21. Primes in arithmetic progressions

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1 2. Primes i arithmetic progressios 2. Euler s theorem ad the zeta fuctio 2.2 Dirichlet s theorem 2.3 Dual groups of abelia groups 2.4 No-vaishig o Re(s) = 2.5 Aalytic cotiuatios 2.6 Dirichlet series with positive coefficiets Dirichlet s theorem is a stregtheig of Euclid s theorem that there are ifiitely may primes p. Dirichlet s theorem allows us to add the coditio that p = for fixed a ivertible modulo fixed N, ad still be assured that there are ifiitely-may primes meetig this coditio. The most itelligible proof of this result uses a bit of aalysis, i additio to some iterestig algebraic ideas. The aalytic idea already arose with Euler s proof of the ifiitude of primes, which we give below. New algebraic ideas due to Dirichlet allowed him to isolate primes i differet cogruece classes modulo N. I particular, this issue is a opportuity to itroduce the dual group, or group of characters, of a fiite abelia group. This idea was oe impetus to the developmet of a more abstract otio of group, ad also of group represetatios studied by Schur ad Frobeious.. Euler s theorem ad the zeta fuctio To illustrate how to use special fuctios of the form Z(s) = called Dirichlet series to prove thigs about primes, we first give Euler s proof of the ifiitude of primes. [] = a s [] Agai, the 2000 year old elemetary proof of the ifiitude of primes, ascribed to Euclid perhaps because his texts survived, proceeds as follows. Suppose there were oly fiitely may primes altogether, p,..., p. The N = + p... p caot be divisible by ay p i i the list, yet has some prime divisor, cotradictio. This viewpoit 29

2 292 Primes i arithmetic progressios The simplest Dirichlet series is the Euler-Riema zeta fuctio [2] ζ(s) = This coverges absolutely ad (uiformly i compacta) for real s >. For real s > = s s = dx x s ζ(s) + dx x s = + s This proves that lim s + ζ(s) = + The relevace of this to a study of primes is the Euler product expasio [3] ζ(s) = = s = p prime p s To prove that this holds, observe that = s = p prime ( + p s + p 2s + ) p 3s +... by uique factorizatio ito primes. [4] Summig the idicated geometric series gives ζ(s) = p prime p s Sice sums are more ituitive tha products, take a logarithm log ζ(s) = log( p s ) = ( p s + 2p 2s + ) 3p 3s +... p p by the usual expasio (for x < ) log( x) = x + x2 2 + x Takig a derivative i s gives ζ (s) ζ(s) = p prime, m p ms Note that, for each fixed p >, m p ms () p s = p s does ot give much idicatio about how to make the argumet more quatitative. Use of ζ(s) seems to be the way. [2] Studied by may other people before ad sice. [3] Valid oly for s >. [4] Maipulatio of this ifiite product of ifiite sums is ot completely trivial to justify.

3 Garrett: Abstract Algebra 293 coverges absolutely for real s > 0. Euler s argumet for the ifiitude of primes is that, if there were oly fiitely-may primes, the the right-had side of ζ (s) ζ(s) = p ms p prime, m would coverge for real s > 0. However, we saw that ζ(s) + as s approaches from the right. Thus, log ζ(s) +, ad d ds (log ζ(s)) = ζ (s)/ζ(s) as s +. This cotradicts the covergece of the sum over (supposedly fiitely-may) primes. Thus, there must be ifiitely may primes. /// 2. Dirichlet s theorem I additio to Euler s observatio (above) that the aalytic behavior [5] of ζ(s) at s = implied the existece of ifiitely-may primes, Dirichlet foud a algebraic device to focus attetio o sigle cogruece classes modulo N. [2.0.] Theorem: (Dirichlet) Fix a iteger N > ad a iteger a such that gcd(a, N) =. The there are ifiitely may primes p with p = [2.0.2] Remark: If gcd(a, N) >, the there is at most oe prime p meetig the coditio p = a mod, sice ay such p would be divisible by the gcd. Thus, the ecessity of the gcd coditio is obvious. It is oteworthy that beyod this obvious coditio there is othig further eeded. [2.0.3] Remark: For a =, there is a simple purely algebraic argumet usig cyclotomic polyomials. For geeral a the most itelligible argumet ivolves a little aalysis. Proof: A Dirichlet character modulo N is a group homomorphism χ : (Z/N) C exteded by 0 to all of Z/, that is, by defiig χ(a) = 0 if a is ot ivertible modulo N. This extesioby-zero the allows us to compose χ with the reductio-mod-n map Z Z/N ad also cosider χ as a fuctio o Z. Eve whe exteded by 0 the fuctio χ is still multiplicative i the sese that χ(m) = χ(m) χ() where or ot oe of the values is 0. The trivial character χ o modulo N is the character which takes oly the value (ad 0). The stadard cacellatio trick is that χ(a) = { ϕ(n) (for χ = χo ) 0 (otherwise) where ϕ is Euler s totiet fuctio. The proof of this is easy, by chagig variables, as follows. For χ = χ o, all the values for a ivertible mod N are, ad the others are 0, yieldig the idicated sum. For χ χ o, [5] Euler s proof uses oly very crude properties of ζ(s), ad oly of ζ(s) as a fuctio of a real, rather tha complex, variable. Give the status of complex umber ad complex aalysis i Euler s time, this is ot surprisig. It is slightly more surprisig that Dirichlet s origial argumet also was a real-variable argumet, sice by that time, a hudred years later, complex aalysis was well-established. Still, util Riema s memoir of 858 there was little reaso to believe that the behavior of ζ(s) off the real lie was of ay iterest.

