Dirichlet s Theorem on Arithmetic Progressions

Transcription

1 Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty lies i the simplicity of its statemet. Theorem. (Dirichlet). Let a, m Z, with (a, m) =. The there are ifiitely may prime umbers i the sequece of itegers a, a + m, a + 2m,..., a + km,... for k N. A sixth grader kows eough mathematics to uderstad this particular formulatio of the theorem. However, may deep ideas of algebra ad aalysis are required to prove it. I order to motivate some of the ideas we will itroduce, we will sketch how to show there are ifiitely may primes of the form 4k +, the special case a =, m = 4 of Theorem.. We shall follow Kapp s expositio i our sketch [2]. Defie the (real valued) Riema zeta fuctio as ζ(s) = =, s >. () s Throughout this paper, p shall deote a prime umber, uless otherwise idicated. It is possible to write the zeta fuctio as the ifiite product ζ(s) = p To see why this is true, otice that for fiite N, p N p s = S. (2) p s where S is the set of atural umbers whose prime factors do ot exceed N. Lettig N we obtai the result. With this product formula for ζ(s), it is possible to show (ad we will do so i the proof of Theorem.) that log ζ(s) = p s + g(s) (3) ps

2 where g(s) is bouded as s. Defie a fuctio : Z, 0, } by 0 if a is eve, (a) = if a mod 4, if a 3 mod 4. This fuctio will allow us to distiguish primes of the form 4k + ad 4k + 3 from oe aother. Notice that (m) = (m)() for all itegers m ad. Now let L(s, ) = = () s. (4) Sice is multiplicative for all itegers (we say is strictly multiplicative i this case), oe ca write, just like i the case of the zeta fuctio, L(s, ) = p. (5) (p)p s This product formula ca be used to show that log L(s, ) = p (p) p s + g (s, ) (6) where g (s, ) is a fuctio that remais bouded as s. Combiig (3) ad (6) we get log ζ(s) + log L(s, ) = 2 ( ) p + s 2 + g(s) + g (s, ), (7) s p (4) log ζ(s) log L(s, ) = 2 p 3(4) p s + ( 2 s + g(s) g (s, ) ). (8) From the Taylor expasio of arcta x, L(, ) = π/4 > 0. But ζ(s) diverges as s, so the left had sides of (7) ad (8) ted to ifiity as s. Sice both /2 s + g(s) + g (s, ) ad /2 s + g(s) g (s, ) remai bouded, it follows that both sums p (4) /ps ad p 3(4) /ps diverge as s. This proves there are ifiitely may primes of the form 4k +. As a bous, we obtaied the existece of ifiitely may primes of the form 4k + 3. There were two crucial ideas that made this last proof possible. First, it was imperative that L(, ) was fiite ad o-zero, so that its logarithm remai bouded i (8). The other key idea was the use of the fuctio to filter out the primes of the form 4k + from all other primes. To prove Dirichlet s theorem, we ll eed fuctios like that will filter out primes of the form a+km. We thus direct our attetio to such fuctios: group characters. 2

3 2 Group Characters Let G be a fiite abelia group. A group character is a homomorphism : G C. The characters of a group form themselves a group uder poitwise multiplicatio. We call this group the dual of G ad deote it Ĝ. If G is a cyclic group of order, the it is easy to describe Ĝ. Let g be a geerator of G. The (g) = w for some w C. Sice is a homomorphism, = () = (g ) = (g) = w. Hece w is a th root of uity. Coversely, let w be a th root of uity. The we ca defie a character of Ĝ by settig (g) = w. Notice that (a) = (a) for all a G. We have a bijective correspodece betwee the group of th roots of uity µ ad Ĝ. I fact, it is easy to see that this correspodece gives a isomorphism. Sice µ = Ĝ, it follows that G = Ĝ. Now let G be ay fiite abelia group. The structure theorem for fiite abelia groups tells us G ca be writte as a direct product of cyclic groups, G = C C k. Let g i be a geerator of C i. Every elemet of G ca be writte as a product of g i s to the appropriate powers, so a character of G is completely determied by the images of the g i s. These images must agai be roots of uity. Coversely, we ca defie a character i of Ĝ by sedig g i to a th i root of uity w i ad all other geerators g j to the idetity elemet. It is easy to see that i this case G = Ĝ as well. I particular, a group ad its dual have the same order. 2. Examples of Groups Characters Example 2.. The trivial character : G C defied by (a) = for all a G. Example 2.2. Dirichlet Characters modulo m: Let G = (Z/mZ). The G is a fiite abelia group with φ(m) elemets (here φ is the Euler totiet fuctio). The Dirichlet characters ca be exteded to all of Z by settig (a) = 0 if (a, m) > ad lettig (a + m) = (a) for all itegers a. These extesios are ot themselves group characters (a character ca t take the value 0), but they are multiplicative fuctios o Z. Through a abuse of laguage, we will ofte times refer to these extesios as Dirichlet characters modulo m. Example 2.3. The pricipal Dirichlet character modulo m is the extesio to Z (as a multiplicative fuctio) of the trivial character of (Z/mZ) : if (a, m) =, 0 (a) = We shall ofte write for the pricipal character istead of 0 3

