From 2c3065622cf7fb8d7382475d52f9b835021ef9cc Mon Sep 17 00:00:00 2001
From: Jan Snellman <jan.snellman@liu.se>
Date: Thu, 16 May 2024 19:32:32 +0200
Subject: [PATCH] sjuk
---
homepage/index.org | 7 +-
homepage/sitemap.org | 2 +-
public/index.html | 46 +-
public/literature/Multiplicative.pdf | 3775 ++++++++++++++++++++++++++
public/sitemap.html | 6 +-
5 files changed, 3815 insertions(+), 21 deletions(-)
create mode 100644 public/literature/Multiplicative.pdf
diff --git a/homepage/index.org b/homepage/index.org
index 303dccc..5935fa6 100644
--- a/homepage/index.org
+++ b/homepage/index.org
@@ -23,7 +23,12 @@ Skriftlig tentamen.
| TATA54/TEN1 | Talteori | 2024-08-22 | 14-18 | Linköping | 2024-07-23 - 2024-08-12 |
* Allra senaste nytt VT2024 <<allra-senaste-nytt>>
-** 2024-05-06
+** 2024-05-16
+Jag är inlagd på sjukhus... Det är väl osäkert om det går att hitta vikarier
+till lektionerna i morgon och på måndag. Fråga någon av studierektorerna, eller proprefekt Jesper Thoren.
+
+Tentamen skall kunna bli av utan problem!
+** 2024-05-13
Till på fredag så tittar vi på 13.4.3abc, 13.4.5, 13.4.11.
Till sista lektionen (måndag 20 maj) så tittar vi på 14.1.17ab, 14.1.23.
** 2024-05-06
diff --git a/homepage/sitemap.org b/homepage/sitemap.org
index c1e5cb8..aa4e7f7 100644
--- a/homepage/sitemap.org
+++ b/homepage/sitemap.org
@@ -2,7 +2,6 @@
- [[file:newlectures/newlecture.org][Föreläsningar i Talteori]]
- [[file:newlectures/new-lect-0.org][Talteori översiktsföreläsning]]
-- [[file:index.org][TATA54 Talteori]]
- [[file:lectures/Henselfaktorisering.org][Henselfaktorisering]]
- [[file:senaste-nytt.org][senaste-nytt]]
- [[file:labs/lecture5.org][lecture5]]
@@ -11,5 +10,6 @@
- [[file:labs/lecture2.org][lecture2]]
- [[file:labs/lecture1.org][lecture1]]
- [[file:labs/lecture0.org][lecture0]]
+- [[file:index.org][TATA54 Talteori]]
- [[file:labs/kedjebraklab.org][Sagemath-övningar på kedjebråk]]
- [[file:labs/HenselLyftLabHT2023.org][SageMath-övningar på Hensellyft, primitiva rötter, och KRS]]
\ No newline at end of file
diff --git a/public/index.html b/public/index.html
index 468ff96..2d1ed2b 100644
--- a/public/index.html
+++ b/public/index.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="sv">
<head>
-<!-- 2024-05-13 -->
+<!-- 2024-05-16 -->
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>TATA54 Talteori</title>
@@ -74,20 +74,21 @@
</li>
<li><a href="#orgda7b5cb">Allra senaste nytt VT2024 </a>
<ul>
-<li><a href="#orgec1fbe3">2024-05-06</a></li>
+<li><a href="#orgef16020">2024-05-16</a></li>
+<li><a href="#orgd3edf39">2024-05-13</a></li>
<li><a href="#orgbb2c19c">2024-05-06</a></li>
<li><a href="#org6916c72">2024-04-29</a></li>
<li><a href="#org2fc933f">2024-04-23</a></li>
<li><a href="#org0e96bfa">2024-04-22</a></li>
<li><a href="#org796ce06">2024-04-14</a>
<ul>
-<li><a href="#org8e146d9">Uppgifter att räkna till nästa gång</a></li>
+<li><a href="#orga54ad49">Uppgifter att räkna till nästa gång</a></li>
<li><a href="#org87e8642">Datorlaboration</a></li>
</ul>
</li>
<li><a href="#orgff00c35">2024-04-07</a>
<ul>
-<li><a href="#orga54ad49">Uppgifter att räkna till nästa gång</a></li>
+<li><a href="#orgcd7a1cf">Uppgifter att räkna till nästa gång</a></li>
<li><a href="#orgcb6d72a">Beviset för kvadratisk reciprocitet</a></li>
</ul>
</li>
@@ -260,9 +261,22 @@ Skriftlig tentamen.
<h2 id="orgda7b5cb">Allra senaste nytt VT2024 <a id="org53d52c1"></a></h2>
<div class="outline-text-2" id="text-orgda7b5cb">
</div>
-<div id="outline-container-orgec1fbe3" class="outline-3">
-<h3 id="orgec1fbe3">2024-05-06</h3>
-<div class="outline-text-3" id="text-orgec1fbe3">
+<div id="outline-container-orgef16020" class="outline-3">
+<h3 id="orgef16020">2024-05-16</h3>
+<div class="outline-text-3" id="text-orgef16020">
+<p>
+Jag är inlagd pÃ¥ sjukhus… Det är väl osäkert om det gÃ¥r att hitta vikarier
+till lektionerna i morgon och på måndag. Fråga någon av studierektorerna, eller proprefekt Jesper Thoren.
+</p>
+
+<p>
+Tentamen skall kunna bli av utan problem!
+</p>
+</div>
+</div>
+<div id="outline-container-orgd3edf39" class="outline-3">
+<h3 id="orgd3edf39">2024-05-13</h3>
+<div class="outline-text-3" id="text-orgd3edf39">
<p>
Till på fredag så tittar vi på 13.4.3abc, 13.4.5, 13.4.11.
Till sista lektionen (måndag 20 maj) så tittar vi på 14.1.17ab, 14.1.23.
@@ -315,9 +329,9 @@ Till på fredag 12.4.5b, 12.4.6b
<h3 id="org796ce06">2024-04-14</h3>
<div class="outline-text-3" id="text-org796ce06">
</div>
-<div id="outline-container-org8e146d9" class="outline-4">
-<h4 id="org8e146d9">Uppgifter att räkna till nästa gång</h4>
-<div class="outline-text-4" id="text-org8e146d9">
+<div id="outline-container-orga54ad49" class="outline-4">
+<h4 id="orga54ad49">Uppgifter att räkna till nästa gång</h4>
+<div class="outline-text-4" id="text-orga54ad49">
<p>
12.2.1bf, 12.2.2cd, 12.2.3c, 12.2.11
</p>
@@ -338,9 +352,9 @@ gränsvärdet av a(k,m) då m går mot oändligheten?
<h3 id="orgff00c35">2024-04-07</h3>
<div class="outline-text-3" id="text-orgff00c35">
</div>
-<div id="outline-container-orga54ad49" class="outline-4">
-<h4 id="orga54ad49">Uppgifter att räkna till nästa gång</h4>
-<div class="outline-text-4" id="text-orga54ad49">
+<div id="outline-container-orgcd7a1cf" class="outline-4">
+<h4 id="orgcd7a1cf">Uppgifter att räkna till nästa gång</h4>
+<div class="outline-text-4" id="text-orgcd7a1cf">
<p>
11.1.1d, 11.1.4, 11.1.25ab, 11.2.1cd, 11.2.2, 11.2.3,11.2.4
</p>
@@ -468,7 +482,7 @@ H_L_tree(f,p,r).plot(layout='tree')
</div>
-<figure id="org331bb9b">
+<figure id="org73a37ac">
<img src="img/Henseluppgift.png" alt="Henseluppgift.png">
</figure>
@@ -1694,7 +1708,7 @@ Another text is <a href="https://wstein.org/ent/">Elementary Number Theory</a> a
</ul>
-<figure id="orgd034a26">
+<figure id="org3b9ce29">
<img src="img/sieveE.jpg" alt="sieveE.jpg">
</figure>
@@ -1715,7 +1729,7 @@ Another text is <a href="https://wstein.org/ent/">Elementary Number Theory</a> a
</main>
<footer id="postamble" class="status">
<p class="author">Författare: Jan Snellman Jan Snellman</p>
-<p class="date">Created: 2024-05-13</p>
+<p class="date">Created: 2024-05-16</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
</footer>
</body>
diff --git a/public/literature/Multiplicative.pdf b/public/literature/Multiplicative.pdf