4 294 Primes i arithmetic progressios there is a ivertible b mod N such that χ(b) (ad is ot 0, either, sice b is ivertible). The the map a a b is a bijectio of Z/N to itself, so That is, χ(a) = χ(a b) = ( χ(b)) χ(a) χ(b) = χ(b) χ(a) = 0 Sice χ(b), it must be that χ(b) 0, so the sum is 0, as claimed. Dirichlet s dual trick is to sum over characters evaluated at fixed a i (Z/N). We claim that We will prove this i the ext sectio. χ(a) = Gratig that, we have also, for b ivertible modulo N, χ { ϕ(n) (for a = mod N) 0 (otherwise) χ(a) χ(a)χ(b) = χ χ χ(ab ) = { ϕ(n) (for a = b mod N) 0 (otherwise) Give a Dirichlet character χ modulo N, the correspodig Dirichlet L-fuctio is L(s, χ) = χ() s Sice we have the multiplicative property χ(m) = χ(m)χ(), each such L-fuctio has a Euler product expasio L(s, χ) = χ(p) p s This follows as it did for ζ(s), by = p prime, p N L(s, χ) = p prime, p N with gcd(,n)= ( + χ(p)p s + χ(p) 2 p 2s +... ) = χ() s p prime, p N by summig geometric series. Takig a logarithmic derivative (as with zeta) gives L (s, χ) L(s, χ) = p N prime, m χ(p) m p ms = p N prime χ(p) p s + χ(p) p s p N prime, m 2 χ(p) m p ms The secod sum o the right will tur out to be subordiate to the first, so we aim our attetio at the first sum, where m =. To pick out the primes p with p =, use the sum-over-χ trick to obtai ϕ(n) () p s (for p = ) χ(a) χ(p) p s = 0 (otherwise)

5 Thus, = p= Garrett: Abstract Algebra 295 χ(a) L (s, χ) L(s, χ) = ϕ(n) p s + χ(a) χ(a) p N prime, m p N prime, m 2 χ(p) m p ms χ(p) m p ms We do ot care about whether cacellatio does or does ot occur i the secod sum. All that we care is that it is absolutely coverget for Re(s) > 2. To see this we do ot eed ay subtle iformatio about primes, but, rather, domiate the sum over primes by the correspodig sum over itegers 2. Namely, p N prime, m 2 χ(p) m p ms 2, m 2 where σ = Re(s). This coverges for Re(s) > 2. That is, for s +, log mσ = 2 (log )/ 2σ σ 2 σ 2 log 2σ χ(a) L (s, χ) L(s, χ) = ϕ(n) p= p s + (somethig cotiuous at s = ) We have isolated primes p =. Thus, as Dirichlet saw, to prove the ifiitude of primes p = it would suffice to show that the left-had side of the last iequality blows up at s =. I particular, for the trivial character χ o mod N, with values χ(b) = { (for gcd(b, N) = ) 0 (for gcd(b, N) > ) the associated L-fuctio is barely differet from the zeta fuctio, amely L(s, χ o ) = ζ(s) ( p ) s Sice oe of those fiitely-may factors for primes dividig N is 0 at s =, L(s, χ o ) still blows up at s =. By cotrast, we will show below that for o-trivial character, lim s + L(s, χ) is fiite, ad p N lim L(s, χ) 0 s + Thus, for o-trivial character, the logarithmic derivative is fiite ad o-zero at s =. Puttig this all together, we will have lim χ(a) L (s, χ) s + L(s, χ) = + The ecessarily lim ϕ(n) s + p= p s = + ad there must be ifiitely may primes p =. /// [2.0.4] Remark: The o-vaishig of the o-trivial L-fuctios at, which we prove a bit further belo, is a crucial techical poit.