4 Example 2.4. Let m = 4 i Example 2.2. The (Z/4Z) = Z/2Z, so the dual of (Z/4Z) has oe o-trivial character; it is give by if a mod 4, (a) = if a 3 mod 4, Example 2.5. Let m = p i Example 2.2; here p is a odd prime umber. The dual of (Z/pZ) will be cyclic of order p. Hece there will be a character of order 2, that is 2 = 0. If a is a quadratic residue modulo p, the (a) is forced to be. If a is a quadratic o-residue, the( (a) ) is forced to be. Thus we ca idetify with the familiar Legedre a symbol, (a) =. p Remark. The characters of Ĝ are strictly multiplicative, that is, (ab) = (a)(b) for all a, b G. This follows from the defiitio of group homomorphism. 2.2 Orthogoality Relatios As we said earlier, group characters are essetial to the proof of Dirichlet s theorem because they let us filter out the primes of the form a + km. But how is this the case? It turs out the characters of a group satisfy certai orthogoality relatios. These relatios hold the key to the process of filterig primes. Theorem 2.. Let Ĝ. The (a) = if =, 0 if. (9) Proof. If =, the sum adds up to the umber of elemets i G. Otherwise, choose b G such that (b). The (b) (a) = (a)(b) = (ab) = (a). The last equality follows from the fact that as a rages through the elemets of G, so does ab. Hece we have ((b) ) (a) = 0. Sice (b), (9) follows. By applyig Theorem 2. to the dual group Ĝ ad sice G = Ĝ, we get the followig result. 4

5 Corollary 2.2. Let a G. The Ĝ (a) = Ĝ if a =, 0 if a. (0) Cosider ow two characters, ψ Ĝ. Sice Ĝ is a group, ψ Ĝ, ad a ψ (a) = a (a)ψ (a) = a (a)ψ(a). By Theorem 2., we have (a)ψ(a) = if = ψ, Similarly, (ab ) = (a)(b ) = (a)(b), so that by Corollary 2.2, we get if a = b, Ĝ (a)(b) = (2) Ĝ Equatios () ad (2) are refered to as the orthogoality relatios for group characters. A special case of these relatios, which is of iterest to us, occurs whe G = (Z/mZ). Corollary 2.3 (Orthogoality relatios for Dirichlet Characters). Let ad ψ be Dirichlet characters modulo m, ad let a, b be itegers. The if = ψ, m (a)ψ(a) = φ(m) φ(m) a=0 (a)(b) = if a b mod m, These last two relatios shall do us a great service whe we try to filter out primes of the form a + km from the zeta fuctio. This is all the character theory we will eed. If the reader is iterested i a more thorough treatmet of it, we recommed Serre s book [4]. For a treatmet closer to ours, Irelad ad Rose [] would be a good book to look at. We ow tur our attetio to series like (4). A careful study of them, together with our kowledge of group characters is eough to prove Theorem.. 3 Dirichlet Series A series = with a ad s complex is called a Dirichlet series. We will be primarily cocered with series where a is a Dirichlet character modulo m. First, we must kow somethig about a Dirichlet series regio of covergece. We follow Kapp s [2] treatmet o Dirichlet series for the followig theorems. 5 a s () (3) (4)

