Dirichlet s Theorem on Arithmetic Progressions

Transcription

1 Dirichlet s Theorem o Arithmetic Progressios The ifiitude of rimes Prime umbers are the buildig blocs of all coutig umbers: the fudametal theorem of arithmetic states that every ositive iteger ca be writte as a uique roduct of rime umbers, u to the order of the rime factors. Thus rime umbers are of great imortace, esecially i umber theory, ad for cetury after cetury, mathematicias have wored to uderstad rime umbers better. Oe of the questios that occurred to early mathematicias was: how may rimes are there? There are ifiitely may of them, as first show by Euclid i his boo of Elemets. This roof is usually foud i boos i a form restated i the moder laguage of mathematics; I assume that the reader has see the roof i that form. I Euclid s time, umbers were thought of as legths of lie segmets, so I shall state the roof i the laguage closer to Euclid s origial oe. Theorem.. The umber of rime umbers is more tha ay assiged multitude of rime umbers. A B C E D F Euclid s argumet. Say a umber i.e. a lie segmet of legth X is measured by aother umber Y if X is roortioal to Y. Let A, B, ad C be the assiged rime umbers. Tae the least umber DE measured by A, B, ad C. Add the uit legth segmet DF to DE. The EF is either rime or ot rime. If it is rime, the there are ow the rime umbers A, B, C ad EF, more i umber tha the assiged umber three, that is, A, B ad C. If it is ot rime, the it is measured by some rime umber. Let it be measured by the rime umber G. Sice A, B ad C measure DE, G also measures DE. But it also measures EF, so G measures the remaiig segmet, the uit-legth DF, which is absurd. So G is ot oe of the

2 umbers A,B or C. Sice G is rime, we ow have the rime umbers A, B, C ad G, agai more i umber tha the assiged umber three. We ca aly this to ay umber, ot just three, so there are more rime umbers tha ay assiged multitude of rime umbers. Whe mathematicias became more familiar with the otio of ifiity, they the bega to as: how ifiite is this ifiite umber of rimes? For examle, they would cosider a ifiite sequece of ositive itegers X x 0, x, x,... as, iome sese, bigger more dese i distributio, for examle tha aother ifiite sequece of ositive itegers Y y 0, y, y,... if i x i diverges but i y i coverges. Whe Euler gave aother roof that there are ifiitely may rimes, he also showed us that there are ideed quite a lot of rimes: Theorem.. The sum of the recirocals of the rimes diverges, i.e. diverges. We shall defer the roof util later. After establishig that there is a ifiite umber of rimes, eole bega to as about the ifiitude of rimes i aother way: If we are give ay arithmetic sequece, do we also have a ifiite umber of rimes i the sequece? Euler first stated that if the first term of a arithmetic sequece is, the the sequece cotais a ifiite umber of rimes. Now cosider a geeral arithmetic sequece. If the first term ad the commo differece have a commo factor that is ot equal to, the clearly every umber i the sequece would be divisible by, so this must be excluded for the questio to have a chace of a aswer i the affirmative. With this case excluded, the statemet i the questio was first cojectured to be true by Gauss. It turs out that if this case is excluded, the ideed there is a ifiite umber of rimes i the sequece. This was first roved by Dirichlet i 835, ad is ow ow as Dirichlet s theorem o arithmetic rogressios: Theorem.3. If a is corime to d, the the arithmetic rogressio a, a + d, a + d,... cotais a ifiite umber of rimes. Put aother way, we may say that if two umbers a ad b are corime, the there are a ifiite umber of rimes of the form a + b, where > 0. There is o simle elemetary argumet ow that roves this theorem i full geerality. However, for a few simle cases, we may mimic Euclid s roof for the ifiitude of rimes i order to write out a roof for these cases. Theorem.4. There exist a ifiite umber of rimes of the form Proof. Suose that there are fiitely may rimes of the form Call them,,...,. Let N : The N 5 mod 6. Sice for ay m ad we have 6 + 6m + mod 6