6 296 Primes i arithmetic progressios 3. Dual groups of abelia groups Before worryig about the o-vaishig of L-fuctios at s = for o-trivial characters χ, we explai Dirichlet s iovatio, the use of group characters to isolate primes i a specified cogruece class modulo N. These ideas were the predecessors of the group theory work of Frobeious ad Schur 50 years later, ad oe of the acestors of represetatio theory of groups. The dual group or group of characters Ĝ of a fiite abelia group G is by defiitio Ĝ = {group homomorphisms χ : G C } This Ĝ is itself a abelia group uder the operatio o characters defied for g G by (χ χ 2 )(g) = χ (g) χ 2 (g) [3.0.] Propositio: Let G be a cyclic group of order with specified geerator g. The Ĝ is isomorphic to the group of complex th roots of uity, by (g ζ) ζ That is, a th root of uity ζ gives the character χ such that I particular, Ĝ is cyclic of order. χ(g l ) = ζ l Proof: First, the value of a character χ o g determies all values of χ, sice g is a geerator for G. Ad sice g = e, χ(g ) = χ(g ) = χ(e) = it follows that the oly possible values of χ(g ) are th roots of uity. At the same time, for a th root of uity ζ the formula χ(g l ) = ζ l does give a well-defied fuctio o G, sice the ambiguity o the right-had side is by chagig l by multiples of, but g l does oly deped upo l mod. Sice the formula gives a well-defied fuctio, it gives a homomorphism, hece, a character. /// [3.0.2] Propositio: Let G = A B be a direct sum of fiite abelia groups. The there is a atural isomorphism of the dual groups Ĝ  B by ((a b) χ (a) χ 2 (b)) χ χ 2 Proof: The idicated map is certaily a ijective homomorphism of abelia groups. To prove surjectivity, let χ be a arbitrary elemet of Ĝ. The for a A ad b B χ (a) = χ(a 0) χ 2 (a) = χ(0 b) gives a pair of characters χ ad χ 2 i  ad B. Usurprisigly, χ χ 2 maps to the give χ, provig surjectivity. ///

7 Garrett: Abstract Algebra 297 [3.0.3] Corollary: Ivokig the Structure Theorem for fiite abelia groups, write a fiite abelia group G as G Z/d... Z/d t for some elemetary divisors d i. [6] The Ĝ Z/d... Ẑ/d t Z/d... Z/d t G I particular, Ĝ = G Proof: The leftmost of the three isomorphisms is the assertio of the previous propositio. The middle isomorphism is the sum of isomorphisms of the form (for d 0 ad iteger) Ẑ/d Z/d prove just above i the guise of cyclic groups. /// [3.0.4] Propositio: Let G be a fiite abelia group. For g e i G, there is a character χ Ĝ such that χ(g). [7] Proof: Agai expressig G as a sum of cyclic groups G Z/d... Z/d t give g e i G, there is some idex i such that the projectio g i of g to the i th summad Z/d i is o-zero. If we ca fid a character o Z/d i which gives value o g i, the we are doe. Ad, ideed, sedig a geerator of Z/d i to a primitive d th i root of uity seds every o-zero elemet of Z/d i to a complex umber other tha. /// [3.0.5] Corollary: (Dual versio of cacellatio trick) For g i a fiite abelia group, χ(g) = χ b G { G (for g = e) 0 (otherwise) Proof: If g = e, the the sum couts the characters i Ĝ. From just above, Ĝ = G O the other had, give g e i G, by the previous propositio let χ be i Ĝ such that χ (g). The map o Ĝ χ χ χ is a bijectio of Ĝ to itself, so χ b G χ(g) = χ b G (χ χ )(g) = χ (g) χ(g) χ b G [6] We do ot eed to kow that d... d t for preset purposes. [7] This idea that characters ca distiguish group elemets from each other is just the tip of a iceberg.

