A PRIMER OF ANALYTIC NUMBER THEORY: From Pythagoras to ...

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166 7. Euler’s Product the error E(x) = � So, � p
7.1 Mertens’ Theorems 167 Notice that the theorem implies that � (1 − 1/p) ∼ C log(x) . p
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A PRIMER OF ANALYTIC NUMBER THEORY
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This book is dedicated to all the f
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Preface Good evening. Now, I’m no
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Preface xi Analysis. There are quit
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Preface xiii doing the exercises. M
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2 1. Sums and Differences Figure 1.
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10 1. Sums and Differences Table 1.
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12 1. Sums and Differences square n
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14 1. Sums and Differences One can
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16 1. Sums and Differences Exercise
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18 1. Sums and Differences write (1
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20 1. Sums and Differences imitatin
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22 1. Sums and Differences If we ta
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Chapter 2 Products and Divisibility
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26 2. Products and Divisibility Tab
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28 2. Products and Divisibility Iam
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30 2. Products and Divisibility in
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32 2. Products and Divisibility s(n
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34 2. Products and Divisibility Tab
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36 2. Products and Divisibility bef
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38 2. Products and Divisibility but
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40 2. Products and Divisibility So,
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42 2. Products and Divisibility Exe
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44 3. Order of Magnitude 15 12.5 10
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46 3. Order of Magnitude Exercise 3
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48 3. Order of Magnitude Figure 3.2
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50 3. Order of Magnitude Figure 3.3
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52 3. Order of Magnitude Table 3.2.
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54 3. Order of Magnitude We get 0
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56 3. Order of Magnitude The second
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58 3. Order of Magnitude as n = �
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60 3. Order of Magnitude Exercise 3
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62 3. Order of Magnitude Lemma. �
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Chapter 4 Averages So far, we’ve
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66 4. Averages 4.1. Divisor Average
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68 4. Averages n values, N(1) + N(2
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70 4. Averages error by making the
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72 4. Averages 0.5 0.4 0.3 0.2 0.1
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74 4. Averages Exercise 4.2.3. By v
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76 4. Averages The √ n different
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78 4. Averages 1 1 4 1 9 1 n^2 1 2
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80 4. Averages adding up the “y-c
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82 4. Averages 30 25 20 15 10 5 20
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84 I1. Calculus Exercise I1.1.1. Of
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86 I1. Calculus From the definition
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88 I1. Calculus then f (x) is diffe
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90 I1. Calculus a b Figure I1.3. An
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92 I1. Calculus F(x) f(x) Figure I1
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94 I1. Calculus Exercise I1.3.3. Co
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Chapter 5 Primes 5.1. A Probability
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98 5. Primes 1 0.8 0.6 0.4 0.2 2 3
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100 5. Primes Table 5.1. Primes nea
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102 5. Primes We can now make a pre
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104 5. Primes Table 5.2. Numerical
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106 5. Primes Proof. We start with
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108 5. Primes So, 2n + 1 �(2n + 1
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110 5. Primes Finally, since log(n
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112 I2. Series 4 3 2 1 0.2 0.4 0.6
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114 I2. Series We can get a polynom
Page 132 and 133: 116 I2. Series and the Taylor serie
Page 134 and 135: 118 I2. Series Every term shifts do
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Page 138 and 139: 122 I2. Series I2.3. Laurent Series
Page 140 and 141: 124 I2. Series So, we get the Laure
Page 142 and 143: 126 I2. Series Because 1 − x n+1
Page 144 and 145: 128 I2. Series clever. Oresme obser
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Page 148 and 149: 132 I2. Series 1. Show by induction
Page 150 and 151: 134 I2. Series Exercise I2.6.6. Use
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Page 154 and 155: 138 I2. Series Show that the n + 1s
Page 156 and 157: 140 I2. Series Basically, the integ
Page 158 and 159: 142 I2. Series Use v(m) = 0 and fac
Page 160 and 161: 144 I2. Series This implies that
Page 162 and 163: Chapter 6 Basel Problem It was disc
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Page 168 and 169: 152 6. Basel Problem hand, for each
Page 170 and 171: 154 6. Basel Problem Theorem. �(2
Page 172 and 173: 156 6. Basel Problem sum. These are
Page 174 and 175: 158 6. Basel Problem so f = e C ·
Page 176 and 177: 160 7. Euler’s Product This subtr
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Page 180 and 181: 164 7. Euler’s Product Lemma. Pro