6 Theorem 3.. Let a = s be a Dirichlet series. If the series coverges for a particular s = s 0, the it coverges uiformly o the ope half-plae Re s > Re s 0. Furthermore, the sum is aalytic i this regio. We will eed Abel s summatio formula to prove the theorem. Suppose u } ad v } are sequeces of complex umbers such that = u v coverges. Let U = u i; if U v 0 as, the u v = U (v v + ) = = Proof of Theorem 3.. We have a = a s s 0. Let u s s = a 0 ad v s =. We kow 0 s s 0 U } is coverget by hypothesis, ad v 0 uiformly o the half-plae Re s > Re s 0. Thus U v 0 as i this regio. Say U U as. The u v = U (v v + ) U v v + U v v +. If we ca show that v v + = coverges uiformly o the s s 0 ( + ) s s 0 half-plae Re s > Re s 0, we will be doe. For t +, we have (s s 0) t (s s 0) sup t + = sup t + d ( (s s 0 ) t ) (s s 0) dt s s 0 s s 0 t s s 0+, +Re(s s 0) (5) ad so v v + s s 0 +Re(s s 0). Hece v v + s s 0 +Re(s s 0), ad this last expressio coverges uiformly whe Re(s s 0 ) > 0. The aalyticity of the sum follows from the aalyticity of each term i the half-plae. Corollary 3.2. If the Dirichlet series = a / s coverges absolutely at s = s 0, the it coverges uiformly ad absolutely i the half-plae Re s Re s 0. Proof. With absolute covergece, we deduce a a = a ad s s 0 s s 0 s 0 sice the sum o the right had side of these relatios coverges, we get the result by a simple applicatio of the Weierstrass M-test. 6

7 3. Dirichlet L-series ad Euler Products Dirichlet series that have Dirichlet characters modulo m (exteded to Z) as their coefficiets are called L-fuctios. () L(s, ) =. (6) s Whe we studied primes of the form 4k+, we came across a example of L-fuctio. Notice though that a geeral L-fuctio ca have s ad () take complex values. Remark. L(s, ) looks like a zeta fuctio with complex s that is missig all itegers that are divisible by m, sice () = 0 for such. Lemma 3.3. The zeta fuctio ζ(s) is meromorphic i the half-plae Re s > 0. Its oly pole is s = ad it is simple. Proof. We have = = ζ(s) = s + s s = s + s = = s + ( = s + s = + = t s dt + ( s t s ) t dt s ) dt. Notice that + ( s t s ) dt is a aalytic fuctio for Re s > 0. To show the sum of such itegrals (as rages from to ) is aalytic, all we eed is covergece o compact sets for which Re s > 0. Now, + s t s dt + s t s dt sup ( s t s), t + s ad this last expressio is at most by (5). The series coverges for +Re s +Re s Re s > 0. Hece the desired series of itegrals coverges i this regio as well. Our ext goal is to obtai a product expasio for L(s, ) like that of the zeta fuctio. We use the crucial fact that Dirichlet characters are strictly multiplicative. Lemma 3.4. The Dirichlet series = () s () s = p coverges absolutely for Res >. Furthermore,. (7) (p)p s 7

8 Proof. is a bouded fuctio. This gives the desired absolute covergece for Re s >. To see why the product expasio holds ote that for Re s > ad a fixed prime umber q, q p=2, p prime q (p)p = s = p=2, p prime q p=2, p prime ( + (p)p s + 2 (p)p 2s + ) ( + (p)p s + (p 2 )p 2s + ) = S (8) () s (9) where S is the set of atural umbers whose prime factors do ot exceed q. This meas the partial product (8) is equal to a coverget ifiite sum. Now fix a atural umber N. We have N = () s = r p=2, p prime (p)p s S >N () s (20) where r is the largest prime umber less tha or equal to N, ad ow S is the set of atural umbers whose prime factors do ot exceed r. Lettig q i (8) ad N i (20) we see that the product expasio ad the series coverge or diverge together. Sice we kow the series coverges for Re s >, the product expasio must also coverge i that regio. Furthermore, lettig q i (8), we obtai (7) With the above three lemmas i had, we ca exted our remark about L(s, ). Applyig Lemma 3.4 to the pricipal character, we have L(s, ) = p = s p m p m ( p s )ζ(s). This last equality follows from the fact that we have exteded the zeta fuctio to the regio Re s > 0 i Lemma 3.3. Sice the product over p m is fiite, it follows that L(s, ) is meromorphic i the regio Re s > 0 ad its oly pole is simple at s =. Note, however, that the product expasio of L(s, ) is oly valid i the regio Re s >. The product expressio (7) is a example of a Euler product of first degree. If is ot the pricipal character, the we ca go further ad show that the series L(s, ) is coverget ad aalytic i the regio Re s > 0. Theorem 3.5. Let be a Dirchlet character modulo m differet from the pricipal character. The the series L(s, ) coverges ad is aalytic i Re s > 0. Proof. We exteded Dirchlet characters to Z by settig (a) = 0 whe (a, m) > ad by lettig (a + m) = (a) for all itegers a. Usig the exteded characters, it follows, by Theorem 2., that m (m + a) = 0 (2) = 8