new file mode 100644
index 0000000..8461096
--- /dev/null
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+/F47 10.91 Tf 0 0 Td[(Lemma)-383(1.)]TJ/F48 10.91 Tf 59.7 0 Td[(If)]TJ/F32 10.91 Tf 12.02 0 Td[(f)]TJ/F48 10.91 Tf 11 0 Td[(is)-409(multiplic)53(ative,)-423(then)-411(either)]TJ/F32 10.91 Tf 140.01 0 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\0501\051)-372(=)-373(1)]TJ/F48 10.91 Tf 40.49 0 Td[(or)]TJ/F32 10.91 Tf 14.65 0 Td[(f)]TJ/F34 10.91 Tf 10.58 0 Td[()]TJ/F15 10.91 Tf 12.56 0 Td[(0)]TJ/F48 10.91 Tf 5.45 0 Td[(.)-617(F)76(urther,)-422(if)]TJ/F32 10.91 Tf 64.77 0 Td[(f)]TJ/F31 7.97 Tf 5.35 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-166(f)]TJ/F31 7.97 Tf 10.19 -1.64 Td[(2)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-166(:::;)-166(f)]TJ/F33 7.97 Tf 24.13 -1.78 Td[(k)]TJ/F48 10.91 Tf -426.88 -11.18 Td[(ar)51(e)-357(multiplic)53(ative,)-357(then)-358(the)-358(usual)-357(pr)52(o)51(duct)]TJ/F32 10.91 Tf 198.49 0 Td[(f)]TJ/F31 7.97 Tf 5.35 -1.63 Td[(1)]TJ/F32 10.91 Tf 4.73 1.63 Td[(f)]TJ/F31 7.97 Tf 5.34 -1.63 Td[(2)]TJ/F34 10.91 Tf 6.55 1.63 Td[()-166()-167()]TJ/F32 10.91 Tf 14.54 0 Td[(f)]TJ/F33 7.97 Tf 5.35 -1.63 Td[(n)]TJ/F48 10.91 Tf 9.53 1.63 Td[(is)-358(also)-357(multiplic)53(ative.)]TJ/F15 10.91 Tf -249.88 -26.45 Td[(Using)-307(the)-308(prime)-308(decomp)-27(osition)-308(of)]TJ/F32 10.91 Tf 163.31 0 Td[(n)]TJ/F34 10.91 Tf 9.58 0 Td[(2)]TJ/F40 10.91 Tf 10.3 0 Td[(N)]TJ/F15 10.91 Tf 7.88 0 Td[(,)-313(deriv)29(e)-308(the)-308(follo)29(wing)-308(represen)28(tations)-307(of)]TJ/F32 10.91 Tf 190.12 0 Td[( )-58(;)-166()]TJ/F15 10.91 Tf 20.23 0 Td[(and)]TJ/F32 10.91 Tf 20.94 0 Td[()]TJ/F15 10.91 Tf 6.61 0 Td[(.)]TJ/F47 10.91 Tf -428.97 -26.46 Td[(Lemma)-383(2.)]TJ/F48 10.91 Tf 59.7 0 Td[(The)-358(functions)]TJ/F32 10.91 Tf 69.69 0 Td[( )]TJ/F15 10.91 Tf 6.01 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)]TJ/F32 10.91 Tf 4.24 0 Td[(;)-166()]TJ/F15 10.91 Tf 11.47 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)]TJ/F48 10.91 Tf 8.15 0 Td[(and)]TJ/F32 10.91 Tf 21.19 0 Td[()]TJ/F15 10.91 Tf 6.61 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)]TJ/F48 10.91 Tf 8.14 0 Td[(ar)51(e)-357(given)-358(by)-357(the)-358(formulas:)]TJ/F32 10.91 Tf -147.26 -28 Td[( )]TJ/F15 10.91 Tf 6 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-278(=)]TJ/F33 7.97 Tf 23.73 13.63 Td[(r)]TJ/F36 10.91 Tf -4.94 -3.27 Td[(Y)]TJ/F33 7.97 Tf 0.12 -23.45 Td[(i)]TJ/F31 7.97 Tf 2.88 0 Td[(=1)]TJ/F15 10.91 Tf 10.94 13.09 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[( )]TJ/F33 7.97 Tf 6.98 -1.64 Td[(i)]TJ/F15 10.91 Tf 5.81 1.64 Td[(+)-222(1\051)]TJ/F32 10.91 Tf 20.6 0 Td[(;)-500()]TJ/F15 10.91 Tf 15.11 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-278(=)]TJ/F33 7.97 Tf 23.73 13.63 Td[(r)]TJ/F36 10.91 Tf -4.94 -3.27 Td[(Y)]TJ/F33 7.97 Tf 0.12 -23.45 Td[(i)]TJ/F31 7.97 Tf 2.88 0 Td[(=1)]TJ/F32 10.91 Tf 13.95 20.6 Td[(p)]TJ/F33 7.97 Tf 5.49 4.97 Td[( )]TJ/F19 5.98 Tf 5.44 -1.21 Td[(i)]TJ/F31 7.97 Tf 3.16 1.21 Td[(+1)]TJ/F33 7.97 Tf -8.6 -8.23 Td[(i)]TJ/F34 10.91 Tf 22.34 3.26 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1)]TJ
+ET
+229.1 -91.36 44.2 0.44 re f
+1 0 0 1 237.37 -101.35 cm
+BT
+/F32 10.91 Tf 0 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(i)]TJ/F34 10.91 Tf 5.81 1.64 Td[(�)]TJ/F15 10.91 Tf 10.9 0 Td[(1)]TJ/F32 10.91 Tf 14.92 7.48 Td[(;)-500()]TJ/F15 10.91 Tf 15.1 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-277(=)]TJ/F32 10.91 Tf 18.79 0 Td[(n)]TJ/F17 5.98 Tf 7.74 7.76 Td[(1)]TJ
+ET
+89.54 13.8 3.65 0.36 re f
+1 0 0 1 89.54 8.86 cm
+BT
+/F17 5.98 Tf 0 0 Td[(2)]TJ/F33 7.97 Tf 4.85 3.12 Td[( )]TJ/F31 7.97 Tf 4.68 0 Td[(\050)]TJ/F33 7.97 Tf 3.29 0 Td[(n)]TJ/F31 7.97 Tf 5.14 0 Td[(\051)]TJ/F32 10.91 Tf 3.79 -4.5 Td[(:)]TJ/F48 10.91 Tf -348.66 -28.87 Td[(Conclude)-358(that)]TJ/F32 10.91 Tf 69.97 0 Td[( )]TJ/F48 10.91 Tf 9.91 0 Td[(and)]TJ/F32 10.91 Tf 21.19 0 Td[()]TJ/F48 10.91 Tf 10.53 0 Td[(ar)52(e)-358(multiplic)53(ative,)-357(while)]TJ/F32 10.91 Tf 116.95 0 Td[()]TJ/F48 10.91 Tf 10.51 0 Td[(is)-357(not.)]TJ/F47 10.91 Tf -239.06 -32.58 Td[(Examples.)]TJ/F15 10.91 Tf 60.9 0 Td[(The)-334(follo)29(wing)-332(are)-334(further)-331(examples)-334(of)-333(w)28(ell-kno)28(wn)-333(m)28(ultiplicativ)30(e)-334(functions.)]TJ/F34 10.91 Tf 5.6 -16.01 Td[()]TJ/F32 10.91 Tf 9.09 0 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(n)]TJ/F15 10.91 Tf 6.54 0 Td[(\051,)-333(the)-333(M)500(obius)-332(function;)]TJ/F34 10.91 Tf -26.45 -12.95 Td[()]TJ/F32 10.91 Tf 9.09 0 Td[(e)]TJ/F15 10.91 Tf 5.08 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-278(=)]TJ/F32 10.91 Tf 18.79 0 Td[()]TJ/F31 7.97 Tf 4.85 -1.64 Td[(1)]TJ/F33 7.97 Tf 4.23 0 Td[(;n)]TJ/F15 10.91 Tf 7.99 1.64 Td[(,)-333(the)-333(Diric)28(hlet)-333(iden)29(tit)28(y)-333(in)]TJ/F49 10.91 Tf 123.51 0 Td[(A)]TJ/F15 10.91 Tf 8.41 0 Td[(;)]TJ/F34 10.91 Tf -192.74 -12.96 Td[()]TJ/F32 10.91 Tf 9.09 0 Td[(I)]TJ/F15 10.91 Tf 5.65 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-278(=)-277(1)-334(for)-332(all)]TJ/F32 10.91 Tf 59.73 0 Td[(n)]TJ/F34 10.91 Tf 9.58 0 Td[(2)]TJ/F40 10.91 Tf 10.3 0 Td[(N)]TJ/F15 10.91 Tf 7.88 0 Td[(;)]TJ/F34 10.91 Tf -113.02 -12.95 Td[()]TJ/F32 10.91 Tf 9.09 0 Td[(id)]TJ/F15 10.91 Tf 9.44 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-277(=)]TJ/F32 10.91 Tf 18.79 0 Td[(n)]TJ/F15 10.91 Tf 10.18 0 Td[(for)-332(all)]TJ/F32 10.91 Tf 31.85 0 Td[(n)]TJ/F34 10.91 Tf 9.58 0 Td[(2)]TJ/F40 10.91 Tf 10.3 0 Td[(N)]TJ/F15 10.91 Tf 7.88 0 Td[(.)]TJ -172.45 -16.01 Td[(T)83(aking)-423(the)-423(sum-functions)-423(of)-423(these,)-446(w)28(e)-424(obtain)-423(the)-423(relations:)]TJ/F32 10.91 Tf 296.13 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.64 Td[()]TJ/F15 10.91 Tf 10.26 1.64 Td[(=)]TJ/F32 10.91 Tf 13.16 0 Td[(e)]TJ/F15 10.91 Tf 5.08 0 Td[(,)]TJ/F32 10.91 Tf 7.89 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.64 Td[(e)]TJ/F15 10.91 Tf 9.1 1.64 Td[(=)]TJ/F32 10.91 Tf 13.16 0 Td[(I)]TJ/F15 10.91 Tf 5.65 0 Td[(,)]TJ/F32 10.91 Tf 7.9 0 Td[(S)]TJ/F33 7.97 Tf 6.68 -1.69 Td[(I)]TJ/F15 10.91 Tf 9.47 1.69 Td[(=)]TJ/F32 10.91 Tf 13.15 0 Td[( )]TJ/F15 10.91 Tf 6.01 0 Td[(,)]TJ -428.97 -12.95 Td[(and)]TJ/F32 10.91 Tf 21.15 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(id)]TJ/F15 10.91 Tf 10.76 1.78 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[()]TJ/F15 10.91 Tf 6.62 0 Td[(.)