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Dirichlet L-Functions and Dedekind Ζ-Functions

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Dirichlet L-Functions and Dedekind Ζ-Functions DIRICHLET L-FUNCTIONS AND DEDEKIND ζ-FUNCTIONS FRIMPONG A. BAIDOO Abstract. We begin by introducing Dirichlet L-functions which we use to prove Dirichlet's theorem on arithmetic progressions. From there, we discuss algebraic number fields and introduce the tools needed to define the Dedekind zeta function. We then use it to prove the class number formula for imaginary quadratic fields. Contents 1. Introduction1 2. The Riemann Zeta Function2 3. Dirichlet Characters4 4. Dirichlet L-functions6 5. Dirichlet's Theorem on Arithmetic Progressions8 6. Algebraic Number Fields9 7. The Ring of Integers 10 8. Ideals of the Ring of Integers 11 9. The Dedekind Zeta Function and The Class Number Formula 13 10. Imaginary Quadratic Fields 16 Acknowledgements 20 References 20 1. Introduction L-functions are a class of functions that generalize the Riemann zeta function. In this paper, we will look at two kinds of L-functions, namely, Dirichlet L-functions and Dedekind zeta functions. The manner in which we will exposit on these two types of L-functions will essentially be the same. First, we provide preliminary information necessary to make sense of the L-function. After this, we define the L-function and then apply it to a problem in number theory to show how the L-function reveals arithmetic information. In section 2, we introduce the Riemann zeta function, the prototype of all L- functions, study its pole and, in the process, prove Euclid's theorem that there are infinitely many prime numbers. In Sections 3 and 4, we aim to define the Dirichlet L-function. In section 5, we give a proof of Dirichlet's theorem on arithmetic progressions, which states that for two coprime positive integers a and n, there will be infinitely many prime numbers of the form a+mn, where m is a natural number. The next three sections then describe algebraic number fields and the aspects of it Date: 30th August 2016. 1 2 FRIMPONG A. BAIDOO necessary for providing context to the Dedekind zeta function. In section 9, we then define the Dedekind zeta function, describe the ideal class group and then highlight the Dedekind zeta function's role in the class number formula. Finally, following what was done in [5], we use the Dedekind zeta function to prove the class number formula for imaginary quadratic fields. 2. The Riemann Zeta Function We begin by taking a brief look at the prototypical L-function, the Riemann Zeta function, highlighting properties we will be using. The Riemann zeta function is given by 1 X 1 (2.1) ζ(s) = ; for s = σ + it where σ; t 2 and σ > 1 ks R k=1 and one can use the Fundamental Theorem of Arithmetic to prove that the above series can be written as an infinite product Y 1 (2.2) ζ(s) = ; where P is the set of prime numbers. 1 − 1 p2P ps The zeta function is important because it contains a wealth of information re- garding the field of rational numbers, Q (or, more precisely, the prime numbers). One such piece of information regarding the behaviour of prime numbers is the following Theorem 2.3 (Euclid). There are infinitely many prime numbers. P −s We will prove this theorem by showing that as s ! 1, the function p2P p is 1 P −1 asymptotic the function log s−1 . Thus the series p2P p diverges, proving the theorem. For this, we first establish a fact regarding the Riemann zeta function. 1 Lemma 2.4. ζ(s) = s−1 + (s) where (s) is some function that is holomorphic in the half-plane σ > 0. This lemma tells us that it is possible to define the zeta function so that it is holomorphic in σ > 0 except for s = σ = 1 where it will have a simple pole. By contrast, equations (2.1) and (2.2) only tell us that the zeta function is holomorphic in σ > 1. 