9 for ay a. Let s > 0 for ow. We use Abel s summatio formula with u = () ad v = / s. Equatio (2) says U } is bouded; say U U. It is easy to see that U v 0 as. Hece =M () s = u v = U (v v + ) U =M =M v v + = U M s for ay fiite M. The last equality follows because v v + = (v v + ) for s > 0. As M, the last expressio teds to zero. Therefore the series ()/s is coverget for s real ad positive. By Theorem 3., the series is coverget ad aalytic i the regio Re s > 0. As a cosequece of Theorem 3.5, we see that whe is ot the pricipal character, L(s, ) is well defied at s =. We will eed to show that i fact it is ot zero to prove Dirichlet s theorem. Theorem 3.6. For o-pricipal, L(, ) 0. We will postpoe the proof of this theorem util we prove Dirichlet s theorem. 4 Dirichlet s theorem We are ow i a positio to prove Theorem.. For the first part of the proof we will loosely follow Kapp s [2] treatmet. Proof of Theorem.. First, we will show that for a Dirichlet character modulo m, log L(s, ) = p =M (p) p s + g(s, ) (22) for real s >, where g(s, ) is a fuctio that remais bouded as s. Eve if s is real, L(s, ) could still be complex valued, so if we wat to take its logarithm, we better choose a brach. For a give p ad s, defie the value of the logarithm of the p th factor i the L-fuctio s Euler product by ad let g(s,, p) = log =2 log z z = (p) = + (p)p s p s =2 (p ) p s (p ). For this choice of brach, ad for z /2, we have ps =2 z z =2 z 2 ) ( z + 2( 2 = z 2 2)+ 2 =0 9 =0

10 Now set z = (p) p. Sice (p) s p s 2 g(s,, p) = we obtai (p) (p) 2 log (p)p s for s. p s p s p 2 Fially, set g(s, ) = p g(s,, p). Now p g(s,, p) p p 2 which coverges. 2 Thus, g(s, ) remais bouded as s. By addig all the logarithms of the factors i the L-fuctio s Euler product, we obtai a brach of log L(s, ), log (p)p = p p (p) p s + g(s, ) This establishes (22). Now we use group characters to filter out the primes of the form a + km. Recall from our discussio of Dirichlet characters modulo m the followig orthogoality relatio if a b mod m, (a)(b) = φ(m) If we multiply (22) by (a) ad sum over all we get (a) log L(s, ) = (a) p (p) p s + (a)g(s, ) Usig the orthogoality relatio, we rearrage this last equatio to give φ(m) p = (a) log L(s, ) (a)g(s, ). (23) s p a(m) We kow L(s, ) has a pole at s = ; i fact L(s, ) as s for real s. O the other had, by Theorem 3.6, we kow log L(s, ) is bouded at s = for o-pricipal. Hece the sum (a) log L(s, ) has oly oe ubouded term as s. This meas the sum must itself be ubouded as s. This is why Theorem 3.6 is so importat. Had two or more terms of the previous sum bee ubouded, they could have cacelled, depedig o the sig of (a). The term (a)g(s, ) is bouded as s because g(s, ) is. Thus, overall, the right had side of (23) is ubouded as s approaches. This meas there must be ifiitely may primes cotributig to the sum o the left had side. This cocludes the proof of the theorem. 4. The missig lik We have to prove Theorem 3.6. Let be a Dirichlet character modulo m. Defie ζ m (s) by ζ m (s) = L(s, ). (24) 0