3 6 + 6m mod 6 ad m mod 6, N must have at least oe rime factor q such that q 5 mod 6.,,...,, as otherwise q N ad q 6..., But q caot be oe of which imlies that q 6... N, i.e. q, which is imossible. So we foud aother rime of the form 6 + 5, a cotradictio. Hece there are ifiitely may rimes of the form Utilisig the Legedre symbol ad / or quadratic recirocity, we ca also rove some of the other cases of the theorem, still mimicig Euclid s idea. We shall demostrate with oe usig the roerties of Legedre symbol aloe, ad aother that also uses quadratic recirocity. Theorem.5. There exist a ifiite umber of rimes of the form 8. Proof. Suose that there are fiitely may rimes of the form 8. Call them,,...,. Let N : Sice N/ is odd, N must be divisible by some odd rime q, so 4... mod q. Hece we have q. Sice for a odd rime we have /8, we ca coclude that q mod 8 or q 7 mod 8. Sice N/ is odd, N/ is the roduct of all the odd rime factors of N. But N/ mod 8, so there must be at least oe such factor of the form 8, as otherwise N/ mod 8. Let this factor be r. But r caot be oe of,,...,, as otherwise r N ad r 4..., which imlies that r, which is imossible as r is odd. So we foud aother rime of the form 8, a cotradictio. Hece there are ifiitely may rimes of the form 8. Theorem.6. There exist a ifiite umber of rimes of the form 6 +. Proof. Suose that there are fiitely may rimes of the form 6 +. Call them,,...,. Let N : The N mod 6. Suose q is a rime factor dividig N. Theice N :... +, rovided that q 3 we have a solutio to the cogruece i.e. 3x 3 mod, givig we fid that 3 3 3x + 0 mod,. Usig / ad quadratic recirocity, 3 / so i fact 3. This meas that mod 3. But is odd, so mod 6. As before, caot be oe of,,...,, so we have our cotradictio. 3

4 However, there is o ow elemetary roof similar to the above oes for the geeral case. To rove the geeral theorem, istead of followig Euclid s idea, Dirichlet followed ad we shall follow Euler s idea, that is, by rovig that the sum of the recirocals of such sequeces diverges. For the remaider of this essay, the mai focus will be to rove Dirichlet s theorem, ad we shall develo ideas to the extet that they are eeded i achievig our goal. To reare ourselves for ivestigatio ito the qualitative roerties of the ifiitude of rimes, we first cosider arithmetic fuctios. Arithmetic fuctios Defiitio.. A arithmetic fuctio or a umber-theoretic fuctio is a fuctio f from the ositive itegers to the comlex umbers. That is, it is a sequece of comlex umbers. Defiitio.. A arithmetic fuctio is called multilicative if fm fmf for all airs of corime m,. A fuctio is called comletely multilicative if fm fmf for all m,. We first cosider two imortat arithmetic fuctios: the Möbius fuctio ad the Euler totiet fuctio. Recall that a square-free iteger is oe ot divisible by ay erfect square other tha. Defiitio.3. The Möbius fuctio is the fuctio µ : Z + {, 0, } defied by if or if is square-free with a eve umber of distict rime factors; µ : if is square-free with a odd umber of distict rime factors; 0 if is ot square-free. Notice that the Möbius fuctio is multilicative. Theorem.4. Suose. The µd d { if 0 if >. That is, µd d. Proof. This is clearly true for. Now assume that >. Write α α... α. µd is zero if d has a square i its factors, so the sum is comrised of terms where d ad where d is a divisor of that is a roduct of distict rimes. So we have µd µ + µ + + µ + µ + + µ + + µ... d

