Primes in arithmetic progressions

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1 (April 2, 20) Primes i arithmetic progressios Paul Garrett garrett@math.um.edu garrett/. Dirichlet s theorem 2. Dual groups of abelia groups 3. Ugly proof of o-vaishig o Re(s) = 4. Aalytic cotiuatios 5. Ladau s lemma: Dirichlet series with positive coefficiets Dirichlet s 837 theorem combies ideas from Euclid s argumet for the ifiitude of primes with harmoic aalysis o fiite abelia groups, ad some subtler thigs, to show that there are ifiitely may primes p = a mod N for fixed a ivertible modulo fixed N. 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 isolatio of primes i differet cogruece classes modulo N. Dirichlet s result itroduces 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 Frobeius. The subtle elemet is o-vaishig of L-fuctios, explaied below. For expediecy, we first give a ugly proof that explais little, does ot apply i may iterestig situatios, but has few prerequisites. There are two better lies of argumet, both of which give mechaisms whereby L-fuctios ought ot vaish. A form of the simpler oe was used by Dirichlet, expressig products of Dirichlet L-fuctios as zeta fuctios of umber fields. The less simple oe is at most 50 years old, ad uses Eisestei series. Both better viewpoits will be explaied subsequetly.. Dirichlet s theorem I additio to Euler s observatio that the aalytic behavior [] 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. This sectio gives the cetral argumet, ad i doig so ucovers several issues take up i followig sectios. [.0.] Theorem: (Dirichlet) Give a iteger N > ad a iteger a such that gcd(a, N) =, there are ifiitely may primes p with p = a mod N [.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 gcd coditio is ecessary. The poit is that this obvious ecessary coditio is also sufficiet. [] Euler s proof uses oly simple 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 there was little reaso to believe that the behavior of ζ(s) off the real lie played a critical role.

2 [.0.3] Remark: For a =, there is a simple purely algebraic argumet usig cyclotomic polyomials. For geeral a the itelligible argumet ivolves a little aalysis. Proof: A character modulo N is a group homomorphism χ : (Z/N) C Give such a character, exted it by 0 to all of Z/N, by defiig χ(a) = 0 for a ot ivertible modulo N. The compose χ with the reductio-mod-n map Z Z/N ad 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() whether or ot either of the values is 0. The pulled-back-to-z versio of χ, with the extesio by 0, is a Dirichlet character. The trivial Dirichlet character χ o modulo N is the character which takes oly the value (ad 0). Recall the stadard cacellatio trick, that applies more geerally to arbitrary fiite groups: a mod N χ(a) = ϕ(n) (for χ = χo ) where ϕ is Euler s totiet fuctio. Dirichlet s dual trick is to sum over characters evaluated at fixed a i (Z/N) : we claim that χ(a) = χ ϕ(n) (for a = mod N) We will prove this i the ext sectio. Gratig this, for b ivertible modulo N, χ(a)χ(b) χ = χ χ(ab ) = ϕ(n) (for a = b mod N) Give a Dirichlet character χ modulo N, the correspodig Dirichlet L-fuctio is L(s, χ) = χ() s By the multiplicative property χ(m) = χ(m)χ(), each such L-fuctio has a Euler product expasio L(s, χ) = p prime, p N χ(p) p s prove as for ζ(s), by expadig geometric series. Take a logarithmic derivative, as with zeta: d log L(s, χ) = ds p N m p ms = p N χ(p) log p p s + p N, m 2 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 =. 2

3 To pick out the primes p with p = a mod N, use Dirichlet s sum-over-χ trick to obtai χ (a) χ(p) log p p s = ϕ(n) log p p s (for p = a mod N) Thus, χ (a) d log L(s, χ) = ds = ϕ(n) p=a mod N log p p s + χ (a) χ (a) p N, m p N, m 2 p ms p ms We do ot care about cacellatio i the secod sum. All that we eed is its absolute covergece for Re(s) > 2, eedig o subtle iformatio about primes. Domiate the sum over primes by the correspodig sum over itegers 2. Namely, p ms log mσ = (log )/ 2σ log σ 2 σ 2σ 2, m p N, m 2 where σ = Re(s). This coverges for Re(s) > 2. That is, for s +, χ (a) d log L(s, χ) = ϕ(n) ds p=a mod N log p p s + (somethig cotiuous at s = ) We have isolated the primes p = a mod N. Thus, as Dirichlet saw, to prove the ifiitude of primes p = a mod N 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 essetially 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 =, like a o-zero costat multiple of /(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) d log L(s, χ) = + s + d The ecessarily lim s + p=a mod N 3 log p p s = +