-443(These)-328(examples)-328(suggest)-327(that)-327(the)-327(sum-function)-326(is)-327(m)27(ultiplicativ)30(e,)-328(pro)28(vided)-326(the)]TJ -56.74 -12.95 Td[(original)-332(function)-331(is)-333(to)-28(o.)-444(In)-333(fact,)]TJ/F47 10.91 Tf 0 -20.34 Td[(Theorem)-383(1.)]TJ/F32 10.91 Tf 68.48 0 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)]TJ/F48 10.91 Tf 8.14 0 Td[(is)-358(multiplic)54(ative)-358(i )-358(its)-357(sum-function)]TJ/F32 10.91 Tf 174.25 0 Td[(S)]TJ/F33 7.97 Tf 6.68 -1.77 Td[(f)]TJ/F15 10.91 Tf 5.45 1.77 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)]TJ/F48 10.91 Tf 8.15 0 Td[(is)-357(multiplic)53(ative.)]TJ/F46 10.91 Tf -299.25 -20.33 Td[(Pr)22(oof:)]TJ/F15 10.91 Tf 44.56 0 Td[(Let)]TJ/F32 10.91 Tf 20.6 0 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-429(b)-28(e)-430(m)28(ultiplicativ)30(e,)-454(and)-429(let)]TJ/F32 10.91 Tf 137.03 0 Td[(x;)-167(y)]TJ/F34 10.91 Tf 21.61 0 Td[(2)]TJ/F40 10.91 Tf 12.06 0 Td[(N)]TJ/F15 10.91 Tf 12.56 0 Td[(suc)27(h)-429(that)-429(\050)]TJ/F32 10.91 Tf 54.59 0 Td[(x;)-167(y)]TJ/F15 10.91 Tf 16.82 0 Td[(\051)-439(=)-438(1.)-733(F)84(urther,)-453(let)]TJ/F32 10.91 Tf -337.14 -12.95 Td[(x)]TJ/F31 7.97 Tf 6.23 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.74 1.64 Td[(;)-166(x)]TJ/F31 7.97 Tf 11.08 -1.64 Td[(2)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-167(:::;)-165(x)]TJ/F33 7.97 Tf 25.02 -1.78 Td[(k)]TJ/F15 10.91 Tf 8.38 1.78 Td[(and)]TJ/F32 10.91 Tf 20.83 0 Td[(y)]TJ/F31 7.97 Tf 5.35 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-167(y)]TJ/F31 7.97 Tf 10.2 -1.64 Td[(2)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-167(:::;)-165(y)]TJ/F33 7.97 Tf 24.14 -1.64 Td[(m)]TJ/F15 10.91 Tf 11.24 1.64 Td[(b)-28(e)-299(all)-298(divisors)-297(of)]TJ/F32 10.91 Tf 80.76 0 Td[(x)]TJ/F15 10.91 Tf 9.49 0 Td[(and)]TJ/F32 10.91 Tf 20.83 0 Td[(y)]TJ/F15 10.91 Tf 5.74 0 Td[(,)-305(resp)-28(ectiv)28(ely)83(.)-433(Then)-298(\050)]TJ/F32 10.91 Tf 101.4 0 Td[(x)]TJ/F33 7.97 Tf 6.24 -1.64 Td[(i)]TJ/F32 10.91 Tf 3.38 1.64 Td[(;)-166(y)]TJ/F33 7.97 Tf 10.2 -1.64 Td[(j)]TJ/F15 10.91 Tf 4.38 1.64 Td[(\051)-277(=)-278(1,)-305(and)]TJ/F34 10.91 Tf -383.82 -12.95 Td[(f)]TJ/F32 10.91 Tf 5.45 0 Td[(x)]TJ/F33 7.97 Tf 6.24 -1.64 Td[(i)]TJ/F32 10.91 Tf 3.38 1.64 Td[(y)]TJ/F33 7.97 Tf 5.35 -1.64 Td[(j)]TJ/F34 10.91 Tf 4.38 1.64 Td[(g)]TJ/F33 7.97 Tf 5.46 -1.64 Td[(i;j)]TJ/F15 10.91 Tf 13.25 1.64 Td[(are)-333(all)-333(divisors)-332(of)]TJ/F32 10.91 Tf 85.64 0 Td[(xy)]TJ/F15 10.91 Tf 11.97 0 Td[(.)]TJ/F34 10.91 Tf -113.94 -30.57 Td[(\051)]TJ/F32 10.91 Tf 17.58 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.44 1.78 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(x)]TJ/F15 10.91 Tf 6.24 0 Td[(\051)]TJ/F34 10.91 Tf 6.67 0 Td[()]TJ/F32 10.91 Tf 5.45 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.44 1.78 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(y)]TJ/F15 10.91 Tf 5.74 0 Td[(\051)-277(=)]TJ/F33 7.97 Tf 24.35 13.64 Td[(k)]TJ/F36 10.91 Tf -5.57 -3.28 Td[(X)]TJ/F33 7.97 Tf 1.03 -23.45 Td[(i)]TJ/F31 7.97 Tf 2.88 0 Td[(=1)]TJ/F32 10.91 Tf 13.67 13.09 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(x)]TJ/F33 7.97 Tf 6.23 -1.64 Td[(i)]TJ/F15 10.91 Tf 3.38 1.64 Td[(\051)]TJ/F33 7.97 Tf 10.2 13.64 Td[(m)]TJ/F36 10.91 Tf -4.14 -3.28 Td[(X)]TJ/F33 7.97 Tf 0.53 -23.45 Td[(j)]TJ/F31 7.97 Tf 3.88 0 Td[(=1)]TJ/F32 10.91 Tf 13.17 13.09 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(y)]TJ/F33 7.97 Tf 5.35 -1.64 Td[(j)]TJ/F15 10.91 Tf 4.38 1.64 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.79 10.36 Td[(X)]TJ/F33 7.97 Tf 3.32 -23.45 Td[(i;j)]TJ/F32 10.91 Tf 14.25 13.09 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(x)]TJ/F33 7.97 Tf 6.23 -1.64 Td[(i)]TJ/F15 10.91 Tf 3.39 1.64 Td[(\051)]TJ/F32 10.91 Tf 4.24 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(y)]TJ/F33 7.97 Tf 5.34 -1.64 Td[(j)]TJ/F15 10.91 Tf 4.39 1.64 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.78 10.36 Td[(X)]TJ/F33 7.97 Tf 3.32 -23.45 Td[(i;j)]TJ/F32 10.91 Tf 14.26 13.09 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(x)]TJ/F33 7.97 Tf 6.23 -1.64 Td[(i)]TJ/F32 10.91 Tf 3.38 1.64 Td[(y)]TJ/F33 7.97 Tf 5.35 -1.64 Td[(j)]TJ/F15 10.91 Tf 4.38 1.64 Td[(\051)-278(=)]TJ/F32 10.91 Tf 18.79 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.45 1.78 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(xy)]TJ/F15 10.91 Tf 11.97 0 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(:)]TJ/F15 10.91 Tf -401.79 -31.03 Td[(Hence)]TJ/F32 10.91 Tf 32.42 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.77 Td[(f)]TJ/F15 10.91 Tf 5.45 1.77 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-333(is)-333(m)27(ultiplicativ)30(e.)]TJ -43.4 -16.01 Td[(Con)28(v)28(ersely)83(,)-465(if)]TJ/F32 10.91 Tf 69.95 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.45 1.78 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-438(is)-439(m)27(ultiplicativ)31(e,)-466(then)-438(let)]TJ/F32 10.91 Tf 137.3 0 Td[(n)]TJ/F31 7.97 Tf 6.55 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-167(n)]TJ/F31 7.97 Tf 11.4 -1.64 Td[(2)]TJ/F34 10.91 Tf 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Td[(n)]TJ/F31 7.97 Tf 6.55 -1.64 Td[(1)]TJ/F15 10.91 Tf 4.73 1.64 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F31 7.97 Tf 6.55 -1.64 Td[(2)]TJ/F15 10.91 Tf 4.73 1.64 Td[(\051.)-518(The)-358(statemen)27(t)-358(is)]TJ -338.11 -12.95 Td[(trivial)-356(for)]TJ/F32 10.91 Tf 49.69 0 Td[(n)]TJ/F15 10.91 Tf 10.03 0 Td[(=)-318(1:)]TJ/F32 10.91 Tf 25.84 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\0501\051)-319(=)]TJ/F32 10.91 Tf 29.38 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.45 1.78 Td[(\0501\051\050=)-318(1)-358(or)-358(0.\051)-518(Assume)-359(that)-357(it)-358(is)-358(true)-357(for)-357(all)]TJ/F32 10.91 Tf 214.26 0 Td[(m)]TJ/F31 7.97 Tf 9.58 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 1.64 Td[(m)]TJ/F31 7.97 Tf 9.58 -1.64 Td[(2)]TJ/F32 10.91 Tf 8.21 1.64 Td[(<)-319(n)]TJ/F15 10.91 Tf 18.51 0 Td[(.)