1 Proof. We will prove that (s) = ζ(s) − s−1 defines a function that is holomorphic in the half-plane σ > 0. First notice that 1 1 Z 1 X Z n+1 = t−s dt = t−s dt s − 1 1 n=1 n We thus have that 1 1 X Z n+1 1 1 (2.5) ζ(s) − = − dt s − 1 ns ts n=1 n and it suffices to show that the right hand side of the equation (2.5) converges absolutely on the half-plane σ > 0. DIRICHLET L-FUNCTIONS AND DEDEKIND ζ-FUNCTIONS 3 Because Z n+1 1 1 Z n+1 Z n s s − s dt = s+1 dx dt n n t n t x One can check that Z n+1 1 1 jsj s − s dt ≤ σ+1 n n t n Thus we can use the comparison test to show that the right hand side of (2.5) converges absolutely for all σ > 0. Remark 2.6. It is actually possible to extend the zeta function to the entire complex plane. However, we do not need to do this for the proof of Dirichlet's theorem. Proof of theorem 2.3. The presence of a pole at s = 1 means that the zeta func- tion is non-zero on some neighbourhood of this point. We can therefore take the logarithm of both sides of the equation Y 1 1 (2.7) = + (s) 1 − 1 s − 1 p2P ps for s 6= 1 within this neighbourhood. Taylor expanding the logarithm of the left hand side of (2.7) will then give X 1 X X 1 1 (2.8) + = log + (s) ps pks s − 1 p2P p2P k≥2 since (s) will be holomorphic (and thus bounded) on this neighbourhood, the right 1 hand side of (2.8) will be asymptotic to log s−1 as s ! 1. So now we only need to show that the double series on the left hand side of (2.8) remains bounded in order to prove Theorem 2.3. It can be seen that the double series is actually a series whose terms are geometric series. Thus 0 1 X X 1 X 1 (2.9) = @ pks A ps(ps − 1) p2P k≥2 p2P P 1 The series on the right is contained within the series ns(ns−1) , which converges 1 absolutely in the half-plane σ > 2 . Thus once we ensure that the region on which 1 (2.8) is defined falls within a circle of radius 2 centered at s = 1, we immediately prove that X 1 1 ∼ log ps s − 1 p2P The asymptotic relation from the proof of Theorem 2.3, allows us to define the density of subsets of the prime numbers. 4 FRIMPONG A. BAIDOO Definition 2.10. The analytic density (or Dirichlet density) of a subset Pa of the prime numbers P is defined to be the number k such that X 1 1 ∼ k log ps s − 1 p2Pa as s ! 1. Notice that this immediately implies that 0 ≤ k ≤ 1. Remark 2.11. It can be shown that if a subset of the primes has a density defined in the most "natural sense" (the limit of the ratio of the number of elements in the subset to the number of prime numbers), then it will have an analytic density of the same value. However, the converse is false and this makes the analytic density a better way to define the density of subsets of the primes. We now state a more precise version of Dirichlet's theorem that we will prove. Theorem 2.12 (Dirichlet's Theorem on Arithmetic Progressions). Let a and n be two positive integers that are coprime and let Pa denote the set of prime numbers 1 p ≡ a (mod n). Then the set Pa has analytic density φ(n) , where φ(n) is the Euler totient function. This theorem is stronger than the one stated in the introduction because, it not only says that Pa is an infinite set (if Pa were finite, its density would be 0) but also shows, as we will soon be able to see, that the set P distributes itself equally among all the Pa, for a coprime to n. Example 2.13. For n = 5 and a 2 f0; 1; 2; 3; 4g, the portion of the arithmetic progressions less than 500 is shown on each line of the figure below with the prime numbers in the arithmetic progression circled. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216 221 226 231 236 241 246 2 7 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137 142 147 152 157 162 167 172 177 182 187 192 197 202 207 212 217 222 227 232 237 242 247 3 8 13 18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138 143 148 153 158 163 168 173 178 183 188 193 198 203 208 213 218 223 228 233 238 243 248 4 9 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164 169 174 179 184 189 194 199 204 209 214 219 224 229 234 239 244 249 For each a 6= 0, Dirichlet's theorem not only states that the sequence of circled primes for fixed a never ends but also shows that the probability of (almost) every prime number falling into one of the four arithmetic progressions is one-fourth.
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