4 ad there must be ifiitely may primes p = a mod N. /// [.] What remais to be doe? The o-vaishig of the o-trivial L-fuctios at, which we prove a bit further below, is the crucial techical poit. We prove Dirichlet s dual cacellatio trick i the ext sectio: this is a immediate cosequece of Fourier aalysis o fiite abelia groups. We will also check that the L-fuctios L(s, χ) have aalytic cotiuatios to regios icludig s =. 2. Dual groups of abelia groups Dirichlet s use of group characters to isolate primes i a specified cogruece class modulo N was a big iovatio i 837. These ideas were 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) Recall the basic result o Fourier expasios o fiite abelia groups: [2.0.] Theorem: For a fiite abelia group G with dual group Ĝ, ay complex-valued fuctio f o G has a Fourier expasio f(g) = f(χ) χ(g) (for all g G) G χ Ĝ where the Fourier coefficiets f(χ) are f(χ) = g G f(g) χ(g) The characters are a orthogoal basis for L 2 (G). I particular, Fourier coefficiets are uique. (This is really about commutig uitary operators o fiite-dimesioal complex vector spaces, ad the mai poit is the spectral theorem for uitary operators.) /// [2.0.2] Corollary: Let G be a fiite abelia group. For g e i G, there is a character χ Ĝ such that χ(g). [2] Proof: Suppose that χ(g) = for all χ Ĝ. That is, χ(g) = χ(e) for all χ. The, for ay coefficiets c χ, c χ χ(e) = c χ χ(g) χ χ Sice every fuctio o the group has such a Fourier expasio, this says that every fuctio o G has the same value at g as at e. Thus, g = e. /// [2.0.3] Corollary: For a fiite abelia group G, G = Ĝ [2] This idea that characters ca distiguish group elemets from each other is just the tip of a iceberg. 4

5 Proof: The characters form a orthogoal basis for L 2 (G), so the umber of characters is the dimesio of L 2 (G), which is G. /// [2.0.4] Remark: I fact, usig the structure theorem for fiite abelia groups, oe ca show that G ad its dual are isomorphic, but this isomorphism is ot caoical. [2.0.5] Corollary: (Dual versio of cacellatio trick) For g i a fiite abelia group, χ(g) = χ Ĝ Ĝ (for g = e) Proof: If g = e, the the sum couts the characters i Ĝ. O the other had, give g e i G, let χ be i Ĝ such that χ (g), from a previous corollary. The map o Ĝ χ χ χ is a bijectio of Ĝ to itself, so (χ χ )(g) = χ (g) χ(g) χ Ĝ χ(g) = χ Ĝ χ Ĝ which gives ( χ (g)) χ(g) = 0 χ Ĝ Sice χ (g) 0, it must be that the sum is 0. /// 3. Ugly proof of o-vaishig o Re(s) = Dirichlet s argumet for the ifiitude of primes p = a mod N (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. The argumet here is uillumiatig, but has low prerequisites. [3.0.] 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 [3] Re(s) =, direct argumets ivolve tricks similar to what we do here. [3] 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. 5

6 First, 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 m,p θ m,p = (the argumet of χ(p m )) R The trick [4] is that for ay real θ 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, [5] λ(s) e 0 = 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 mp 2ms = log ζ(2s) 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 below, 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. /// [3.0.2] Remark: Agai, the above argumet is quick, but uillumiatig. We will give better proofs later. [4] [5] Presumably foud after cosiderable foolig aroud. Miraculously... 6

7 4. 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 [6] the L-fuctios to Re(s) > 0. [7] Sice we eed oly this slight aalytic cotiuatio, we ca give a simpler argumet tha would be eeded to aalytically cotiue these L-fuctios to the etire plae. [4.0.] Theorem: The Dirichlet L-fuctios L(s, χ) = χ() s = p χ(p) p s 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 ζ(s) s = = s dx x s = p N s = ( s ( + ) s s ) Sice ( ) s ( + ) s with a uiform O-term, we obtai s = s + O( s+ ) ζ(s) s = O( ) = holomorphic for Re(s) > 0 s+ [6] 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. [7] Already prior to Riema s 859 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. 7

8 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 + = =0 =0 A (b b + ) =0 Takig a = χ() ad b = / s gives L(s, χ) = ( ) χ(l) ( s ( + ) s ) =0 l= The differece / s /( + ) s is s/ s+ up to higher-order terms, so this expressio gives a holomorphic fuctio for Re(s) > 0. /// 5. Dirichlet series with positive coefficiets Now we prove Ladau s result o Dirichlet series with positive coefficiets. (More precisely, the coefficiets are o-egative.) [5.0.] 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. 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 (Abel s theorem), we evaluate the derivatives of f(s) at s = as f (i) () = ( log ) i a ad Cauchy s formulas yield, for s < + 2ε, f(s) = i 0 = f (i) () i! a s = ( ) 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 (log ) i a 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! 8 (ε + ) i

9 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). /// 9