8 298 Primes i arithmetic progressios which gives ( χ (g)) χ(g) = 0 Sice χ (g) 0, it must be that the sum is 0. /// χ b G 4. No-vaishig o Re(s) = Dirichlet s argumet for the ifiitude of primes p = (for gcd(a, N) = ) requires that L(, χ) 0 for all. We prove this ow, gratig that these fuctios have meromorphic extesios to some eighborhood of s =. We also eed to kow that for the trivial character χ o mod N the L-fuctio L(s, χ o ) has a simple pole at s =. These aalytical facts are prove i the ext sectio. [4.0.] Theorem: For a Dirichlet character other tha the trivial character χ o mod N, L(, χ) 0 Proof: To prove that the L-fuctios L(s, χ) do ot vaish at s =, ad i fact do ot vaish o the whole lie [8] Re(s) =, ay direct argumet ivolves a trick similar to what we do here. [9] For χ whose square is ot the trivial character χ o modulo N, the stadard trick is to cosider λ(s) = L(s, χ o ) 3 L(s, χ) 4 L(s, χ 2 ) The, lettig σ = Re(s), from the Euler product expressios for the L-fuctios oted earlier, i the regio of covergece, ( ) 3 + 4χ(p m ) + χ 2 (p m ) cos θ m,p + cos 2θ m,p λ(s) = exp mp ms = exp mp mσ m,p where for each m ad p we let The trick [0] is that for ay real θ m,p θ m,p = (the argumet of χ(p m )) R cos θ + cos 2θ = cos θ + 2 cos 2 θ = cos θ + 2 cos 2 θ = 2( + cos θ) 2 0 Therefore, all the terms iside the large sum beig expoetiated are o-egative, ad, [] λ(s) e 0 = [8] No-vaishig of ζ(s) o the whole lie Re(s) = yields the Prime Number Theorem: let π(x) be the umber of primes less tha x. The π(x) x/ l x, meaig that the limit of the ratio of the two sides as x is. This was first prove i 896, separately, by Hadamard ad de la Vallée Poussi. The same sort of argumet also gives a aalogous asymptotic statemet about primes i each cogruece class modulo N, amely that π a,n (x) x/[ϕ(n) l x], where gcd(a, N) = ad ϕ is Euler s totiet fuctio. [9] A more atural (ad digified) but cosiderably more demadig argumet for o-vaishig would etail followig the Maaß-Selberg discussio of the spectral decompositio of SL(2, Z)\SL(2, R). [0] Presumably foud after cosiderable foolig aroud. [] Miraculously...

9 Garrett: Abstract Algebra 299 I particular, if L(, χ) = 0 were to be 0, the, sice L(s, χ o ) has a simple pole at s = ad sice L(s, χ 2 ) does ot have a pole (sice χ 2 χ o ), the multiplicity 4 of the 0 i the product of L-fuctios would overwhelm the three-fold pole, ad λ() = 0. This would cotradict the iequality just obtaied. For χ 2 = χ o, istead cosider λ(s) = L(s, χ) L(s, χ o ) = exp ( p,m ) + χ(p m ) mp ms If L(, χ) = 0, the this would cacel the simple pole of L(s, χ o ) at, givig a o-zero fiite value at s =. The series iside the expoetiatio is a Dirichlet series with o-egative coefficiets, ad for real s p,m + χ(p m ) mp ms p, m eve + mp ms = p, m + 2mp 2ms = p, m = log ζ(2s) mp2ms Sice ζ(2s) has a simple pole at s = 2 the series log (L(s, χ) L(s, χ o )) = p,m + χ(p m ) mp ms log ζ(2s) ecessarily blows up as s + 2. But by Ladau s Lemma (i the ext sectio), a Dirichlet series with o-egative coefficiets caot blow up as s s o alog the real lie uless the fuctio represeted by the series fails to be holomorphic at s o. Sice the fuctio give by λ(s) is holomorphic at s = /2, this gives a cotradictio to the suppositio that λ(s) is holomorphic at s = (which had allowed this discussio at s = /2). That is, L(, χ) 0. /// 5. Aalytic cotiuatios Dirichlet s origial argumet did ot emphasize holomorphic fuctios, but by ow we kow that discussio of vaishig ad blowig-up of fuctios is most clearly ad simply accomplished if the fuctios are meromorphic whe viewed as fuctios of a complex variable. For the purposes of Dirichlet s theorem, it suffices to meromorphically cotiue [2] Re(s) > 0. [3] the L-fuctios to [5.0.] Theorem: The Dirichlet L-fuctios L(s, χ) = χ() s = p χ(p) p s [2] A extesio of a holomorphic fuctio to a larger regio, o which it may have some poles, is called a meromorphic cotiuatio. There is o geeral methodology for provig that fuctios have meromorphic cotiuatios, due i part to the fact that, geerically, fuctios do ot have cotiuatios beyod some atural regio where they re defied by a coverget series or itegral. Ideed, to be able to prove a meromorphic cotiuatio result for a give fuctio is tatamout to provig that it has some deeper sigificace. [3] Already prior to Riema s 858 paper, it was kow that the Euler-Riema zeta fuctio ad all the L- fuctios we eed here did ideed have meromorphic cotiuatios to the whole complex plae, have o poles uless the character χ is trivial, ad have fuctioal equatios similar to that of zeta, amely that π s/2 Γ(s/2)ζ(s) is ivariat uder s s.

