theorem2sqreultblem 26136* Lemma for 2sqreultb 26147. (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)))

Theorem2sqreunnlem1 26137* Lemma 1 for 2sqreunn 26145. (Contributed by AV, 11-Jun-2023.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Theorem2sqreunnltlem 26138* Lemma for 2sqreunnlt 26148. (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))

Theorem2sqreunnltblem 26139* Lemma for 2sqreunnltb 26149. (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)))

Theorem2sqreulem2 26140 Lemma 2 for 2sqreu 26144 etc. (Contributed by AV, 25-Jun-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝐶 ∈ ℕ0) → (((𝐴↑2) + (𝐵↑2)) = ((𝐴↑2) + (𝐶↑2)) → 𝐵 = 𝐶))

Theorem2sqreulem3 26141 Lemma 3 for 2sqreu 26144 etc. (Contributed by AV, 25-Jun-2023.)
((𝐴 ∈ ℕ0 ∧ (𝐵 ∈ ℕ0𝐶 ∈ ℕ0)) → (((𝜑 ∧ ((𝐴↑2) + (𝐵↑2)) = 𝑃) ∧ (𝜓 ∧ ((𝐴↑2) + (𝐶↑2)) = 𝑃)) → 𝐵 = 𝐶))

Theorem2sqreulem4 26142* Lemma 4 for 2sqreu 26144 et. (Contributed by AV, 25-Jun-2023.)
(𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))       𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑

Theorem2sqreunnlem2 26143* Lemma 2 for 2sqreunn 26145. (Contributed by AV, 25-Jun-2023.)
(𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))       𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑

Theorem2sqreu 26144* There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two nonnegative integers. See 2sqnn0 26126 for the existence of such a decomposition. (Contributed by AV, 4-Jun-2023.) (Revised by AV, 25-Jun-2023.)
(𝜑 ↔ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))       ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0𝑎 ∈ ℕ0 𝜑))

Theorem2sqreunn 26145* There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two positive integers. See 2sqnn 26127 for the existence of such a decomposition. (Contributed by AV, 11-Jun-2023.) (Revised by AV, 25-Jun-2023.)
(𝜑 ↔ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))       ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))

Theorem2sqreult 26146* 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 ∈ ℕ0𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0𝑎 ∈ ℕ0 𝜑))

Theorem2sqreultb 26147* 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 ∈ ℕ0𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0𝑎 ∈ ℕ0 𝜑)))

Theorem2sqreunnlt 26148* 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))

Theorem2sqreunnltb 26149* 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)))

Theorem2sqreuop 26150* 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 26144. (Contributed by AV, 2-Jul-2023.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st𝑝) ≤ (2nd𝑝) ∧ (((1st𝑝)↑2) + ((2nd𝑝)↑2)) = 𝑃))

Theorem2sqreuopnn 26151* 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 26145. (Contributed by AV, 2-Jul-2023.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ × ℕ)((1st𝑝) ≤ (2nd𝑝) ∧ (((1st𝑝)↑2) + ((2nd𝑝)↑2)) = 𝑃))

Theorem2sqreuoplt 26152* There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers. Ordered pair variant of 2sqreult 26146. (Contributed by AV, 2-Jul-2023.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st𝑝) < (2nd𝑝) ∧ (((1st𝑝)↑2) + ((2nd𝑝)↑2)) = 𝑃))

Theorem2sqreuopltb 26153* 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 26147. (Contributed by AV, 3-Jul-2023.)
(𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st𝑝) < (2nd𝑝) ∧ (((1st𝑝)↑2) + ((2nd𝑝)↑2)) = 𝑃)))

Theorem2sqreuopnnlt 26154* 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 26148. (Contributed by AV, 3-Jul-2023.)
((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ × ℕ)((1st𝑝) < (2nd𝑝) ∧ (((1st𝑝)↑2) + ((2nd𝑝)↑2)) = 𝑃))

