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

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A PRIMER OF ANALYTIC NUMBER THEORY From Pythagoras to Riemann JEFFREY STOPPLE University of California, Santa Barbara
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Page 6 and 7: cambridge university press Cambridg
Page 9: Contents Preface page ix Chapter 1.
Page 12 and 13: x Preface This number, 216, is the
Page 14 and 15: xii Preface introducing analysis, o
Page 17 and 18: Chapter 1 Sums and Differences Imet
Page 19 and 20: 1.1 Polygonal Numbers 3 Figure 1.3.
Page 21 and 22: 1.1 Polygonal Numbers 5 Exercise 1.
Page 23 and 24: 1.1 Polygonal Numbers 7 We will do
Page 25 and 26: The formulas 1.1 Polygonal Numbers
Page 27 and 28: 1.2 The Finite Calculus 11 1.2. The
Page 29 and 30: 1.2 The Finite Calculus 13 The diff
Page 31 and 32: 1.2 The Finite Calculus 15 Theorem
Page 33 and 34: 1.2 The Finite Calculus 17 The firs
Page 35 and 36: 1.2 The Finite Calculus 19 no power
Page 37 and 38: 1.2 The Finite Calculus 21 geometry
Page 39 and 40: 1.2 The Finite Calculus 23 Suppose
Page 41 and 42: 2.1 Conjectures 25 This section con
Page 43 and 44: p k1 1 pk2 2 ...pkm m and , then 2.
Page 45 and 46: 2.1 Conjectures 29 Ages; Boethius d
Page 47 and 48: 2.1 Conjectures 31 The primes 3, 7,
Page 49 and 50: Ibn Khaldun wrote in Muqaddimah tha
Page 51 and 52: 2.2 Theorems 35 was overlooked unti
Page 53 and 54: 2.2 Theorems 37 left side, starting
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2.3 Structure 39 a prime. But, a fe
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2.3 Structure 41 out one of the pri
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Chapter 3 Order of Magnitude In his
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The constant C = 8 will work as 3.1
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Exercise 3.1.4. Show that as x →
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3.2 Harmonic Numbers 49 Proof. This
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3.2 Harmonic Numbers 51 Table 3.1.
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3.3 Factorials 53 Lemma. log(n!) =
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3.4 Estimates for Sums of Divisors
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3.5 Estimates for the Number of Div
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3.6 Very Abundant Numbers 59 3.6. V
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3.7 Highly Composite Numbers 61 Tha
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6 15 6 16 3.7 Highly Composite Numb
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30 25 20 15 10 5 4. Averages 65 3 4
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4.1 Divisor Averages 67 Observe tha
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n 3 2 1 1 2 3 n 4.1 Divisor Average
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7 6 5 4 4.2 Improved Estimate 71 4
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Exercise 4.2.1. We know that 4.2 Im
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d 4.2 Improved Estimate 75 y n x n
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4.3 Second-Order Harmonic Numbers 7
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4.4 Averages of Sums 79 Table 4.1.
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n log(n)/n 2 = log(n)/n → 0asn
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Interlude 1 Calculus The techniques
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I1.2 More on Landau’s Big Oh 85 2
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I1.2 More on Landau’s Big Oh 87 E
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c (i) (iii) I1.3 Fundamental Theore
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I1.3 Fundamental Theorem of Calculu
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5.1 A Probability Argument 97 getti
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5.1 A Probability Argument 99 sum o
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1 0.8 0.6 0.4 0.2 5.1 A Probability
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15 10 5 80 60 40 20 600 400 200 5.1
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5.2 Chebyshev’s Estimates 105 Exe
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5.2 Chebyshev’s Estimates 107 Bec
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Lemma. For any real numbers a and b
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Interlude 2 Series In the previous
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1 0.8 0.6 0.4 0.2 -0.2 I2.1 Taylor
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I2.2 Taylor Series 115 In fact, the
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I2.2 Taylor Series 117 for x 2 exp(
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I2.2 Taylor Series 119 1.5 1 0.5 -3
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I2.2 Taylor Series 121 Exercise I2.
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I2.3 Laurent Series 123 many terms
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I2.4 Geometric Series 125 can be de
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I2.5 Harmonic Series 127 By the fou
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I2.6 Convergence 129 I2.6. Converge
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I2.6 Convergence 131 For another ex
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I2.6 Convergence 133 and say that t
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I2.6 Convergence 135 In the case wh
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I2.6 Convergence 137 The theorem gi
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I2.6 Convergence 139 a sequence whi
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f (x) = � ∞ k=0 akx k . Then,
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I2.7 Abel’s Theorem 143 According
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I2.7 Abel’s Theorem 145 1,.... Th
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6.1 Euler’s First Proof 147 6.1.
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So, the coefficient of x 4 is also
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6.2 Bernoulli Numbers 151 and looks
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6.3 Euler’s General Proof 153 whe
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6.3 Euler’s General Proof 155 Pro
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that as n goes to infinity, 6.3 Eul
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Chapter 7 Euler’s Product Euler
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7. Euler’s Product 161 �(x) is,
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Lemma. p
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7.1 Mertens’ Theorems 165 Notice
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7.1 Mertens’ Theorems 167 Notice
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7.1 Mertens’ Theorems 169 Proof.