10 300 Primes i arithmetic progressios have meromorphic cotiuatios to Re(s) > 0. For χ o-trivial, L(s, χ) is holomorphic o that half-plae. For χ trivial, L(s, χ o ) has a simple pole at s = ad is holomorphic otherwise. Proof: First, to treat the trivial character χ o mod N, recall, as already observed, that the correspodig L-fuctio differs i a elemetary way from ζ(s), amely L(s, χ o ) = ζ(s) ( p ) s Thus, we aalytically cotiue ζ(s) istead of L(s, χ o ). To aalytically cotiue ζ(s) to Re(s) > 0 observe that the sum for ζ(s) is fairly well approximated by a more elemetary fuctio ( s s ζ(s) s = = s dx p N x s = Sice ( ) s (+) s with a uiform O-term, we obtai ζ(s) s = s = = s + O( s+ ) (+) s ) s O( ) = holomorphic for Re(s) > 0 s+ The obvious aalytic cotiuatio of /(s ) allows aalytic cotiuatio of ζ(s). A relatively elemetary aalytic cotiuatio argumet for o-trivial characters uses partial summatio. That is, let {a } ad {b } be sequeces of complex umbers such that the partial sums A = i= a i are bouded, ad b 0. The it is useful to rearrage (takig A 0 = 0 for otatioal coveiece) a b = (A A )b = A b A b + = A (b b + ) = = Takig a = χ() ad b = / s gives L(s, χ) = =0 =0 i= =0 ( ) χ() ( s ( + ) s ) The differece / s /( + ) s is s/ s+ up to higher-order terms, so this expressio gives a holomorphic fuctio for Re(s) > 0. /// =0 6. Dirichlet series with positive coefficiets Now we prove Ladau s result o Dirichlet series with positive coefficiets. (More precisely, the coefficiets are o-egative.) [6.0.] Theorem: (Ladau) Let f(s) = be a Dirichlet series with real coefficiets a 0. Suppose that the series defiig f(s) coverges for Re(s) > σ o. Suppose further that the fuctio f exteds to a fuctio holomorphic i a eighborhood of s = σ o. The, i fact, the series defiig f(s) coverges for Re(s) > σ o ε for some ε > 0. = a s

11 Garrett: Abstract Algebra 30 Proof: First, by replacig s by s σ o we lighte the otatio by reducig to the case that σ o = 0. Sice the fuctio f(s) give by the series is holomorphic o Re(s) > 0 ad o a eighborhood of 0, there is ε > 0 such that f(s) is holomorphic o s < + 2ε, ad the power series for the fuctio coverges icely o this ope disk. Differetiatig the origial series termwise, we evaluate the derivatives of f(s) at s = as f (i) () = ( log ) i a = ( ) i (log ) i a ad Cauchy s formulas yield, for s < + 2ε, f(s) = i 0 f (i) () i! (s ) i I particular, for s = ε, we are assured of the covergece to f( ε) of f( ε) = i 0 f (i) () i! ( ε ) i Note that ( ) i f (i) () is a positive Dirichlet series, so we move the powers of a little to obtai f( ε) = i 0 ( ) i f (i) () i! (ε + ) i The series ( ) i f (i) () = (log ) i a has positive terms, so the double series (coverget, with positive terms) f( ε) =,i a (log ) i ( + ε) i i! ca be rearraged to f( ε) = a ( ) (log ) i ( + ε) i = i! i a (+ε) = a ε That is, the latter series coverges (absolutely). ///