5 Defiitio.5. The Euler totiet fuctio is the fuctio ϕ : Z + Z + defied as the umber of ositive itegers ot exceedig that are corime to. Theorem.6. Let. The ϕd. d Proof. Defie S : {,,..., }, Ad { : gcd, d, 0 < }. S is thus a disjoit uio of Ad, so Ad. d Now gcd, d gcd/d, /d, ad 0 < 0 < /d /d. So if we let q /d, the q satisfies gcdq, /d, 0 < q /d. There are ϕ/d umbers q satisfyig this, ad sice the two coditios corresod exactly to that for the sets Ad, ϕ. d Sice d d ϕ ϕ, the result follows. d d We ow establish a li betwee the Möbius fuctio ad the Euler totiet fuctio: Theorem.7. Let. The ϕ d µd d. Proof. ϕ gcd, d gcd, µd by.4 µd d d d /d µd q where agai q /d d µd d. Theorem.8. With the emty roduct equal to, we have ϕ. 5

6 Proof. Write α α... α. Alyig the revious theorem, ϕ d µd d d α...α µd d µd d d... + i i µ i i i + i,j. sice µd 0 if d is ot square-free + i,j µ i j + µ... i j... i j Euler s argumet for the ifiitude of rimes I this sectio, we shall first wor towards rovig Theorem.. Recall that Theorem. states that the sum of the recirocals of all the rimes diverges. Defiitio 3.. For s >, we defie the Riema zeta fuctio to be ζs. The variable is covetioally deoted s i deferece to Riema s 859 aer that fouded the study of this fuctio as a comlex fuctio, ow famous for the Riema Hyothesis stated i it. The itegral test quicly establishes the covergece of the series for s > ad divergece otherwise. Far before Riema s time, however, this fuctio was studied by Euler as a real fuctio. Euler was the first to comute exact values of ζs for s, as well as umerous other eve-umber values of s. He also deduced a roduct formula for ζs, the geeralised versio of which ow bears his ame. Theorem 3. Euler Product. If f is multilicative, the f + f + f +. rime Proof. Exad artial roducts o the right had side, ad aly the fudametal theorem of arithmetic: + f + f + f f f t t <y 6 t

7 ,,..., t f f,,..., t f : P <y t f ft t t where,,..., t are the rimes less tha y ad P is the largest rime factor of. Every ositive iteger less tha y has o factors more tha y, so f f f. : P <y The right had side teds to zero as y, so the result follows. Corollary 3.3. If f is comletely multilicative, the y f rime f. Proof. Sice f is comletely multilicative, f f. Hece by the revious theorem, f rime rime rime + f + f + + f + f + f. Corollary 3.4. For s >, ζs rime s. Proof. Aly the above corollary to the fuctio f : s, which is obviously comletely multilicative. We ca ow rove Theorem., which we restate here: Theorem 3.5. The sum of the recirocals of the rimes diverges, i.e. diverges. Euler himself wrote this argumet: 7

8 Euler s argumet. Euler maiulated the harmoic series formally. Usig the Euler roduct formula, log log log < < C for some costat C, fiite because is fiite. Sice made the coclusio that log log+. is asymtotic to log, Euler This is ot rigorous by modertadards, but ca be modified to be so. Oe obvious way to iterret the last equatio is to treat it as meaig that the artial sum of the recirocals of rimes is asymtotic to log log. This is ideed the case. We build o Euler s idea to costruct a roof that also gives a estimate for the sum: Proof of 3.5. Firstly, every ositive iteger i ca be writte as a roduct of a squarefree iteger ad a square. Suose i a i b i, where a i i is squarefree, ad b i i. The i i i a i b i a i i + 8 sice a i are squarefree.

9 Secodly, by the itegral defiitio of the atural logarithm, Thirdly, for all x > 0, Ad the sum log < i i. + x < + x + x! + ζ x! ex. is fiite. Combiig everythig above, log < i + < ζ e / ζ ex, i ad the taig logarithm of both sides, log log log ζ < So the artial sum grows at least lie log log. Remar 3.6. The Meissel-Mertes costat, or just Mertes costat, M, is defied to be the limit of the differece betwee the sum of rime recirocals ad log log : M : lim log log γ + [ log + ]. Ad so we have roved that the sum of the recirocals of all the rimes diverges. To adat the roof to ay arithmetic rogressio with corime leadig term ad commo differece, Dirichlet itroduced Dirichlet characters ad Dirichlet series.. 4 Characters ad their roerties We shall use fuctios called Dirichlet characters, which are comletely multilicative fuctios to which Theorem 3.3 ca be alied. Before we loo ito Dirichlet characters, we first cosider characters ad character grous. Defiitio 4.. Let G be a grou. A character of G is a fuctio f : G C satisfyig fg g fg fg for all g, g G. That is, f is a grou homomorhism. This is ow a secial case of the grou of reresetatios of a grou, for the case where the reresetatios are oe-dimesioal. Such characters cosidered by Dirichlet are oe of the acestors of grou reresetatio ad character theory. We ow derive some roerties of characters of fiite grous. So from ow o, let the grous cocered be fiite grous. 9