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Page 202 and 203: 186 7. Euler’s Product We can fin
Page 204 and 205: 188 I3. Complex Numbers Exercise I3
Page 206 and 207: 190 I3. Complex Numbers using the s
Page 208 and 209: 192 I3. Complex Numbers 2 1 0 -1 -2
Page 210 and 211: 194 8. The Riemann Zeta Function Of
Page 212 and 213: 196 8. The Riemann Zeta Function Ac
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Page 216 and 217: 200 8. The Riemann Zeta Function 2
Page 218 and 219: 202 8. The Riemann Zeta Function Eu
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Page 228 and 229: 212 8. The Riemann Zeta Function Wh
Page 230 and 231: 214 8. The Riemann Zeta Function -1
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Chapter 9 Symmetry When the stars t
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218 9. Symmetry s 10 -1 -0.5 0.5 1
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220 9. Symmetry Exercise 9.1.3. The
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222 9. Symmetry according to the Bi
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224 9. Symmetry So we’re half don
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226 9. Symmetry Suppose now that Re
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228 9. Symmetry and so, f (t) = �
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230 10. Explicit Formula It was Rie
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232 10. Explicit Formula (Of course
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234 10. Explicit Formula total �(
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236 10. Explicit Formula You showed
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238 10. Explicit Formula 0.4 0.2 -0
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240 10. Explicit Formula Exercise 1
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242 10. Explicit Formula Formula (1
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244 10. Explicit Formula Theorem (R
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246 10. Explicit Formula actually m
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248 10. Explicit Formula by Möbius
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250 10. Explicit Formula So, when w
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252 10. Explicit Formula 10 8 6 4 2
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Interlude 4 Modular Arithmetic Trad
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256 I4. Modular Arithmetic The poin
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258 I4. Modular Arithmetic Theorem
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Chapter 11 Pell’s Equation In Cha
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262 11. Pell’s Equation solution,
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264 11. Pell’s Equation Proof. We
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266 11. Pell’s Equation 2 2 ≡ 1
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268 11. Pell’s Equation The funct
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270 11. Pell’s Equation Proof in
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272 11. Pell’s Equation Then, for
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Chapter 12 Elliptic Curves In the p
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276 12. Elliptic Curves in geometry
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278 12. Elliptic Curves Diophantus
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280 12. Elliptic Curves is just a h
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282 12. Elliptic Curves 12.2.2. The
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284 12. Elliptic Curves His exposit
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286 12. Elliptic Curves 12.2.5. Eul
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288 12. Elliptic Curves Table 12.1.
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290 12. Elliptic Curves For primes
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292 12. Elliptic Curves for �(s)
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294 12. Elliptic Curves congruent n
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296 13. Analytic Theory of Algebrai
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328 Solutions (1.1.4) For the n = 1
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330 Solutions According to the Fund
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332 Solutions (1.2.14) The function
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334 Solutions Subtraction gives 1 3
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336 Solutions The Catalan-Dickson c
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338 Solutions (c) Again, it is easi
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340 Solutions This is weaker than w
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342 Solutions (4.0.3) Divide both s
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344 Solutions (4.3.4) Using the giv
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346 Solutions t = 1/17, y =−1/17
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348 Solutions (I2.1.5) (I2.1.6) (I2
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350 Solutions (I2.5.2) The estimate
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352 Solutions Because this is just
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354 Solutions If log 10 (x) > 10 9
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356 Solutions (8.2.3) Observe that
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358 Solutions (8.4.2) For x > 0, th
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360 Solutions (10.2.1) With x = 12.
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362 Solutions 17.5 15 12.5 10 7.5 2
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364 Solutions Export["primemusic.ai
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366 Solutions (I4.0.10) We have m =
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368 Solutions (12.1.1) 1. The line
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370 Solutions way to do this. bsd[x
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372 Solutions Table S.1. Examples o
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374 Solutions get that the sum abov
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376 Bibliography Gregorovius, F. (1
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#, number of elements in a set, 101
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asymptotics, 64 bounds, 47-52 defin
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Taylor polynomial definition, 114 e
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