-518(Then)]TJ -398.46 -12.95 Td[(for)-332(our)]TJ/F32 10.91 Tf 36.12 0 Td[(n)]TJ/F31 7.97 Tf 6.55 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 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10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F31 7.97 Tf 5.68 -1.64 Td[(3)]TJ/F15 10.91 Tf 4.73 1.64 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(:)]TJ/F15 10.91 Tf -383.49 -30.1 Td[(With)-332(resp)-28(ect)-334(to)-333(the)-333(Diric)28(hlet)-332(pro)-27(duct,)-333(the)-333(iden)29(tit)28(y)-333(elemen)27(t)]TJ/F32 10.91 Tf 286.06 0 Td[(e)]TJ/F34 10.91 Tf 8.11 0 Td[(2)]TJ/F49 10.91 Tf 10.3 0 Td[(A)]TJ/F15 10.91 Tf 12.04 0 Td[(is)-334(easy)-334(to)-333(\014nd:)]TJ/F32 10.91 Tf -155.01 -24.83 Td[(e)]TJ/F15 10.91 Tf 5.08 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-278(=)]TJ/F36 10.91 Tf 18.79 15.38 Td[()]TJ/F15 10.91 Tf 13.16 -8.77 Td[(1)-913(if)]TJ/F32 10.91 Tf 25.42 0 Td[(n)]TJ/F15 10.91 Tf 9.58 0 Td[(=)-277(1)]TJ -35 -12.95 Td[(0)-913(if)]TJ/F32 10.91 Tf 25.42 0 Td[(n)-278(>)]TJ/F15 10.91 Tf 21.09 0 Td[(1)]TJ/F32 10.91 Tf 5.46 0 Td[(:)]TJ/F15 10.91 Tf -261.29 -18.57 Td[(Indeed,)-423(c)27(hec)28(k)-406(that)]TJ/F32 10.91 Tf 94.08 0 Td[(e)]TJ/F34 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Td[(the)-350(iden)29(tit)28(y)]TJ -376.36 -12.95 Td[(elemen)27(t,)-366(but)-358(has)-360(the)-359(nice)-360(prop)-27(ert)28(y)-359(of)-359(transforming)-358(eac)27(h)-359(function)]TJ/F32 10.91 Tf 315.92 0 Td[(f)]TJ/F15 10.91 Tf 10.44 0 Td[(in)29(to)-360(its)-360(so-called)]TJ/F48 10.91 Tf 82.5 0 Td[(sum-)]TJ -408.86 -12.95 Td[(function)]TJ/F32 10.91 Tf 42.67 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.44 1.78 Td[(.)-444(De\014ne)]TJ/F32 10.91 Tf 119.7 -21.67 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.44 1.78 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.78 10.36 Td[(X)]TJ/F33 7.97 Tf 1.96 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.36 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.91 13.8 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(d)]TJ/F15 10.91 Tf 5.67 0 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(;)]TJ/F15 10.91 Tf -254.47 -31.43 Td[(to)-333(b)-27(e)-334(the)]TJ/F48 10.91 Tf 46.97 0 Td[(sum-function)]TJ/F15 10.91 Tf 65.8 0 Td[(of)]TJ/F32 10.91 Tf 12.43 0 Td[(f)]TJ/F34 10.91 Tf 9.54 0 Td[(2)]TJ/F49 10.91 Tf 10.31 0 Td[(A)]TJ/F15 10.91 Tf 8.4 0 Td[(,)-333(and)-333(note)-333(that)]TJ/F32 10.91 Tf 75.76 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.77 Td[(f)]TJ/F15 10.91 Tf 9.08 1.77 Td[(is)-334(also)-333(arithmetic.)-444(Chec)28(k)-333(that:)]TJ/F32 10.91 Tf -105.45 -18.84 Td[(I)]TJ/F34 10.91 Tf 8.08 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(f)]TJ/F15 10.91 Tf 9.54 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(f)]TJ/F34 10.91 Tf 8.94 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(I)]TJ/F15 10.91 Tf 8.68 0 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 9.08 1.78 Td[(for)-333(all)]TJ/F32 10.91 Tf 31.85 0 Td[(f)]TJ/F34 10.91 Tf 9.55 0 Td[(2)]TJ/F49 10.91 Tf 10.3 0 Td[(A)]TJ/F32 10.91 Tf 8.41 0 Td[(:)]TJ/F15 10.91 Tf -289.44 -18.84 Td[(It)-367(is)-368(in)28(teresting)-367(to)-367(\014nd)-366(the)]TJ/F48 10.91 Tf 132.69 0 Td[(Dirichlet)-388(inverse)]TJ/F32 10.91 Tf 83.4 0 Td[(g)]TJ/F34 10.91 Tf 9.25 0 Td[(2)]TJ/F49 10.91 Tf 10.93 0 Td[(A)]TJ/F15 10.91 Tf 12.41 0 Td[(of)-367(this)-367(simple)-368(function)]TJ/F32 10.91 Tf 111.92 0 Td[(I)]TJ/F15 10.91 Tf 9.66 0 Td[(in)-367(our)-366(set)]TJ/F49 10.91 Tf 50.3 0 Td[(A)]TJ/F15 10.91 Tf 8.41 0 Td[(,)]TJ -428.97 -12.95 Td[(i.e.)-597(suc)27(h)-384(that)]TJ/F32 10.91 Tf 69.82 0 Td[(g)]TJ/F34 10.91 Tf 8.39 0 Td[()]TJ/F32 10.91 Tf 8.25 0 Td[(I)]TJ/F15 10.91 Tf 9.61 0 Td[(=)]TJ/F32 10.91 Tf 12.44 0 Td[(I)]TJ/F34 10.91 Tf 8.45 0 Td[()]TJ/F32 10.91 Tf 8.25 0 Td[(g)]TJ/F15 10.91 Tf 9.56 0 Td[(=)]TJ/F32 10.91 Tf 12.44 0 Td[(e)]TJ/F15 10.91 Tf 5.08 0 Td[(.)-597(This)-385(naturally)-382(leads)-384(to)-384(the)-385(in)29(tro)-27(duction)-383(of)-384(the)-384(so-called)]TJ/F48 10.91 Tf -152.29 -12.96 Td[(M)511(obius)-356(function)]TJ/F32 10.91 Tf 80.6 0 Td[()]TJ/F15 10.91 Tf 6.58 0 Td[(:)]TJ/F47 10.91 Tf -87.18 -18.84 Td[(De\014nition)-384(2.)]TJ/F15 10.91 Tf 73.19 0 Td[(The)]TJ/F48 10.91 Tf 22.42 0 Td[(M)511(obius)-357(function)]TJ/F32 10.91 Tf 80.61 0 Td[()]TJ/F15 10.91 Tf 9.6 0 Td[(:)]TJ/F40 10.91 Tf 6.06 0 Td[(N)]TJ/F34 10.91 Tf 10.91 0 Td[(!)]TJ/F40 10.91 Tf 13.94 0 Td[(C)]TJ/F15 10.91 Tf 11.52 0 Td[(is)-333(de\014ned)-332(b)28(y)]TJ/F32 10.91 Tf -151.09 -31.37 Td[()]TJ/F15 10.91 Tf 6.58 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.79 22.36 Td[(8)]TJ 0 -9.82 Td[(<)]TJ 0 -19.63 Td[(:)]TJ/F15 10.91 Tf 25.44 20.18 Td[(1)-1899(if)]TJ/F32 10.91 Tf 36.18 0 Td[(n)]TJ/F15 10.91 Tf 9.57 0 Td[(=)-278(1)]TJ -45.75 -12.95 Td[(0)-1899(if)]TJ/F32 10.91 Tf 36.18 0 Td[(n)]TJ/F15 10.91 Tf 10.18 0 Td[(is)-333(not)-333(square-free)]TJ -57.13 -12.96 Td[(\050)]TJ/F34 10.91 Tf 4.25 0 Td[(�)]TJ/F15 10.91 Tf 8.48 0 Td[(1\051)]TJ/F33 7.97 Tf 9.7 3.96 Td[(r)]TJ/F15 10.91 Tf 14.52 -3.96 Td[(if)]TJ/F32 10.91 Tf 10 0 Td[(n)]TJ/F15 10.91 Tf 9.57 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.63 Td[(1)]TJ/F32 10.91 Tf 4.73 1.63 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.63 Td[(2)]TJ/F34 10.91 Tf 6.55 1.63 Td[()-166()-167()]TJ/F32 10.91 Tf 14.54 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.63 Td[(r)]TJ/F32 10.91 Tf 4.56 1.63 Td[(;)-499(p)]TJ/F33 7.97 Tf 13.97 -1.63 Td[(j)]TJ/F34 10.91 Tf 6.81 1.63 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(distinct)-332(primes)]TJ/F32 10.91 Tf 71.06 0 Td[(:)]TJ/F47 10.91 Tf -345.63 -20 Td[(Lemma)-383(3.)]TJ/F48 10.91 Tf 59.7 0 Td[(The)-358(Dirichlet)-356(inverse)-357(of)]TJ/F32 10.91 Tf 118.07 0 Td[(I)]TJ/F48 10.91 Tf 9.56 0 Td[(is)-357(the)-358(M)511(obius)-356(function)]TJ/F32 10.91 Tf 110.7 0 Td[()]TJ/F34 10.91 Tf 9.6 0 Td[(2)]TJ/F49 10.91 Tf 10.31 0 Td[(A)]TJ/F48 10.91 Tf 8.4 0 Td[(.)]TJ/F46 10.91 Tf -326.34 -20.2 Td[(Pr)22(oof:)]TJ/F15 10.91 Tf 42.46 0 Td[(The)-333(lemma)-334(means)-334(that)]TJ/F32 10.91 Tf 114.61 0 Td[()]TJ/F34 10.91 Tf 8.99 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(I)]TJ/F15 10.91 Tf 8.68 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(e)]TJ/F15 10.91 Tf 5.08 0 Td[(,)-333(i.e.)]TJ/F36 10.91 Tf -44.56 -14.47 Td[(X)]TJ/F33 7.97 Tf 1.95 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.36 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.91 13.8 Td[()]TJ/F15 10.91 Tf 6.58 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F15 10.91 Tf 5.68 0 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.78 15.38 Td[()]TJ/F15 10.91 Tf 13.17 -8.77 Td[(1)-333(if)]TJ/F32 10.91 Tf 19.09 0 Td[(n)]TJ/F15 10.91 Tf 9.58 0 Td[(=)-277(1)]TJ -28.67 -12.95 Td[(0)-333(if)]TJ/F32 10.91 Tf 19.09 0 Td[(n)-278(>)]TJ/F15 10.91 Tf 21.09 0 Td[(1)]TJ/F32 10.91 Tf 5.46 0 Td[(:)]TJ/F15 10.91 Tf -266.32 6.34 Td[(\0501\051)]TJ/F31 7.97 Tf 213.88 -28.83 Td[(3)]TJ