10 Proositio 4.. Let G be a grou with idetity e. The fe. Proof. whece fe. fe fe fe, Corollary 4.3. Let G be a grou of order. The for each g G, fg is a -th root of uity. Proof. fg fg fe. Proositio 4.4. If G is a abelia grou of order, the there are characters. Proof. Sice G is abelia, usig the structure theorem, we may write G as G C e C e C e where each i is rime, ad C r is a cyclic grou of order r. If g G, the g g α gα... gα, where g i geerates C e i ad the roduct of the orders of each g i is. Thus i fg i fg i α i. The characters deed o the value of each geerator g i, ad sice the geerators orders multily to, there are at most differet characters. Now suose we are give w, w,..., w such that w e i i i for each i. We ca theet fg i w i, ad costruct a character by defiig fg i fg i α i. It ca be routiely checed that each such costructio is a character ad is distict from all the others, so there are at least differet characters. Thus there are i fact recisely differet characters for G. So suose that the set of characters of G is {f, f,..., f }. Defiitio 4.5. The ricial character, f, is the character such that f g for all g G. Theorem 4.6. The characters of a grou G form a abelia grou, with the grou oeratio defied by f i f j g f i gf j g for all g G, with idetity elemet f ad iverse of f beig f, where deotes comlex cojugatio i.e. the iverse, sice fg is a root of uity. Proof. This routie checig will be sied. Defiitio 4.7. The grou of characters i Theorem 4.6 is called the character grou of G. We shall deote this grou by Ĝ. I algebra laguage, this is the dual grou of the abelia grou G. 0

11 Theorem 4.8 Orthogoality relatios. This is a air of similar assertios. i For a fixed character f Ĝ, ii For a fixed elemet g G, fg g G fg f Ĝ { if f f 0 otherwise. { if g e 0 otherwise. Proof. i If f f the fg. g G g G Otherwise, there must exist h G such that fh. The fh fg fhg fg. g G g G g G Sice fh, g G fg 0. ii If g e the f Ĝ fg f Ĝ. Otherwise, there must exist a character f 0 such that f 0 g. If g g α... gα, theice g e, for some j we have α j 0 mod e j. Theet χg j e πi/e j, χg i for all other i. The as i i, we have f 0 g f Ĝ fg f Ĝ f 0 fg f Ĝ fg. Sice f 0 g, f Ĝ fg 0. Remar 4.9. Theorem 4.8 is called the orthogoality relatios, because if oe relaced f i i by f i f j ad g i ii by g g, we obtai i ii G f i gf j g g G { if f i f j 0 otherwise. { if g g fg fg G 0 otherwise. f Ĝ They are the first ad secod orthogoality relatios for grou characters.