Theorem2sqreuopnnltb 26155* 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 26149. (Contributed by AV, 3-Jul-2023.)
(𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)((1st𝑝) < (2nd𝑝) ∧ (((1st𝑝)↑2) + ((2nd𝑝)↑2)) = 𝑃)))

Theorem2sqreuopb 26156* 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 26149. (Contributed by AV, 3-Jul-2023.)
(𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))))

14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem

Theoremchebbnd1lem1 26157 Lemma for chebbnd1 26160: show a lower bound on π(𝑥) at even integers using similar techniques to those used to prove bpos 25981. (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 25972, 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 25968 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
𝐾 = if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁))       (𝑁 ∈ (ℤ‘4) → (log‘((4↑𝑁) / 𝑁)) < ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁))))

Theoremchebbnd1lem2 26158 Lemma for chebbnd1 26160: Show that log(𝑁) / 𝑁 does not change too much between 𝑁 and 𝑀 = ⌊(𝑁 / 2). (Contributed by Mario Carneiro, 22-Sep-2014.)
𝑀 = (⌊‘(𝑁 / 2))       ((𝑁 ∈ ℝ ∧ 8 ≤ 𝑁) → ((log‘(2 · 𝑀)) / (2 · 𝑀)) < (2 · ((log‘𝑁) / 𝑁)))

Theoremchebbnd1lem3 26159 Lemma for chebbnd1 26160: get a lower bound on π(𝑁) / (𝑁 / log(𝑁)) that is independent of 𝑁. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 = (⌊‘(𝑁 / 2))       ((𝑁 ∈ ℝ ∧ 8 ≤ 𝑁) → (((log‘2) − (1 / (2 · e))) / 2) < ((π𝑁) · ((log‘𝑁) / 𝑁)))

Theoremchebbnd1 26160 The Chebyshev bound: The function π(𝑥) is eventually lower bounded by a positive constant times 𝑥 / log(𝑥). Alternatively stated, the function (𝑥 / log(𝑥)) / π(𝑥) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π𝑥))) ∈ 𝑂(1)

Theoremchtppilimlem1 26161 Lemma for chtppilim 26163. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐴 < 1)    &   (𝜑𝑁 ∈ (2[,)+∞))    &   (𝜑 → ((𝑁𝑐𝐴) / (π𝑁)) < (1 − 𝐴))       (𝜑 → ((𝐴↑2) · ((π𝑁) · (log‘𝑁))) < (θ‘𝑁))

Theoremchtppilimlem2 26162* Lemma for chtppilim 26163. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐴 < 1)       (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧𝑥 → ((𝐴↑2) · ((π𝑥) · (log‘𝑥))) < (θ‘𝑥)))

Theoremchtppilim 26163 The θ function is asymptotic to π(𝑥)log(𝑥), so it is sufficient to prove θ(𝑥) / 𝑥𝑟 1 to establish the PNT. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / ((π𝑥) · (log‘𝑥)))) ⇝𝑟 1

Theoremchto1ub 26164 The θ function is upper bounded by a linear term. Corollary of chtub 25900. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)

Theoremchebbnd2 26165 The Chebyshev bound, part 2: The function π(𝑥) is eventually upper bounded by a positive constant times 𝑥 / log(𝑥). Alternatively stated, the function π(𝑥) / (𝑥 / log(𝑥)) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑥 ∈ (2[,)+∞) ↦ ((π𝑥) / (𝑥 / (log‘𝑥)))) ∈ 𝑂(1)

Theoremchto1lb 26166 The θ function is lower bounded by a linear term. Corollary of chebbnd1 26160. (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∈ 𝑂(1)

Theoremchpchtlim 26167 The ψ and θ functions are asymptotic to each other, so is sufficient to prove either θ(𝑥) / 𝑥𝑟 1 or ψ(𝑥) / 𝑥𝑟 1 to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟 1

Theoremchpo1ub 26168 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
(𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)

