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3 Zeta and L-Functions 3 Zeta and L-functions In this section we will use analytic methods to (i) develop a formula for class numbers, and (ii) use this to prove Dirichlet’s theorem in arithmetic progressions: that any arithmetic progression: a + m, a +2m, a +3m, . contains infinitely many primes gcd(a, m)=1. This chapter follows [Cohn], though our presentation is reversed from his, together with some supplementary material taken from various other sources. More general treatments are found in [Marcus] and [Neukirch], though they do not do everything we will do here. 3.1 Zeta functions Recall one defines the Riemann zeta function by 1 ⇣(s)= . (3.1) ns X One knows from calculus that this converges for s>1 (compare with 1 x1 s 1 1 1 dx = − = < .) xs 1 s 1 s 1 Z1 − x=1 − Euler observed that (for s>1) one also has the product expansion 1 1 1 1 ⇣(s)= = 1+ + + = . ns ps p2s ··· 1 p s p − X Y ✓ ◆ Y − Here p runs over all primes of N. The last equality just follows from the formula for a geometric n 1 series: 1 a = if a < 1. To see the why product expansion (middle equality) is valid, it’s n=0 1 a | | perhaps easiest to first− notice that it is formally true for s =1, where it says P ⇤ 1 1 1 1 1 1 1 1 1 1 = 1+ + + + 1+ + + + 1+ + + + (3.2) n 2 22 23 ··· 3 32 33 ··· 5 52 53 ··· ··· X ✓ ◆✓ ◆✓ ◆ What does this (formal) infinite product on the right mean? It just means a (formal) limit of the sequence of finite subproducts: 1 1 1 1 1+ + + + = 2 22 23 ··· n n N2 X2 1 1 1 1 1 1 1 1 1 1 1 1 1 1+ + + + 1+ + + + =1+ + + + + + + + 2 22 23 ··· 3 32 33 ··· 2 3 2 3 22 32 22 3 2 32 ··· ✓ ◆✓ ◆ · · · 1 = n n N2,3 2X ⇤When s =1, neither side of the equality actually converges, but the explanation for why both sides should be equal is perhaps more transparent. Here “formally” is not to be confused with rigorously—we mean we can formally manipulate one side to get to the other. 39 1 1 1 1 1 1 1 1 1 1 1+ + + + 1+ + + + 1+ + + + = 2 22 23 ··· 3 32 33 ··· 5 52 53 ··· n ✓ ◆✓ ◆✓ ◆ n N2,3,5 2X . e1 e2 ek where Np1,p2,...,pk = p p p : ei N 0 , i.e., Np1,p2,...,pk is the set of natural numbers 1 2 ··· k 2 [{ } which only contain the primes p ,...p in their prime decomposition. 1 k We now prove rigorously that the formal product expansion for ⇣(s) given above is valid for s>1. Definition 3.1.1. Let p denote the set of primes of N.Letap C for each p. We define { } 2 ap =lim ap. n p !1 n<x Y Y Hence we will say ap converges (diverges) if the limit on the right does. We say ap con- verges absolutely if Q Q lim ap n i !1 i<nY converges for any ordering p ,p ,p ,... of the set of primes p . { 1 2 3 } { } In other words, a product converges absolutely if it converges regardless of the way we order the terms in the product. One can of course similarly define infinite product over any denumerable index set Example 3.1.2. If some ap =0,thenaftersomepoint(nomatterhowthep’s are ordered), we will have a finite subproduct of ap =0.Thus ap will converge to 0 absolutely. Note that if every ap > 0, thenQ log( ap)=Q log ap. An immediate consequence is that ap converges (absolutely) if and only if the series log ap converges (absolutely). Q P Q Proposition 3.1.3. Let (an)n1=1 be a totally multiplicativeP sequence of complex numbers, i.e., amn = 1 aman for any m, n N,andassumea1 =1.If an converges absolutely, then so does 2 p 1 ap and − 1 P 1 Q a = , n 1 a n=1 p p X Y − where the product is taken over all primes p of N. Proof. Suppose an converges absolutely. Let ✏>0. Then for some N N we can say 2 P a <✏. | n| n>NX Let p1,p2,... be any ordering of the set of primes of N. Then there is some K N such that { } 2 p ,...,p contains all N. Observe { 1 K } K K 1 2 = 1+ap + ap + = an. 1 api ··· i=1 − i=1 n Np1,...,pK Y Y 2 X 40 Since Np1,...,pK contains 1,...,N,wehave K 1 1 an an an <✏. − 1 api | | n=1 i=1 − n>N n>N X Y X X Corollary 3.1.4. For any s>1 the Euler product expansion 1 ⇣(s)= s (3.3) 1 p− Y − is valid. s Proof. Apply the proposition with an = n− . The Euler product expansion demonstrates that the zeta function captures information about primes. In fact, it contains a surprising amount of information about primes. The simplest appli- cation of the zeta function to the study of primes is Euler’s proof of the infinitude of primes. Theorem 3.1.5. There are infinitely many primes. Proof. Assume there are finitely many primes, p1,...,pk. Then 1 1 1 1 1 1 ⇣(s)= s s s < 1 p− · 1 p− ···1 p− ! 