12 Now let be the residue class modulo of a iteger, i.e. the set of all itegers cogruet to modulo. I other terms, + Z Z/Z. The set of reduced residue classes modulo is,,..., ϕ, where each i is corime to ad ϕ is the Euler totiet fuctio. Theorem 4.0. The set of reduced residue classes modulo forms a abelia grou of order ϕ, with the grou oeratio defied by m m. The idetity of the grou is ad the iverse of a elemet m is where atisfies m mod. We deote this grou by Z. Defiitio 4.. A Dirichlet character modulo is a fuctio χ : Z + C defied to be { f if, are corime i.e. Z χ : 0 otherwise, where f is a character of the grou Z. The ricial Dirichlet character modulo, χ, is where f f above, i.e. { if, are corime i.e. Z χ : 0 otherwise. Theorem 4.. A Dirichlet character is a comletely multilicative fuctio. Proof. Suose, m Z +. Case i If ad m are both corime to, the χχm ffm f m fm χm. Case ii If is ot corime to, the m is also ot corime to, so χχm 0 χm 0 χm. I both cases, we have χχm χm, so χ is comletely multilicative. For comleteess, we state this agai i the list of roerties of Dirichlet characters below. Theorem 4.3 Proerties of Dirichlet characters. Dirichlet characters modulo have the followig roerties: i χ + m for all ositive itegers m; ii χ is a ϕ-th root of uity if gcd, ; iii χχm χm for all ositive itegers, m; iv There are recisely ϕ Dirichlet characters modulo ; v For a fixed Dirichlet character χ, vi For a fixed ositive iteger, χ χ χ { ϕ if χ χ 0 otherwise; { ϕ if 0 otherwise. Proof. These roerties directly follow from the defiitio of a Dirichlet character ad the roerties of a grou character.

13 5 Dirichlet series We ow cosider Dirichlet series, the aalogue of the sum of the recirocals of rimes i Euler s roof for the ifiitude of rimes. Dirichlet series have umerous roerties aalogous to those for ower series. We shall first defie a Dirichlet series ad ivestigate its roerties. Defiitio 5.. Let s σ + it C. A Dirichlet series is a series of the form fs. Observe that if σ σ 0, the σ 0, so σ. 0 By the comariso test, we see that if the series coverges absolutely for s 0 σ 0 + it 0, the it coverges absolutely for all s σ + it with σ σ 0. Usig this observatio, we have the followig: Theorem 5. Absolute covergece of Dirichlet series. Suose that the series s does ot coverge or diverge for all s. The there exists a real umber σ a, the abscissa of absolute covergece, such that s coverges absolutely for σ > σ a ad does ot coverge absolutely for σ < σ a. Proof. Let L be the subset of real umbers such that if σ L, the s diverges. By assumtio, the series does ot coverge for all s, so L. The series also does ot diverge for all s, ad by our revious observatio, L must be bouded above. Sice L is a subset of the real umbers, L has a least uer boud. Set σ a to be this least uer boud. Suose σ < σ a. The σ L, because otherwise σ would be a uer boud smaller tha the least uer boud, σ a. Suose σ > σ a. The σ / L, because σ is a uer boud for L. We shall also eed to ow where the series coverges, ot just absolutely. First we eed a lemma, which we the aly to the situatio we eed to obtai the corollary that follows. Lemma 5.3. Let x ad let φx be cotiuously differetiable o [,. Let Sx x C, C C. The Cφ Sxφx x x Stφ tdt. Proof. First otice that if x is such that x < + for some iteger, the Sx S. This will also be subsequetly used to ut the sum iside a itegral with itegratio limits beig two cosecutive itegers. Let us set the emty sum S0 to be 0. Startig from the left had side, x Cφ Cφ 3

14 S S φ Sφ S φ Sφ Sφ + Sφ φ + + Sφ Sφ φ + + Sφ φx + Sφx S + Sxφx Sxφx Sxφx x φ tdt S + x Stφ tdt Stφ tdt Stφ tdt. x φ tdt + Sxφx x Stφ tdt Stφ tdt Corollary 5.4. Let x, y ad for some fixed s, s 0 C, s s 0, let φx : x s s 0. The x< y 0 φ x< y φy 0 Sice φ x s s 0 x s s 0, equivaletly we write x< y x< y 0 y s s 0 + s s 0 y x y x x< t x< t φ tdt. 0 0 t s s 0+ dt. Proof. Alyig Lemma 5.3 with Sx x 0 ad φx x s s 0, we obtai x< y y x y 0 s 0 x 0 s 0 4