Theoremchpo1ubb 26169* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
𝑐 ∈ ℝ+𝑥 ∈ ℝ+ (ψ‘𝑥) ≤ (𝑐 · 𝑥)

Theoremvmadivsum 26170* The sum of the von Mangoldt function over 𝑛 is asymptotic to log𝑥 + 𝑂(1). Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)
(𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)

Theoremvmadivsumb 26171* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
𝑐 ∈ ℝ+𝑥 ∈ (1[,)+∞)(abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ≤ 𝑐

Theoremrplogsumlem1 26172* Lemma for rplogsum 26215. (Contributed by Mario Carneiro, 2-May-2016.)
(𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)

Theoremrplogsumlem2 26173* Lemma for rplogsum 26215. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
(𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2)

Theoremdchrisum0lem1a 26174 Lemma for dchrisum0lem1 26204. (Contributed by Mario Carneiro, 7-Jun-2016.)
(((𝜑𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ‘(⌊‘𝑋))))

Theoremrpvmasumlem 26175* Lemma for rpvmasum 26214. Calculate the "trivial case" estimate Σ𝑛𝑥( 1 (𝑛)Λ(𝑛) / 𝑛) = log𝑥 + 𝑂(1), where 1 (𝑥) is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))

Theoremdchrisumlema 26176* Lemma for dchrisum 26180. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))       (𝜑 → ((𝐼 ∈ ℝ+𝐼 / 𝑛𝐴 ∈ ℝ) ∧ (𝐼 ∈ (𝑀[,)+∞) → 0 ≤ 𝐼 / 𝑛𝐴)))

Theoremdchrisumlem1 26177* Lemma for dchrisum 26180. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿𝑛))) ≤ 𝑅)       ((𝜑𝑈 ∈ ℕ0) → (abs‘Σ𝑛 ∈ (0..^𝑈)(𝑋‘(𝐿𝑛))) ≤ 𝑅)

Theoremdchrisumlem2 26178* Lemma for dchrisum 26180. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿𝑛))) ≤ 𝑅)    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑀𝑈)    &   (𝜑𝑈 ≤ (𝐼 + 1))    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐽 ∈ (ℤ𝐼))       (𝜑 → (abs‘((seq1( + , 𝐹)‘𝐽) − (seq1( + , 𝐹)‘𝐼))) ≤ ((2 · 𝑅) · 𝑈 / 𝑛𝐴))

Theoremdchrisumlem3 26179* Lemma for dchrisum 26180. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿𝑛))) ≤ 𝑅)       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))

Theoremdchrisum 26180* If 𝑛 ∈ [𝑀, +∞) ↦ 𝐴(𝑛) is a positive decreasing function approaching zero, then the infinite sum Σ𝑛, 𝑋(𝑛)𝐴(𝑛) is convergent, with the partial sum Σ𝑛𝑥, 𝑋(𝑛)𝐴(𝑛) within 𝑂(𝐴(𝑀)) of the limit 𝑇. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝑛 = 𝑥𝐴 = 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑛 ∈ ℝ+) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ ℝ+𝑥 ∈ ℝ+) ∧ (𝑀𝑛𝑛𝑥)) → 𝐵𝐴)    &   (𝜑 → (𝑛 ∈ ℝ+𝐴) ⇝𝑟 0)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿𝑛)) · 𝐴))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))

Theoremdchrmusumlema 26181* Lemma for dchrmusum 26212 and dchrisumn0 26209. Apply dchrisum 26180 for the function 1 / 𝑦. (Contributed by Mario Carneiro, 4-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))

Theoremdchrmusum2 26182* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded, provided that 𝑇 ≠ 0. Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1))

Theoremdchrvmasumlem1 26183* An alternative expression for a Dirichlet-weighted von Mangoldt sum in terms of the Möbius function. Equation 9.4.11 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 3-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿𝑚)) · ((log‘𝑚) / 𝑚))))

Theoremdchrvmasum2lem 26184* Give an expression for log𝑥 remarkably similar to Σ𝑛𝑥(𝑋(𝑛)Λ(𝑛) / 𝑛) given in dchrvmasumlem1 26183. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → (log‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))

