September 28, 2021 An Introduction to Number Theory J. J. P. Veerman List of Figures p 1 Eratosthenes’ sieve up to n = 30. All multiples of a less than 31 are cancelled. The remainder are the primes less than n = 31. 6 2 A directed path g passing through all points of Z2. 14 R x 3 On the left, the function 2 lnt dt in blue, p(x) in red, and R x x=lnx in green. On the right, we have 2 lnt dt − x=lnx in blue, p(x) − x=lnx in red. 31 ¥ R ¥ −s 4 Proof that ∑n=1 f (n) is greater than 1 x dx if f is positive and decreasing. 36 ¥ 5 Proof that ∑n=1 f (n) (shaded in blue and green) minus f (1) R ¥ −s (shaded in blue) is less than 1 x dx if f is positive and decreasing to 0. 39 6 The origin is marked by “×”. The red dots are visible from ×; between any blue dot and × there is a red dot. The picture shows exactly one quarter of {−4;··· ;4g2n(0;0) ⊂ Z2. 40 7 The general solution of the inhomogeneous equation (~r;~x) = c in R2. 47 8 A one parameter family ft of maps from the circle to itself. For every t 2 [0;1] the map ft is constructed by truncating the map x ! 2x mod 1 as indicated in this figure. 67 3 4 List of Figures 9 Three branches of the Gauss map. 92 10 The line y = wx and (in red) successive iterates of the rotation Rw . Closest returns in this figure are q in f2;3;5;8g. 101 11 The geometry of successive closest returns. 101 12 Drawing y = w1x and successive approximations (an+1 is taken to be 3). The green arrows correspond to en−1, en, and en+1. 102 13 Black: thread from origin with golden mean slope; red: pulling the thread down from the origin; green: pulling the thread up from the origin. 105 14 Plots of the points (n;n) in polar coordinates, for n ranging from 1 to 50, 180, 330, and 3000, respectively. 108 15 Plots of the prime points (p; p) (p prime) in polar coordinates with p ranging between 2 and 3000, and between 2 and 30000, respectively. 109 16 The Gaussian integers are the lattice points in the complex plane; both real and imaginary parts are integers. For an arbitrary point z 2 C — marked by x in the figure, a nearby integer is k1 + ik2 where k1 is the closest integer to Re(z) and k2 the closest integer to Im(z). In this case that is 2 + 3i. 124 p 17 A depiction of Z[ −6] in the complex plane;p real parts are integers and imaginary parts are multiples of 6. 126 18 The part to the right of the intersection with ` : y = x + 1 (dashed) of bad path (in red) is reflected. The reflected part in indicated in green. The path becomes a monotone path from (0;0) to (n − 1;n + 1). 127 19 The Gaussian primes described in Proposition 7.33. There are approximately 950 within a radius 40 of the origin (left figure) and about 3300 within a radius 80 (right figure). 130 20 Possible values of rg−1 in the proof of Proposition 7.27. 132 21 A comparison between approximating the Lebesgue integral (left) and the Riemann integral (right). 140 22 The pushforward of a measure n. 141 − + 23 The functions m(Xc ) and m(Xc ). 142 List of Figures 5 24 This map is not uniquely ergodic 144 25 The first two stages of the construction of the singular measure np. The shaded parts are taken out. 147 26 The first two stages of the construction of the middle third Cantor set. 150 27 The inverse image of a small interval dy is T −1(dy) 156 p 28 w is irrational and q is a convergent of w. Then x + qw modulo 1 is close to x. Thus adding qw modulo 1 amounts to a translation by a small distance. Note: “om” in the figure stands for w. 159 1 1 29 `(I) is between 3 and 2 of `(J). So there are two disjoint images −1 of I under Rw that fall in J. 160 30 An example of the system described in Corollary 9.9. 162 31 Illustration of the fact that for a concave function f , we have f (wx + (1 − w)y) ≥ w f (x) + (1 − w) f (y) (Jensen’s inequality). 170 32 Plot of the function ln(x)ln(1 + x) 171 33 Left, a curve. Then two simple, closed curves with opposite orientation. The curve on the right is a union of two simple, closed curves. 