11 We kow L(s, ) has a simple pole at s = ad all other L(s, ) are aalytic for Re s > 0. Suppose there is some o-pricipal such that L(, ) = 0. The fuctio ζ m (s) would the be aalytic o Re s > 0. We will prove this is ot the case. The theorem will follow. Suppose p is a prime ot dividig m. Let f(p) be the order of the image p of p i (Z/mZ). Defie g(p) = φ(m)/f(p), that is, the order of the quotiet of (Z/mZ) by (p). Lemma 4.. If p m the ( (p) ) ( = ) g(p). (25) p s p f(p)s Proof. Let µ deote the group of th roots of uity. The ( ) ( wx) = w f(p) w x = e iπ( ) (w x) w µ f(p) w µ f(p) w µ f(p) w µ f(p) = ( ) ( ) w µ f(p) (x w) = x f(p). For ay w µ f(p), there are φ(m)/f(p) = g(p) Dirichlet characters modulo m such that (p) = w. Lettig x = we obtai the desired result. p s Lemma 4.2. The fuctio ζ m (s) has a product expasio for Re s > give by ζ m (s) = ( ) g(p). (26) p f(p)s p m Proof. We have ζ m (s) = L(s, ) = p m ( ) = ( ) g(p), (p)p s p f(p)s p m where the last step follows from Lemma 4.. Notice that is give by a Dirichlet series with positive coefficiets. Hece, p f(p)s Lemma 4.2 shows ζ m (s) is itself a Dirichlet series with positive coefficiets. This is the key fact we shall exploit to obtai our cotradictio. Lemma 4.3. Let a / s be a Dirichlet series with coefficiets a 0. Suppose the series coverges for Re s > s 0 for some real s 0, ad that it exteds aalytically to a aalytic fuctio i a eighborhood aroud the poit s 0. The there is a ɛ > 0 for which the series coverges for Re s > (s 0 ɛ).

12 Proof. We will follow Serre s [3, p. 67] ideas i this proof, though our proof is ot as geeral as his. We may assume without loss of geerality that s 0 = 0. Just replace s with s s 0. For coveiece, deote the series above by f(s). Because f(s) is aalytic i the regio Re s > 0 ad i a eighborhood aroud 0, there is a ɛ > 0 such that f(s) is aalytic i the disc s + ɛ. This meas the Taylor series of f(s) must coverge i this disc. The p th derivative of f(s) is give by f (p) (s) = f (p) () = a ( log ) p = = The Taylor series of f(s) aroud s = is give by s ( ) p a (log ) p f(s) = p= f (p) () (s ) p, s + ɛ. p! Now for s = ɛ, we obtai f( ɛ) = p= f (p) () ( + ɛ) p ( ) p. p! But ( ) p f (p) a (log ) p () = is a coverget series with positive terms. This meas the = followig double sum coverges: f( ɛ) = p But this sum ca be re-expressed as a p! ( + ɛ) p (log ) p. f( ɛ) = = = a s p= a )(+ɛ) e(log s a ɛ p! ( + ɛ)p (log ) p This is the Dirichlet series we started with evaluated at ɛ! Therefore, the series coverges at s = ɛ, ad thus, by Theorem 3., for all Re(s) > ɛ. 2

13 So far we have argued that if L(, ) = 0 for some o-pricipal, the ζ m (s) must be aalytic for Re s > 0. We saw that ζ m (s) is a Dirichlet series with positive coefficiets. Moreover, sice all L(s, ) are coverget for Re(s) >, ζ m (s) coverges i this regio as well. Sice ζ m (s) is aalytic i the regio Re(s) > 0, Lemma 4.3 tells us we ca push back the regio of covergece of ζ m (s) to Re(s) > 0. Our cotradictio is at had. Let s be a real umber greater tha. Cosider the p th factor i the product expasio of ζ m (s) (which is valid for Re s > ) ( ) g(p) = ( + p f(p)s + p 2f(p)s + ) g(p) p f(p)s + p f(p)g(p)s + p 2f(p)g(p)s + = + p φ(m)s + p 2φ(m)s + = p. φ(m)s This shows that for s >, ζ m (s) = L(s, ) = ( ) = (p)p s p p p ( )g(p) p f(p)s ( ) = p φ(m)s (,m)= φ(m)s (27) Thus, ζ m (s) has all its coefficiets greater tha those of the series i (27). These coefficiets of ζ m (s) remai uchaged if we take s betwee 0 ad. But the series (27) diverges for s = > 0. Hece ζ φ(m) m (s) is ubouded for this value of s, ad thus the series ζ m (s) diverges for a value of s whose real part is greater tha zero. This is a cotradictio, sice we showed ζ m (s) coverges for Re s > 0. This completes the proof of Theorem 3.6. Remark. Eve though the product expasio of ζ m (s) is oly valid for Re s >, we were able to use the expasio to look at the coefficiets of the series represetatio for ζ m (s), which is valid for Re s > 0 Refereces [] K. Irelad, M. Rose, A Classical Itroductio to Moder Number Theory Secod Editio. Spriger, New York, 990. [2] A. Kapp, Elliptic Curves Priceto UP, New Jersey, 992. [3] J-P. Serre, A course i Arithmetic Spriger, New York, 973. [4] J-P. Serre, Liear Represetatios of Fiite Groups Spriger, New York,