Theoremdchrvmasum2if 26185* Combine the results of dchrvmasumlem1 26183 and dchrvmasum2lem 26184 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))))

Theoremdchrvmasumlem2 26186* Lemma for dchrvmasum 26213. (Contributed by Mario Carneiro, 4-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   ((𝜑𝑚 ∈ ℝ+) → 𝐹 ∈ ℂ)    &   (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑𝑇 ∈ ℂ)    &   ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹𝑇)) ≤ 𝑅)       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾𝑇)) / 𝑑)) ∈ 𝑂(1))

Theoremdchrvmasumlem3 26187* Lemma for dchrvmasum 26213. (Contributed by Mario Carneiro, 3-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   ((𝜑𝑚 ∈ ℝ+) → 𝐹 ∈ ℂ)    &   (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑𝑇 ∈ ℂ)    &   ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹𝑇)) ≤ 𝑅)       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾𝑇))) ∈ 𝑂(1))

Theoremdchrvmasumlema 26188* Lemma for dchrvmasum 26213 and dchrvmasumif 26191. Apply dchrisum 26180 for the function log(𝑦) / 𝑦, which is decreasing above e (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))       (𝜑 → ∃𝑡𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))

Theoremdchrvmasumiflem1 26189* Lemma for dchrvmasumif 26191. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))    &   𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))    &   (𝜑𝐸 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))       (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) ∈ 𝑂(1))

Theoremdchrvmasumiflem2 26190* Lemma for dchrvmasum 26213. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))    &   𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))    &   (𝜑𝐸 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)    &   (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))

Theoremdchrvmasumif 26191* 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 26213.) (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))

Theoremdchrvmaeq0 26192* The set 𝑊 is the collection of all non-principal Dirichlet characters such that the sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑋1 )    &   𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))    &   (𝜑𝐶 ∈ (0[,)+∞))    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)    &   (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))    &   𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿𝑚)) / 𝑚) = 0}       (𝜑 → (𝑋𝑊𝑆 = 0))

Theoremdchrisum0fval 26193* Value of the function 𝐹, the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))       (𝐴 ∈ ℕ → (𝐹𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))

Theoremdchrisum0fmul 26194* The function 𝐹, the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)       (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) · (𝐹𝐵)))

Theoremdchrisum0ff 26195* The function 𝐹 is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)       (𝜑𝐹:ℕ⟶ℝ)

Theoremdchrisum0flblem1 26196* Lemma for dchrisum0flb 26198. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℕ0)       (𝜑 → if((√‘(𝑃𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃𝐴)))

Theoremdchrisum0flblem2 26197* Lemma for dchrisum0flb 26198. Induction over relatively prime factors, with the prime power case handled in dchrisum0flblem1 . (Contributed by Mario Carneiro, 5-May-2016.) Replace reference to OLD theorem. (Revised by Wolf Lammen, 8-Sep-2020.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝐴)    &   (𝜑 → ∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹𝑦))       (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹𝐴))

Theoremdchrisum0flb 26198* The divisor sum of a real Dirichlet character, is lower bounded by zero everywhere and one at the squares. Equation 9.4.29 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑𝐴 ∈ ℕ)       (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹𝐴))

Theoremdchrisum0fno1 26199* The sum Σ𝑘𝑥, 𝐹(𝑥) / √𝑘 is divergent (i.e. not eventually bounded). Equation 9.4.30 of [Shapiro], p. 383. (Contributed by Mario Carneiro, 5-May-2016.)
𝑍 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))    &   (𝜑𝑋𝐷)    &   (𝜑𝑋:(Base‘𝑍)⟶ℝ)    &   (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))) ∈ 𝑂(1))        ¬ 𝜑

Theoremrpvmasum2 26200* A partial result along the lines of rpvmasum 26214. 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))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45797
 Copyright terms: Public domain < Previous  Next >