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+/F15 10.91 Tf 0 0 Td[(This)-333(follo)29(ws)-334(easily)-333(from)-333(the)-333(de\014nition)-331(of)]TJ/F32 10.91 Tf 196.27 0 Td[()]TJ/F15 10.91 Tf 6.58 0 Td[(.)-444(Indeed,)-332(for)]TJ/F32 10.91 Tf 63.06 0 Td[(n)]TJ/F15 10.91 Tf 9.57 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(p)]TJ/F33 7.97 Tf 5.49 4.86 Td[( )]TJ/F17 5.98 Tf 5.44 -1.11 Td[(1)]TJ/F31 7.97 Tf -5.44 -6.88 Td[(1)]TJ/F32 10.91 Tf 10.09 3.13 Td[(p)]TJ/F33 7.97 Tf 5.48 4.86 Td[( )]TJ/F17 5.98 Tf 5.44 -1.11 Td[(2)]TJ/F31 7.97 Tf -5.44 -6.88 Td[(2)]TJ/F34 10.91 Tf 11.91 3.13 Td[()-166()-167()]TJ/F32 10.91 Tf 14.55 0 Td[(p)]TJ/F33 7.97 Tf 5.48 3.95 Td[( )]TJ/F19 5.98 Tf 5.44 -0.99 Td[(r)]TJ/F33 7.97 Tf -5.44 -5.66 Td[(r)]TJ/F32 10.91 Tf 13.13 2.7 Td[(>)]TJ/F15 10.91 Tf 11.51 0 Td[(1)-334(w)28(e)-334(ha)29(v)28(e)]TJ/F36 10.91 Tf -303.15 -18.85 Td[(X)]TJ/F33 7.97 Tf 1.96 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.35 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.92 13.8 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F15 10.91 Tf 5.68 0 Td[(\051)-278(=)]TJ/F36 10.91 Tf 35.27 10.36 Td[(X)]TJ/F33 7.97 Tf -16.48 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.35 0 Td[(j)]TJ/F33 7.97 Tf 2.36 0 Td[(n;d)]TJ/F35 7.97 Tf 11.84 0 Td[(�)]TJ/F31 7.97 Tf 6.59 0 Td[(sq.free)]TJ/F32 10.91 Tf 25.4 13.8 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F15 10.91 Tf 5.68 0 Td[(\051)-278(=)]TJ/F32 10.91 Tf 18.79 0 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\0501\051)-222(+)]TJ/F33 7.97 Tf 33.12 13.64 Td[(r)]TJ/F36 10.91 Tf -5.85 -3.28 Td[(X)]TJ/F33 7.97 Tf 0.16 -23.71 Td[(k)]TJ/F31 7.97 Tf 4.62 0 Td[(=1)]TJ/F36 10.91 Tf 33.01 23.71 Td[(X)]TJ/F31 7.97 Tf -20.21 -23.51 Td[(1)]TJ/F35 7.97 Tf 4.23 0 Td[()]TJ/F33 7.97 Tf 6.59 0 Td[(i)]TJ/F17 5.98 Tf 2.88 -1.1 Td[(1)]TJ/F33 7.97 Tf 4.15 1.1 Td[(<)]TJ/F35 7.97 Tf 6.59 0 Td[()]TJ/F33 7.97 Tf 7.06 0 Td[(<i)]TJ/F19 5.98 Tf 9.47 -1.4 Td[(k)]TJ/F35 7.97 Tf 4.57 1.4 Td[()]TJ/F33 7.97 Tf 6.59 0 Td[(r)]TJ/F32 10.91 Tf 5.87 13.15 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(i)]TJ/F17 5.98 Tf 2.88 -1.1 Td[(1)]TJ/F34 10.91 Tf 6.47 2.74 Td[()-166()-166()]TJ/F32 10.91 Tf 14.54 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(i)]TJ/F19 5.98 Tf 2.88 -1.4 Td[(k)]TJ/F15 10.91 Tf 5.07 3.04 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(:)]TJ/F15 10.91 Tf -367.48 -32.38 Td[(Here)-334(the)-333(last)-333(sum)-334(runs)-332(o)28(v)28(er)-333(all)-333(square-free)-333(divisors)-333(of)]TJ/F32 10.91 Tf 258.73 0 Td[(n)]TJ/F15 10.91 Tf 6.54 0 Td[(.)-445(F)84(or)-333(com)27(binatorial)-331(reasons,)]TJ/F36 10.91 Tf -156.43 -17.83 Td[(X)]TJ/F33 7.97 Tf 1.96 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.36 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.91 13.79 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F15 10.91 Tf 5.68 0 Td[(\051)-278(=)]TJ/F33 7.97 Tf 24.64 13.64 Td[(r)]TJ/F36 10.91 Tf -5.85 -3.27 Td[(X)]TJ/F33 7.97 Tf 0.16 -23.72 Td[(k)]TJ/F31 7.97 Tf 4.62 0 Td[(=0)]TJ/F36 10.91 Tf 12.8 28.73 Td[()]TJ/F32 10.91 Tf 8.43 -8 Td[(r)]TJ -0.4 -14.86 Td[(k)]TJ/F36 10.91 Tf 6.02 22.86 Td[()]TJ/F15 10.91 Tf 8.03 -15.38 Td[(\050)]TJ/F34 10.91 Tf 4.24 0 Td[(�)]TJ/F15 10.91 Tf 8.49 0 Td[(1\051)]TJ/F33 7.97 Tf 9.69 4.51 Td[(k)]TJ/F15 10.91 Tf 8.15 -4.51 Td[(=)-278(\0501)]TJ/F34 10.91 Tf 23.64 0 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051)]TJ/F33 7.97 Tf 9.7 4.51 Td[(r)]TJ/F15 10.91 Tf 7.58 -4.51 Td[(=)-278(0)]TJ/F32 10.91 Tf 16.97 0 Td[(:)]TJ
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+1 0 0 1 11.95 -122.77 cm
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+/F15 10.91 Tf 0 0 Td[(T)83(o)-244(summarize,)-261(w)27(e)-244(ha)29(v)28(e)-244(sho)28(wn)-243(that)-243(the)-243(Diric)29(hlet)-243(in)28(v)28(erse)-244(of)-243(the)-243(function)]TJ/F32 10.91 Tf 334 0 Td[(I)]TJ/F15 10.91 Tf 5.66 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-243(is)-243(the)-244(M)501(obius)]TJ -362.4 -12.95 Td[(function)]TJ/F32 10.91 Tf 42.67 0 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051:)]TJ/F32 10.91 Tf 12.09 0 Td[()]TJ/F34 10.91 Tf 8.88 0 Td[()]TJ/F32 10.91 Tf 7.75 0 Td[(I)]TJ/F15 10.91 Tf 8.68 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(I)]TJ/F34 10.91 Tf 7.95 0 Td[()]TJ/F32 10.91 Tf 7.76 0 Td[()]TJ/F15 10.91 Tf 9.6 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(e)]TJ/F15 10.91 Tf 5.08 0 Td[(.)-442(Unfortunately)86(,)-329(not)-327(all)-327(arithmetic)-327(functions)-326(ha)28(v)28(e)-328(Diric)29(hlet)]TJ -150.86 -12.95 Td[(in)29(v)28(erses)-335(in)]TJ/F49 10.91 Tf 53.18 0 Td[(A)]TJ/F15 10.91 Tf 8.41 0 Td[(.)-444(In)-333(fact,)-333(sho)28(w)-333(that)]TJ/F47 10.91 Tf -61.59 -20.8 Td[(Lemma)-383(4.)]TJ/F48 10.91 Tf 59.7 0 Td[(A)26(n)-358(arithmetic)-357(function)]TJ/F32 10.91 Tf 113.35 0 Td[(f)]TJ/F48 10.91 Tf 10.41 0 Td[(has)-358(a)-358(Dirichlet)-356(inverse)-357(in)]TJ/F49 10.91 Tf 125.33 0 Td[(M)]TJ/F48 10.91 Tf 14.62 0 Td[(i )]TJ/F32 10.91 Tf 13.94 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\0501\051)]TJ/F34 10.91 Tf 16.97 0 Td[(6)]TJ/F15 10.91 Tf 0 0 Td[(=)-278(0)]TJ/F48 10.91 Tf 16.97 0 Td[(.)]TJ/F31 7.97 Tf 3.35 3.96 Td[(2)]TJ/F15 10.91 Tf -369.2 -24.75 Td[(The)-341(\134righ)29(t")-341(notion)-339(to)-341(replace)-340(\134in)28(v)28(erses)-342(in)]TJ/F49 10.91 Tf 203.21 0 Td[(A)]TJ/F15 10.91 Tf 8.4 0 Td[(")-341(turns)-340(out)-340(to)-340(b)-28(e)-341(\134sum-functions",)-341(whic)28(h)-340(is)]TJ -223.56 -12.95 Td[(the)-333(idea)-333(of)-333(the)-333(M)501(obius)-333(in)29(v)28(ersion)-333(theorem.)]TJ/F47 10.91 Tf 0 -20.8 Td[(Theorem)-383(2)]TJ/F15 10.91 Tf 63.73 0 Td[(\050M)501(obius)-333(in)29(v)28(ersion)-333(theorem\051)]TJ/F47 