15 Sy y s s + s s 0 0 y Sx x s s s s 0 0 Sy Sx y s s 0 St t s s 0+ dt x + Sx Sy Sx y s s s s 0 0 Sy Sx y s s + s s 0 0 x< y 0 y s s 0 + s s 0 St t s s 0+ dt y s s 0 y x y x y x x s s 0 y + s s 0 x Sx t s s 0+ dt + s s 0 St Sx t s s 0+ dt x< t 0 t s s 0+ dt. St t s s 0+ dt y x St t s s 0+ dt Theorem 5.5 Coditioal covergece of Dirichlet series. For each Dirichlet series, there is a abscissa of coditioal covergece σ c [, ] such that the series coverges coditioally for all s with σ > σ c, ad diverges for all s with σ < σ c. Proof. Suose fs coverges for s s 0. We wish to show that fs the coverges for all s with σ > σ 0. We use the fact that over the real umbers, a sequece is a Cauchy sequece if ad oly if it coverges. Let each term i our sequece be defied by a m : m s 0. Thus our assumtio that fs 0 coverges meas that for every ɛ > 0 there is a umber x 0 such that a x a y < ɛ for all x, y > x 0, that is, 0 < ɛ for all x, y > x 0. x< y Now suose ɛ > 0 is give, ad cosider the sequece defied by b m : m s. Alyig 5.4, we have b x b y x< y < ɛ ɛ y σ σ 0 + ɛ s s 0 y σ σ 0 + ɛ s s 0 σ σ 0 y x t σ σ 0+ dt x σ σ 0 ɛ s s 0 σ σ 0 y σ σ 0 + ɛ s s 0 σ σ 0 ɛ s s 0 σ σ 0 x σ σ 0 y σ σ 0 x σ σ 0 y σ σ 0 sice s s 0 σ σ 0 5

16 ɛ s s 0 σ σ 0. Sice ɛ was arbitrary, the series coverges. We set σ c : if{re s : coverges}. Theorem 5.6. Suose that fs s coverges at s s 0. The for every δ > 0, the series coverges uiformly i the sector U : {s : arg s s 0 < π δ}. Proof. Let s be i U. By elemetary trigoometry, hece by we have s s 0 σ σ 0 σ σ 0/ cos arg s s 0 σ σ 0 < cosπ/ δ si δ, sice s U ad cosie decreases o [0, π/ x< y < ɛ si δ, ideedet of s, ad thus covergece is uiform for each s U. Theorem 5.7. A Dirichlet series fs with abscissa of covergece σ c is aalytic i the half-lae {s : Re s > σ c }. Proof. Sice each term i the series is aalytic ad the series coverges uiformly i each U for each give δ with s 0 σ c + it 0 secified above, by Weierstrass Theorem, fs is aalytic i U. The regio U is U {s : arg s s 0 < π δ} {s σ + it : σ > σ c}, so fs is aalytic i the required half-lae. For a ower series, there is a sigularity o its circle of covergece. For a Dirichlet series, the aalogue oly holds if each coefficiet is o-egative. Theorem 5.8 Ladau s theorem. Suose that fs is a Dirichlet series with 0 for each, ad that the abscissa of covergece σ c is fiite. The fs has a sigularity at s σ c. 6

17 Proof. Suose that, i fact, fs is aalytic at s σ c. The there exists a δ > 0 such that fs is aalytic i D : {s : s σ c < δ}. Now let σ 0 be a oit o the real axis such that σ 0 > σ c ad such that D : {s : s σ 0 < ɛ} for some ɛ > 0 is totally cotaied i D, with some oits i D o the real axis less tha σ c. Hece fs is aalytic i D ad we may exad fs about σ 0 to get f σ 0 fs s σ 0.! We also have 0 fs ad sice this is uiformly coverget i the regio cocered, we may differetiate term by term ad obtai f log σ 0 σ, 0 ad substitutig ito the ower series exasio, we get fs σ 0 s 0! log σ. 0 Now tae s real, σ 0 ɛ < s σ < σ 0, so that σ 0 s σ 0 σ > 0: fσ σ 0 σ 0 σ 0! log σ 0 σ 0 σ log 0! chagig order of summatioice every term is ositive σ 0 exσ 0 σ log σ 0 σ. σ σ 0 Hece fs coverges for all σ o the secified iterval. But for some σ i the secified iterval, σ < σ c. This cotradicts the fact that σ c was the abscissa of covergece. Theorem 5.9. Give a Dirichlet series fs s, set S a. If S O β for some β > 0, the the abscissa of covergece σ c for fs is less tha or equal to β. 7