1 1/p1 · 1 1/p2 ···1 1/pk 1 − 1 − 2 − k − − − as s 1. On the other hand ! 1 1 ⇣(s)= = ns ! n 1 as s 1. Contradiction. X X ! Exercise 3.1. For any integer k>1,onecanshow1/⇣(k) represents the probability that k “ran- domly chosen” integers are coprime (have gcd 1). Let f(x)=x on [ ⇡,⇡),computetheFourier − coefficients and apply Parseval’s identity. Use this to compute ⇣(2),andhencedeterminetheproba- bility that 2 randomly chosen integers are coprime. (Alternatively, you can try to derive the product expansion sin x 1 x 2 = 1 , x − n⇡ n=1 Y ✓ ⇣ ⌘ ◆ and look at the x2 coefficient to find ⇣(2).) We will briefly discuss some deeper connections of ⇣(s) to the study of primes, but first let us give a generalization of the Riemann zeta function. Definition 3.1.6. Let K be a number field. The Dedekind zeta function for K is 1 ⇣ (s)= , K N(a)s a X for s>1 where a runs over all (nonzero) ideals of . OK 41 As before, one can show this series indeed converges for all s>1, and we have an Euler product expansion 1 ⇣ (s)= K 1 N(p) s p − Y − valid for s>1 as above. Hence the Dedekind zeta function can be used to study the prime ideals of of K. We remark that another way to write the above definition is a ⇣ (s)= n K ns X where an denotes the number of ideals of K with norm n (convince yourself of this). Consequently, the Dedekind zeta function can be used to study the number of ideals of norm n. However, we will be interested in it for its applications to the class number hK of K. 3.2 Interlude: Riemann’s crazy ideas Riemann published a single paper in number theory, On the Number of Primes Less Than a Given Magnitude in 1859, which was 8 pages long, contained no formal proofs, and essentially gave birth to all of analytic number theory. We will summarize the main ideas here. We only defined the Riemann zeta function for real s>1, but in fact Riemann considered it for complex values of s. In general if a>0 and z C, then one defines 2 az = ez ln a where zn ez = . n! This allows one formally to make sense of the definitionX 1 ⇣(s)= ns X for s C, and one can show the sum actually converges provided Re(s) > 1. Riemann showed that 2 ⇣(s) can be extended (uniquely) to a differentiable function on all of C except at s =1, where ⇣(s) has a pole (must be ). However the above series expression is only valid for Re(s) > 1. 1 1 Riemann showed that ⇣(s) has a certain symmetry around the line Re(s)= 2 , namely one has the functional equation ⇣(1 s)=Γ⇤(s)⇣(s) − where Γ⇤(s) is a function closely related to the Γ function. The functional equation says one can compute ⇣(1 s) in terms of ⇣(s), so we can indirectly use the series for ⇣(s) to compute ⇣(s) when − Re(s) < 0. The region 0 < Re(s) < 1 is called the critical strip, and the central line of symmetry 1 Re(s)= 2 is called the critical line. Let ⇢ denote the set of zeroes of ⇣(s) inside of the critical strip. There are countably (infinitely) { } many, and let us order them by their absolute value. Let 1 1 f(x)=⇡(x)+ ⇡(x1/2)+ ⇡(x1/3)+ 2 3 ··· 42 where ⇡(x) is the number of primes less than x. Riemann discovered the following formula for f(x) dt f(x) = Li(x) Li(x⇢) log(2) + 1 − − t(t2 1) ln t ⇢ x X Z − x dt where Li(x)= 0 ln t . Hence this formula relates ⇡(x) with the (values of Li at the) zeroes of ⇣(s). In fact, using Möbius inversion, one can rewrite ⇡(x) in terms of f(x) (and therefore the zeroes of R ⇣(s))as 1 1 ⇡(x)=f(x) f(x1/2) (x1/3) − 2 − 3 −··· Essentially this says the following: if we know exactly where all the zeroes ⇢ of ⇣(s) are we know exactly where the primes are (these are the places on the real line where ⇡(x) jumps).
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