10.91 Tf 131.69 0 Td[(.)]TJ/F48 10.91 Tf 8.94 0 Td[(A)25(ny)-455(arithmetic)-455(function)]TJ/F32 10.91 Tf 121.86 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)]TJ/F48 10.91 Tf 9.21 0 Td[(c)51(an)-456(b)51(e)-456(expr)52(esse)51(d)]TJ -352.73 -12.95 Td[(in)-358(terms)-357(of)-357(its)-357(sum-function)]TJ/F32 10.91 Tf 138.14 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.44 1.78 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-278(=)]TJ/F36 10.91 Tf 18.79 8.18 Td[(P)]TJ/F33 7.97 Tf 11.52 -11.4 Td[(d)]TJ/F35 7.97 Tf 4.35 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 7.46 3.22 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(d)]TJ/F15 10.91 Tf 5.67 0 Td[(\051)]TJ/F48 10.91 Tf 8.15 0 Td[(as)]TJ/F32 10.91 Tf -67.61 -25.22 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.79 10.37 Td[(X)]TJ/F33 7.97 Tf 1.95 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.36 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.91 13.79 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(d)]TJ/F15 10.91 Tf 5.67 0 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.77 Td[(f)]TJ/F15 10.91 Tf 5.44 1.77 Td[(\050)]TJ/F32 10.91 Tf 5.44 7.38 Td[(n)]TJ
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+/F32 10.91 Tf 0 0 Td[(d)]TJ/F15 10.91 Tf 7.31 7.48 Td[(\051)]TJ/F32 10.91 Tf 4.25 0 Td[(:)]TJ/F46 10.91 Tf -266.47 -33.37 Td[(Pr)22(oof:)]TJ/F15 10.91 Tf 42.46 0 Td[(The)-333(statemen)27(t)-333(is)-334(nothing)-331(else)-335(but)-332(the)-333(Diric)29(hlet)-333(pro)-27(duct)]TJ/F32 10.91 Tf 269.48 0 Td[(f)]TJ/F15 10.91 Tf 9.55 0 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[()]TJ/F34 10.91 Tf 9 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 9.08 1.78 Td[(in)]TJ/F49 10.91 Tf 12.73 0 Td[(A)]TJ/F15 10.91 Tf 8.4 0 Td[(:)]TJ/F32 10.91 Tf -287.32 -19.88 Td[()]TJ/F34 10.91 Tf 8.99 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 8.48 1.78 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[()]TJ/F34 10.91 Tf 9 0 Td[()]TJ/F15 10.91 Tf 7.88 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(I)]TJ/F34 10.91 Tf 8.08 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\051)-278(=)-277(\050)]TJ/F32 10.91 Tf 23.03 0 Td[()]TJ/F34 10.91 Tf 9 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(I)]TJ/F15 10.91 Tf 5.65 0 Td[(\051)]TJ/F34 10.91 Tf 6.67 0 Td[()]TJ/F32 10.91 Tf 7.87 0 Td[(f)]TJ/F15 10.91 Tf 9.55 0 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[(e)]TJ/F34 10.91 Tf 7.51 0 Td[()]TJ/F32 10.91 Tf 7.88 0 Td[(f)]TJ/F15 10.91 Tf 9.54 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(f)-51(:)]TJ
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+/F48 10.91 Tf 0 0 Td[(Note:)]TJ/F15 10.91 Tf 30.53 0 Td[(Here)-333(is)-334(a)-333(more)-334(traditional)-330(pro)-27(of)-333(of)-333(the)-333(M)500(obius)-332(In)28(v)28(ersion)-333(F)84(orm)28(ula:)]TJ/F36 10.91 Tf 23.27 -11.76 Td[(X)]TJ/F33 7.97 Tf 1.95 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.36 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.92 13.8 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F15 10.91 Tf 5.68 0 Td[(\051)]TJ/F32 10.91 Tf 4.24 0 Td[(S)]TJ/F33 7.97 Tf 6.69 -1.78 Td[(f)]TJ/F15 10.91 Tf 5.45 1.78 Td[(\050)]TJ/F32 10.91 Tf 5.43 7.38 Td[(n)]TJ
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+/F32 10.91 Tf 0 0 Td[(d)]TJ/F15 10.91 Tf 7.31 7.48 Td[(\051)-913(=)]TJ/F36 10.91 Tf 32.65 10.36 Td[(X)]TJ/F33 7.97 Tf 1.95 -24.16 Td[(d)]TJ/F35 7.97 Tf 4.36 0 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 8.92 13.8 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 5.44 7.38 Td[(n)]TJ
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+/F32 10.91 Tf 0 0 Td[(d)]TJ/F15 10.91 Tf 7.3 7.48 Td[(\051)]TJ/F36 10.91 Tf 6.07 10.36 Td[(X)]TJ/F33 7.97 Tf 0.27 -24.16 Td[(d)]TJ/F17 5.98 Tf 4.35 -1.1 Td[(1)]TJ/F35 7.97 Tf 4.15 1.1 Td[(j)]TJ/F33 7.97 Tf 2.36 0 Td[(d)]TJ/F32 10.91 Tf 6.44 13.8 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F31 7.97 Tf 5.68 -1.64 Td[(1)]TJ/F15 10.91 Tf 4.73 1.64 Td[(\051)]TJ -183 -33.19 Td[(=)]TJ/F36 10.91 Tf 18.57 10.36 Td[(X)]TJ/F33 7.97 Tf -0.12 -24.16 Td[(d)]TJ/F17 5.98 Tf 4.36 -1.1 Td[(1)]TJ/F35 7.97 Tf 4.15 1.1 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 6.96 13.8 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F31 7.97 Tf 5.68 -1.64 Td[(1)]TJ/F15 10.91 Tf 4.73 1.64 Td[(\051)]TJ/F36 10.91 Tf 9.54 10.36 Td[(X)]TJ/F33 7.97 Tf -3.48 -24.16 Td[(d)]TJ/F17 5.98 Tf 4.36 -1.1 Td[(1)]TJ/F35 7.97 Tf 4.15 1.1 Td[(j)]TJ/F33 7.97 Tf 2.36 0 Td[(d)]TJ/F35 7.97 Tf 4.35 0 Td[(j)]TJ/F33 7.97 Tf 2.36 0 Td[(n)]TJ/F32 10.91 Tf 6.95 13.8 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 5.44 7.38 Td[(n)]TJ
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+-30.86 -23.2 6.55 0.44 re f
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+/F32 10.91 Tf 0 0 Td[(d)]TJ/F15 10.91 Tf 7.31 7.48 Td[(\051)-277(=)]TJ/F36 10.91 Tf 18.9 10.36 Td[(X)]TJ/F33 7.97 Tf -0.12 -24.16 Td[(d)]TJ/F17 5.98 Tf 4.36 -1.1 Td[(1)]TJ/F35 7.97 Tf 4.15 1.1 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(n)]TJ/F32 10.91 Tf 6.96 13.8 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(d)]TJ/F31 7.97 Tf 5.68 -1.64 Td[(1)]TJ/F15 10.91 Tf 4.73 1.64 Td[(\051)]TJ/F36 10.91 Tf 7.36 10.36 Td[(X)]TJ/F33 7.97 Tf -1.3 -24.16 Td[(d)]TJ/F17 5.98 Tf 4.36 -1.1 Td[(2)]TJ/F35 7.97 Tf 4.15 1.1 Td[(j)]TJ/F33 7.97 Tf 2.35 0 Td[(m)]TJ/F32 10.91 Tf 9.31 13.8 Td[()]TJ/F15 10.91 Tf 6.57 0 Td[(\050)]TJ/F32 10.91 Tf 5.86 7.38 Td[(m)]TJ
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+1 0 0 1 103.32 0 cm
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+/F32 10.91 Tf 0 0 Td[(d)]TJ/F31 7.97 Tf 5.68 -1.64 Td[(2)]TJ/F15 10.91 Tf 5.93 9.12 Td[(\051)]TJ/F32 10.91 Tf 4.24 0 Td[(;)]TJ/F15 10.91 Tf -363.22 -32.98 Td[(where)]TJ/F32 10.91 Tf 30.93 0 Td[(m)]TJ/F15 10.91 Tf 12.61 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(n=d)]TJ/F31 7.97 Tf 17.68 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 1.64 Td[(;)-333(d)]TJ/F31 7.97 Tf 12.34 -1.64 Td[(2)]TJ/F15 10.91 Tf 7.77 1.64 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[(d=d)]TJ/F31 7.97 Tf 16.81 -1.64 Td[(1)]TJ/F15 10.91 Tf 4.73 1.64 Td[(.)-426(By)-277(prop)-26(ert)28(y)-277(\0501\051,)-288(the)-277(second)-277(sum)-277(is)-278(non-zero)-277(only)-276(when)]TJ/F32 10.91 Tf 268.76 0 Td[(m)]TJ/F15 10.91 Tf 12.61 0 Td[(=)-277(1,)]TJ -412 -12.96 Td[(i.e.)]TJ/F32 10.91 Tf 18.79 0 Td[(d)]TJ/F31 7.97 Tf 5.68 -1.63 Td[(1)]TJ/F15 10.91 Tf 7.76 1.63 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(,)-333(and)-332(hence)-334(the)-333(whole)-333(expression)-333(equals)]TJ/F32 10.91 Tf 193.55 0 Td[(f)]TJ/F15 10.91 Tf 6.51 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051.)]TJ