177 34 A singular point z0 where f is bounded does not contribute to H g f dz. 178 35 The curve g goes around z exactly once in counter-clockwise direction. If d is small enough, z + d also lies inside g. 181 36 The curve w goes around z0 exactly once in counter-clockwise direction. 183 37 g is analytic in DR := fRez ≥ −dRg \ fjzj ≤ Rg (shaded). The red is p p curve is given by C+(s) = Re with s 2 (− 2 ; 2 ). The green curve is p 3p is given by C+(s) = Re with s 2 ( 2 ; 2 ). The blue L− consists of 2 small circular segments plus the segment connecting their left endpoints at a distance d to the left of the the imaginary axis. 184 38 The complex plane with eit , −e−it and e−it on the unit circle. cost is the average of eit and e−it and isint as the average of eit and −e−it . 188 6 List of Figures 39 The functions gi and hi of exercise 10.20 for i 2 f2;8;15;30g. 192 40 The contour C is the concatenation of c (celeste), b1 (blue), r1 (red), g (green), p (purple), −g, r2, and b2. The path r is a semi-circle of radius R. The path p is a small circle of radius r. See exercise 10.21. 192 41 Integration over the shaded triangle of area 1=2 in equation 11.9. 201 42 The functions q(x)=x (green), y(x)=x (red), and p(x)lnx=x (blue) for x 2 [1;1000]. All converge to 1 as x tends to infinity. The x-axis is horizontal. 210 43 The two characters modulo 3 illustrate the orthogonality of the Dirichlet characters. 220 44 The set S consists of the natural numbers contained in intervals shaded in the top figure of the form [22n−1;22n). The bottom picture is the same but with a logarithmic horizontal scale. 231 R 1 k 45 Proof that 0 f (x)dx is between ∑ j=1 f ( j dx) and f (0) − f (1) + k ∑ j=1 f ( j dx) if f is decreasing. 232 46 The function ln(ln(x)) for x 2 [1;1040]. 234 List of Tables 7 Contents List of Figures 3 List of Tables 7 Preface 1 Part 1. Introduction to Number Theory Chapter 1. A Quick Tour of Number Theory 5 §1.1. Divisors and Congruences 6 §1.2. Rational and Irrational Numbers 7 §1.3. Algebraic and Transcendental Numbers 9 §1.4. Countable and Uncountable Sets 12 §1.5. Exercises 15 Chapter 2. The Fundamental Theorem of Arithmetic 21 §2.1. Bezout’s´ Lemma 22 §2.2. Corollaries of Bezout’s´ Lemma 23 §2.3. The Fundamental Theorem of Arithmetic 25 §2.4. Corollaries of the Fundamental Theorem of Arithmetic 27 §2.5. The Riemann Hypothesis 29 §2.6. Exercises 33 9 10 Contents Chapter 3. Linear Diophantine Equations 41 §3.1. The Euclidean Algorithm 41 §3.2. A Particular Solution of ax + by = c 43 §3.3. Solution of the Homogeneous equation ax + by = 0 45 §3.4. The General Solution of ax + by = c 46 §3.5. Recursive Solution of x and y in the Diophantine Equation 47 §3.6. The Chinese Remainder Theorem 48 §3.7. Exercises 50 Chapter 4. Number Theoretic Functions 57 §4.1. Multiplicative Functions 57 §4.2. Additive Functions 60 §4.3. Mobius¨ inversion 60 §4.4. Euler’s Phi or Totient Function 62 §4.5. Dirichlet and Lambert Series 64 §4.6. Exercises 67 Chapter 5. Modular Arithmetic and Primes 75 §5.1. Modular Arithmetic 75 §5.2. Euler’s Theorem and Primitive Roots 76 §5.3. Fermat’s Little Theorem and Primality Testing 79 §5.4. Fermat and Mersenne Primes 81 §5.5. Division in Zb 84 §5.6. Exercises 86 Chapter 6. Continued Fractions 91 §6.1. The Gauss Map 91 §6.2. Continued Fractions 92 §6.3. Computing with Continued Fractions 97 §6.4. The Geometric Theory of Continued Fractions 99 §6.5. Closest Returns 101 §6.6. Exercises 103 Contents 11 Part 2. Currents in Number Theory: Algebraic, Probabilistic, and Analytic Chapter 7. Algebraic Integers 113 §7.1. Rings and Fields 113 §7.2. Primes and Integral Domains 115 §7.3. Norms 118 §7.4. Euclidean Domains 121 §7.5. Example and Counter-Example 123 §7.6. Exercises 126 Chapter 8. Ergodic Theory 135 §8.1. The Trouble with Measure Theory 135 §8.2. Measure and Integration 137 §8.3. The Birkhoff Ergodic Theorem 141 §8.4. Examples of Ergodic Measures 144 §8.5. The Lebesgue Decomposition 146 §8.6. Exercises 148 Chapter 9. Three Maps and the Real Numbers 155 §9.1.