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Lemma. A(�) = � n
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7.1 Mertens’ Theorems 173 Proof i
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7.2 Colossally Abundant Numbers 175
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7.2 Colossally Abundant Numbers 177
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Notice that 7.3 Sophie Germain Prim
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7.3 Sophie Germain Primes 181 with
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40 30 20 10 1200 1000 800 600 400 2
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15 10 5 7.4 Wagstaff’s Conjecture
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Interlude 3 Complex Numbers Unless
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I3. Complex Numbers 189 Here’s so
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2 1 0 -1 I3. Complex Numbers 191 -2
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Chapter 8 The Riemann Zeta Function
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8.1 The Critical Strip 195 8.1. The
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3 2 1 -1 -2 8.1 The Critical Strip
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8.2 Summing Divergent Series 199 Le
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8.2 Summing Divergent Series 201 Eu
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8.2 Summing Divergent Series 203 Th
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8.3 The Gamma Function 205 sum of a
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So, � − 1 2 � 8.3 The Gamma F
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8.4 Analytic Continuation 209 Exerc
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Now, multiply in the x s−1 term t
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8.4 Analytic Continuation 213 For t
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8.4 Analytic Continuation 215 Exerc
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9.1. Stirling’s Formula In Sectio
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9.1 Stirling’s Formula 219 is ver
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9.2 The Transformation Formula 221
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Divide both sides of (9.4) by 2 2m
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For (9.7), observe that m2m (m2 −
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and t = 0 corresponds to � =∞.W
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Chapter 10 Explicit Formula 10.1. H
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We see that 10.1 Hadamard’s Produ
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15 10 5 80 60 40 20 500 400 300 200
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10.2 Von Mangoldt’s Formula 235 f
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10.2 Von Mangoldt’s Formula 237 z
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10.3 The Logarithmic Integral 239 x
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to the series for exp(y). So, 10.3
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10 8 6 4 2 10.4 Riemann’s Formula
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10.4 Riemann’s Formula 245 Instea
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We need to check that � ∞ x dt
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Theorem. The Prime Number Theorem i
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10 8 6 4 2 10.4 Riemann’s Formula
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10.4 Riemann’s Formula 253 Re(s)
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I4. Modular Arithmetic 255 modulo 6
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I4. Modular Arithmetic 257 Lemma. S
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I4. Modular Arithmetic 259 The next
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11.1 The Cattle Problem of Archimed
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11.3 Solutions Modulo a Prime 263 f
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11.3 Solutions Modulo a Prime 265 a
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in s = 1, we get L(1, �12) = �
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11.4 Dirichlet L-Functions 269 This
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11.4 Dirichlet L-Functions 271 Exer
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Exercise 11.4.6. Show that for |x|
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P Q 12.1 Group Structure 275 Figure
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12.2 Diophantine Equations 277 For
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12.2 Diophantine Equations 279 them
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12.2 Diophantine Equations 281 in m
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12.2 Diophantine Equations 283 The
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12.2 Diophantine Equations 285 12.2
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12.3 Solutions Modulo a Prime 287 1
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12.3 Solutions Modulo a Prime 289 I
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12.4 Birch and Swinnerton-Dyer 291
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12.4 Birch and Swinnerton-Dyer 293
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Chapter 13 Analytic Theory of Algeb
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13.1 Binary Quadratic Forms 297 whe
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13.1 Binary Quadratic Forms 299 the
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13.1 Binary Quadratic Forms 301 We
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13.1 Binary Quadratic Forms 303 of
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13.1 Binary Quadratic Forms 305 The
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13.2 Analytic Class Number Formula
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13.3 Siegel Zeros and the Class Num
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Solutions (1.1.1) One possible solu
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(1.2.2) (1.2.3) with solution C = 1
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After some messy algebra, we get So
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(2.1.8) 1. 2. 3. Solutions 333 �(
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Solutions 335 4, 5, 6, 8, and 9 are
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(2.2.1) The sum of the row entries
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Solutions 339 So, it suffices to sh
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(3.6.3) Solutions 341 n = N! = 4032
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(4.2.2) (4.2.3) (4.3.1) H (2) 1 [t]
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Solutions 345 So, for some constant
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Solutions 347 (5.1.2) The smallest
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(I2.2.7) (I2.2.8) (I2.2.9) (I2.2.10
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Solutions 351 and � 1 1 = arctan(
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(7.1.2) According to Exercise 7.1.1
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So, Solutions 355 (1 − 2 1−s )
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Solutions 357 (8.3.3) According to
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(9.1.2) � ∞ � ∞ −∞ −
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Solutions 361 The following defines
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Solutions 363 gammas = {14.13472, 2
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Solutions 365 + 0 1 2 3 4 5 0 0 1 2
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with Solutions 367 u = x 2 + √ 2x
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Solutions 369 Now, 5 · 4 > 31/2. S
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(13.1.6) Solutions 371 F ={12, 11,
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(13.2.1) Take f−3(x) = Solutions
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Bibliography Recreational-Expositor
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Bibliography 377 Hardy, G. H. and L
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380 Index continuity in terms of
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382 Index �(x) definition, 242 Ri
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