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+76.71 -38.46 0.4 7.36 re f
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+77.11 -38.46 6.55 0.4 re f
+83.66 -38.46 0.4 7.36 re f
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Td[(0)]TJ/F15 10.91 Tf 14.69 1.64 Td[(if)]TJ/F32 10.91 Tf 10 0 Td[(n)]TJ/F15 10.91 Tf 9.58 0 Td[(=)-278(1)]TJ/F32 10.91 Tf -39.95 -12.95 Td[(a)]TJ/F33 7.97 Tf 5.77 -1.64 Td[(r)]TJ/F15 10.91 Tf 14.6 1.64 Td[(if)]TJ/F32 10.91 Tf 10 0 Td[(n)]TJ/F15 10.91 Tf 9.58 0 Td[(=)]TJ/F32 10.91 Tf 11.52 0 Td[(p)]TJ/F31 7.97 Tf 5.48 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.74 1.64 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(2)]TJ/F34 10.91 Tf 6.55 1.64 Td[()-166()-166()]TJ/F32 10.91 Tf 14.54 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(r)]TJ/F15 10.91 Tf -91.33 -11.32 Td[(0)-1145(if)]TJ/F32 10.91 Tf 27.94 0 Td[(n)]TJ/F34 10.91 Tf 8.98 0 Td[(�)]TJ/F15 10.91 Tf 10.9 0 Td[(not)-333(sq.free)]TJ/F32 10.91 Tf 49.79 0 Td[(:)]TJ/F47 10.91 Tf -286.47 -19.12 Td[(Solution)-485(11.)]TJ/F15 10.91 Tf 74.35 0 Td[(W)83(e)-422(ha)29(v)28(e)]TJ/F32 10.91 Tf 45.87 0 Td[(')]TJ/F15 10.91 Tf 7.13 0 Td[(\0502)]TJ/F33 7.97 Tf 9.7 3.96 Td[(n)]TJ/F31 7.97 Tf 5.14 0 Td[(+1)]TJ/F34 10.91 Tf 14.38 -3.96 Td[(�)]TJ/F15 10.91 Tf 11.55 0 Td[(1\051)-425(=)-424(2)]TJ/F33 7.97 Tf 32.9 3.96 Td[(n)]TJ/F15 10.91 Tf 5.64 -3.96 Td[(.)-709(If)-421(2)]TJ/F33 7.97 Tf 28.09 3.96 Td[(n)]TJ/F31 7.97 Tf 5.14 0 Td[(+1)]TJ/F34 10.91 Tf 14.38 -3.96 Td[(�)]TJ/F15 10.91 Tf 11.55 0 Td[(1)-425(=)-424(1,)-443(then)]TJ/F32 10.91 Tf 62.34 0 Td[(n)]TJ/F15 10.91 Tf 11.18 0 Td[(=)-425(0)-421(and)]TJ/F32 10.91 Tf 45.35 0 Td[(')]TJ/F15 10.91 Tf 7.13 0 Td[(\0501\051)-425(=)-424(1.)]TJ -391.82 -22.82 Td[(Otherwise,)-333(2)]TJ/F33 7.97 Tf 60.09 3.96 Td[(n)]TJ/F31 7.97 Tf 5.14 0 Td[(+1)]TJ/F34 10.91 Tf 13.74 -3.96 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1)-278(=)]TJ/F32 10.91 Tf 20 0 Td[(p)]TJ/F33 7.97 Tf 5.49 4.87 Td[( )]TJ/F17 5.98 Tf 5.44 -1.11 Td[(1)]TJ/F31 7.97 Tf -5.44 -6.88 Td[(1)]TJ/F32 10.91 Tf 10.09 3.12 Td[(p)]TJ/F33 7.97 Tf 5.49 4.87 Td[( )]TJ/F17 5.98 Tf 5.43 -1.11 Td[(2)]TJ/F31 7.97 Tf -5.43 -6.88 Td[(2)]TJ/F34 10.91 Tf 11.9 3.12 Td[()-167()-166()]TJ/F32 10.91 Tf 14.55 0 Td[(p)]TJ/F33 7.97 Tf 5.49 3.96 Td[( )]TJ/F19 5.98 Tf 5.43 -0.99 Td[(r)]TJ/F33 7.97 Tf -5.43 -5.66 Td[(r)]TJ/F32 10.91 Tf 13.12 2.69 Td[(>)]TJ/F15 10.91 Tf 11.52 0 Td[(1)-333(with)-333(all)]TJ/F32 10.91 Tf 49.09 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.63 Td[(i)]TJ/F15 10.91 Tf 3.38 1.63 Td[('s)-333(o)-28(dd.)-443(F)84(rom)-334(a)-333(previous)-332(form)28(ula,)]TJ/F32 10.91 Tf -202.57 -19.05 Td[(')]TJ/F15 10.91 Tf 7.13 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(p)]TJ/F33 7.97 Tf 5.49 4.86 Td[( )]TJ/F17 5.98 Tf 5.43 -1.1 Td[(1)]TJ/F31 7.97 Tf -5.43 -6.89 Td[(1)]TJ/F32 10.91 Tf 10.08 3.13 Td[(p)]TJ/F33 7.97 Tf 5.49 4.86 Td[( )]TJ/F17 5.98 Tf 5.44 -1.1 Td[(2)]TJ/F31 7.97 Tf -5.44 -6.89 Td[(2)]TJ/F34 10.91 Tf 11.91 3.13 Td[()-166()-167()]TJ/F32 10.91 Tf 14.54 0 Td[(p)]TJ/F33 7.97 Tf 5.49 4.5 Td[( )]TJ/F19 5.98 Tf 5.44 -0.99 Td[(r)]TJ/F33 7.97 Tf -5.44 -6.21 Td[(r)]TJ/F15 10.91 Tf 10.1 2.7 Td[(\051)-277(=)-278(2)]TJ/F33 7.97 Tf 24.24 4.5 Td[(n)]TJ/F15 10.91 Tf 8.67 -4.5 Td[(=)]TJ/F32 10.91 Tf 11.51 0 Td[(p)]TJ/F33 7.97 Tf 5.49 4.86 Td[( )]TJ/F17 5.98 Tf 5.44 -1.1 Td[(1)]TJ/F35 7.97 Tf 4.15 1.1 Td[(�)]TJ/F31 7.97 Tf 6.59 0 Td[(1)]TJ -16.18 -7.99 Td[(1)]TJ/F32 10.91 Tf 20.91 3.13 Td[(p)]TJ/F33 7.97 Tf 5.49 4.86 Td[( )]TJ/F17 5.98 Tf 5.44 -1.1 Td[(2)]TJ/F35 7.97 Tf 4.15 1.1 Td[(�)]TJ/F31 7.97 Tf 6.58 0 Td[(1)]TJ -16.17 -7.99 Td[(2)]TJ/F34 10.91 Tf 22.72 3.13 Td[()-167()-166()]TJ/F32 10.91 Tf 14.55 0 Td[(p)]TJ/F33 7.97 Tf 5.49 4.5 Td[( )]TJ/F19 5.98 Tf 5.44 -0.99 Td[(r)]TJ/F35 7.97 Tf 4.15 0.99 Td[(�)]TJ/F31 7.97 Tf 6.59 0 Td[(1)]TJ/F33 7.97 Tf -16.18 -7.2 Td[(r)]TJ/F15 10.91 Tf 20.91 2.7 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(p)]TJ/F31 7.97 Tf 5.48 -1.64 Td[(1)]TJ/F34 10.91 Tf 7.16 1.64 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051\050)]TJ/F32 10.91 Tf 13.94 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(2)]TJ/F34 10.91 Tf 7.15 1.64 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051)]TJ/F34 10.91 Tf 11.52 0 Td[()-166()-167()]TJ/F15 10.91 Tf 14.54 0 Td[(\050)]TJ/F32 10.91 Tf 4.25 0 Td[(p)]TJ/F33 7.97 Tf 5.48 -1.64 Td[(r)]TJ/F34 10.91 Tf 6.98 1.64 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051)]TJ/F32 10.91 Tf 9.7 0 Td[(:)]TJ/F15 10.91 Tf -386.05 -17.89 Td[(Therefore,)-478(all)]TJ/F32 10.91 Tf 70.49 0 Td[( )]TJ/F33 7.97 Tf 6.97 -1.63 Td[(i)]TJ/F15 10.91 Tf 8.53 1.63 Td[(=)-471(1)]TJ/F32 10.91 Tf 19.08 0 Td[(;)]TJ/F15 10.91 Tf 8.48 0 Td[(2)]TJ/F33 7.97 Tf 5.46 3.96 Td[(n)]TJ/F15 10.91 Tf 10.77 -3.96 Td[(=)-471(\050)]TJ/F32 10.91 Tf 17.87 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.63 Td[(1)]TJ/F34 10.91 Tf 8 1.63 Td[(�)]TJ/F15 10.91 Tf 11.76 0 Td[(1\051\050)]TJ/F32 10.91 Tf 13.94 0 Td[(p)]TJ/F31 7.97 Tf 5.48 -1.63 Td[(2)]TJ/F34 10.91 Tf 8 1.63 Td[(�)]TJ/F15 10.91 Tf 11.76 0 Td[(1\051)]TJ/F34 10.91 Tf 11.51 0 Td[()-167()-166()]TJ/F15 10.91 Tf 14.55 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.63 Td[(r)]TJ/F34 10.91 Tf 7.82 1.63 Td[(�)]TJ/F15 10.91 Tf 11.76 0 Td[(1\051,)-478(and)-448(all)]TJ/F32 10.91 Tf 56.84 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.63 Td[(i)]TJ/F15 10.91 Tf 8.52 1.63 Td[(=)-471(2)]TJ/F33 7.97 Tf 19.08 3.96 Td[(s)]TJ/F19 5.98 Tf 3.92 -1.21 Td[(i)]TJ/F15 10.91 Tf 6.93 -2.75 Td[(+)-299(1)-449(for)-449(some)]TJ/F32 10.91 Tf -368.23 -12.95 Td[(s)]TJ/F33 7.97 Tf 5.11 -1.64 Td[(i)]TJ/F34 10.91 Tf 6.84 1.64 Td[()]TJ/F15 10.91 Tf 11.93 0 Td[(1.)-514(It)-356(is)-356(easy)-357(to)-357(see)-357(that)-356(if)]TJ/F32 10.91 Tf 127.25 0 Td[(s)]TJ/F33 7.97 Tf 5.11 -1.64 Td[(i)]TJ/F15 10.91 Tf 7.27 1.64 Td[(has)-357(an)-355(o)-28(dd)-356(divisor)]TJ/F32 10.91 Tf 92.68 0 Td[(>)]TJ/F15 10.91 Tf 11.93 0 Td[(1,)-362(then)-356(2)]TJ/F33 7.97 Tf 43 3.96 Td[(s)]TJ/F19 5.98 Tf 3.91 -1.22 Td[(i)]TJ/F15 10.91 Tf 6.25 -2.74 Td[(+)-238(1)-356(will)-356(factor,)-361(making)]TJ/F32 10.91 Tf -321.28 -12.95 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(i)]TJ/F15 10.91 Tf 7.02 1.64 Td[(non-prime.)-443(Hence)]TJ/F32 10.91 Tf 88.81 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(i)]TJ/F15 10.91 Tf 6.41 1.64 Td[(=)-278(2)]TJ/F31 7.97 Tf 16.97 3.96 Td[(2)]TJ/F19 5.98 Tf 4.24 2.81 Td[(q)]TJ 3.41 -1.9 Td[(i)]TJ/F15 10.91 Tf 6.58 -4.87 Td[(+)-222(1)-333(for)-333(some)]TJ/F32 10.91 Tf 64.03 0 Td[(q)]TJ/F33 7.97 Tf 4.87 -1.64 Td[(i)]TJ/F34 10.91 Tf 6.41 1.64 Td[()]TJ/F15 10.91 Tf 11.52 0 Td[(0.)]TJ -219.3 -15.42 Td[(W)83(e)-315(ha)28(v)28(e)-315(reduced)-315(the)-314(problem)-314(to)-315(\014nding)-313(all)-314(sets)-316(of)-315(primes)]TJ/F32 10.91 Tf 277.47 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(i)]TJ/F15 10.91 Tf 6.82 1.64 Td[(of)-314(the)-315(ab)-27(o)28(v)28(e)-316(t)28(yp)-27(e)-315(suc)27(h)-314(that)]TJ/F32 10.91 Tf -212.63 -17.88 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(1)]TJ/F32 10.91 Tf 4.73 1.64 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(2)]TJ/F34 10.91 Tf 6.55 1.64 Td[()-167()-166()]TJ/F32 10.91 Tf 14.55 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(r)]TJ/F15 10.91 Tf 6.98 1.64 Td[(+)-222(1)-277(=)-278(2)]TJ/F33 7.97 Tf 36.36 4.5 Td[(n)]TJ/F31 7.97 Tf 5.14 0 Td[(+1)]TJ/F15 10.91 Tf 14.35 -4.5 Td[(=)-277(2\050)]TJ/F32 10.91 Tf 21.21 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(1)]TJ/F34 10.91 Tf 7.15 1.64 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051\050)]TJ/F32 10.91 Tf 13.94 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(2)]TJ/F34 10.91 Tf 7.16 1.64 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051)]TJ/F34 10.91 Tf 11.51 0 Td[()-167()-166()]TJ/F15 10.91 Tf 14.55 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(p)]TJ/F33 7.97 Tf 5.49 -1.64 Td[(r)]TJ/F34 10.91 Tf 6.98 1.64 Td[(�)]TJ/F15 10.91 Tf 10.91 0 Td[(1\051)]TJ/F32 10.91 Tf 9.69 0 Td[(:)]TJ/F15 10.91 Tf -339.86 -17.89 Td[(Order)-350(the)-352(primes)]TJ/F32 10.91 Tf 86.2 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(1)]TJ/F32 10.91 Tf 8.1 1.64 Td[(<)-307(p)]TJ/F31 7.97 Tf 17.33 -1.64 Td[(2)]TJ/F32 10.91 Tf 8.1 1.64 Td[(<)]TJ/F34 10.91 Tf 11.84 0 Td[()-167()-166()]TJ/F32 10.91 Tf 16.09 0 Td[(<)-308(p)]TJ/F33 7.97 Tf 17.34 -1.64 Td[(r)]TJ/F15 10.91 Tf 4.55 1.64 Td[(.)-499(Chec)27(k)-351(consecutiv)28(ely)-352(via)-351(mo)-28(dulo)-350(appropriate)-349(p)-28(o)28(w)28(ers)]TJ -175.04 -12.95 Td[(of)-366(2,)-375(that,)-375(if)-366(they)-366(exist,)-376(the)-366(smallest)-368(primes)-367(m)27(ust)-366(b)-28(e:)]TJ/F32 10.91 Tf 261.37 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(1)]TJ/F15 10.91 Tf 8.38 1.64 Td[(=)-333(3)]TJ/F32 10.91 Tf 17.58 0 Td[(;)-500(p)]TJ/F31 7.97 Tf 13.97 -1.64 Td[(2)]TJ/F15 10.91 Tf 8.38 1.64 Td[(=)-333(5)]TJ/F32 10.91 Tf 17.58 0 Td[(;)-499(p)]TJ/F31 7.97 Tf 13.97 -1.64 Td[(3)]TJ/F15 10.91 Tf 8.37 1.64 Td[(=)-334(17)]TJ/F32 10.91 Tf 23.04 0 Td[(;)-500(p)]TJ/F31 7.97 Tf 13.97 -1.64 Td[(4)]TJ/F15 10.91 Tf 8.38 1.64 Td[(=)-333(257.)]TJ -400.48 -13.74 Td[(Ho)28(w)28(ev)27(er,)-498(if)]TJ/F32 10.91 Tf 60.25 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(5)]TJ/F15 10.91 Tf 9.81 1.64 Td[(also)-466(exists,)-499(then)]TJ/F32 10.91 Tf 84.57 0 Td[(p)]TJ/F31 7.97 Tf 5.49 -1.64 Td[(5)]TJ/F15 10.91 Tf 10.17 1.64 Td[(=)-498(2)]TJ/F31 7.97 Tf 19.38 3.96 Td[(2)]TJ/F17 5.98 Tf 4.23 2.81 Td[(4)]TJ/F15 10.91 Tf 8.04 -6.77 Td[(+)-310(1,)-499(whic)28(h)-465(is)-466(not)-465(prime.)-841(Hence,)]TJ/F32 10.91 Tf 168.48 0 Td[(r)]TJ/F34 10.91 Tf 10.66 0 Td[()]TJ/F15 10.91 Tf 13.93 0 Td[(4,)-498(and)]TJ/F32 10.91 Tf -400.5 -12.95 Td[(n)]TJ/F15 10.91 Tf 9.58 0 Td[(=)-277(1)]TJ/F32 10.91 Tf 16.97 0 Td[(;)]TJ/F15 10.91 Tf 4.85 0 Td[(2)]TJ/F32 10.91 Tf 5.45 0 Td[(;)]TJ/F15 10.91 Tf 4.85 0 Td[(3)]TJ/F32 10.91 Tf 5.45 0 Td[(;)]TJ/F15 10.91 Tf 4.85 0 Td[(4.)]TJ
+ET
+41.49 -340.32 0.4 7.36 re f
+41.89 -333.36 6.55 0.4 re f
+41.89 -340.32 6.55 0.4 re f
+48.44 -340.32 0.4 7.36 re f
+1 0 0 1 -382.59 -363.14 cm
+BT
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Td[(\051)-437(=)-438(ln)]TJ/F32 10.91 Tf 33.18 0 Td[(p)]TJ/F15 10.91 Tf 10.18 0 Td[(if)]TJ/F32 10.91 Tf 11.04 0 Td[(n)]TJ/F15 10.91 Tf 11.23 0 Td[(is)-430(a)-429(p)-27(o)28(w)27(er)-429(of)-429(a)-429(prime)]TJ/F32 10.91 Tf 110.64 0 Td[(p)]TJ/F15 10.91 Tf 5.49 0 Td[(,)-453(and)]TJ/F34 10.91 Tf 30.24 0 Td[(^)]TJ/F15 10.91 Tf 7.27 0 Td[(\050)]TJ/F32 10.91 Tf 4.24 0 Td[(n)]TJ/F15 10.91 Tf 6.55 0 Td[(\051)-437(=)-438(0)-429(otherwise,)-453(and)-428(use)-430(M)501(obius)]TJ -269.61 -12.95 Td[(in)29(v)28(ersion.)]TJ/F47 10.91 Tf 0 -22.82 Td[(Solution)-384(16.)]TJ/F15 10.91 Tf 70.36 0 Td[(W)83(e)-333(ha)28(v)28(e)]TJ/F32 10.91 Tf 43.94 0 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\0501\051)-277(=)-278(1)-333(and)]TJ/F32 10.91 Tf 58.78 0 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\0502\051)-277(=)-278(2.)-444(Let)]TJ/F32 10.91 Tf 61.36 0 Td[(f)]TJ/F15 10.91 Tf 6.52 0 Td[(\0503\051)-277(=)-278(3)-222(+)]TJ/F32 10.91 Tf 47.27 0 Td[(m)]TJ/F15 10.91 Tf 13.22 0 Td[(for)]TJ/F32 10.91 Tf 16.69 0 Td[(m)]TJ/F34 10.91 Tf 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<li><a href="lectures/Henselfaktorisering.html">Henselfaktorisering</a></li>
<li><a href="senaste-nytt.html">senaste-nytt</a></li>
<li><a href="labs/lecture5.html">lecture5</a></li>
@@ -217,13 +216,14 @@
<li><a href="labs/lecture2.html">lecture2</a></li>
<li><a href="labs/lecture1.html">lecture1</a></li>
<li><a href="labs/lecture0.html">lecture0</a></li>
+<li><a href="index.html">TATA54 Talteori</a></li>
<li><a href="labs/kedjebraklab.html">Sagemath-övningar på kedjebråk</a></li>
<li><a href="labs/HenselLyftLabHT2023.html">SageMath-övningar på Hensellyft, primitiva rötter, och KRS</a></li>
</ul>
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<p class="author">Author: Jan Snellman</p>
-<p class="date">Created: 2024-05-13</p>
+<p class="date">Created: 2024-05-16</p>
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