18 Proof. Suose M > N. We the have s M M N M N M N M N M N M N N S S S M N S S SN + s N s + SM M + s + SN S s dx xs+ N s + SM M + s + M+ s N Sx SN dx xs+ N s + SM M + s sice Sx S for x [, + Sx SN dx xs+ N s + SM M + s + O M+ O s dx N xσ+ β N σ β + O M σ β O s M + β σ N β σ + ON β σ + OM β σ σ β which, if σ > β, aroaches zero as N, as required. sice by assumtio S O β Lemma 5.0 Euler-Maclauri formula. Suose fx is a fuctio that is cotiuous ad differetiable for x m, where, m are ositive itegers. The Proof. m m m f f x x x dx + f + fm f x x dx + m fx dx + m f x x x dx. 8

19 [ m [ m m + xf x dx ] m + f + f f + m f m f m m m + m f x dx + fx dx fm f fx dx fx dx f fm f fx dx fm + f, m ] + m f x dx f + f fm f where at we have used itegratio by arts o the exressio iquare bracets. We reviously defied the Riema zeta fuctio for real s > as ζs By Theorem 5., we ow ow that the series for ζs coverges absolutely for all s σ + it with σ >. Theorem 5.. The Riema zeta fuctio, as defied above for σ >, ca be exteded aalytically to a aalytic fuctio i the half lae σ > 0, excet for a simle ole at s with residue. Proof. We aly the Euler-Maclauri formula above with fx x s where s. We get m s s + m s + m x s dx s. m x s x x dx. Sice x x <, the last itegral above gives m x s x x m dx O O σ O σ σ x σ dx σ, m σ so if σ > 0, this coverges absolutely ad uiformly as m. So let m i to obtai s s + x s dx s 9 x s x x dx

20 Lettig, we get ζs s s s s + s s x s x x dx. x s x x dx. Sice the last itegral coverges uiformly, by Weierstrass theorem it is a aalytic fuctio for σ > 0. Thus we have exteded ζs to σ > 0, ad it is clear that ζs, so exteded, has a simle ole at s with residue. To fiish this sectio, we shall rove a theorem for the multilicatio of two Dirichlet series, which will be eeded for our fial roof of the Dirichlet s theorem. Theorem 5. Multilicatio theorem for Dirichlet series. Give two Dirichlet series b fs gs that coverge absolutely for σ > σ 0, the we have for σ > σ 0, c fsgs where c j ajb j ab. Proof. For σ > σ 0, by assumtio the two series fs ad gs coverge absolutely. Hece multilyig the series ad rearragig the terms, aj b ajb fsgs j s s j s. Lettig j, 6 Dirichlet L-series fsgs j ajb j Defiitio 6.. The Dirichlet L-series is defied to be χ Ls, χ. Sice χ, for real σ, χ σ σ, so by Theorem 5., the series coverges absolutely for s with σ >. Firstly, we cosider the series for o-ricial characters. 0 c.

21 Theorem 6.. If χ χ, the the series for Ls, χ has abscissa of covergece σ c 0. Proof. Sice χ χ, by the roerties of Dirichlet characters Theorem 4.3, we have χ 0. Hece y χ for ay ositive iteger y. So the differece of artial sums yields y χ y x s χ x x x σ which goes to 0 as x if σ > 0. So the series has abscissa of covergece at most 0 by Theorem 5.5. But if σ < 0, each term i the series does ot ted to 0, so the series diverges for σ < 0. By Theorem 5.5 agai, the abscissa of covergece is recisely 0. Recall that the Dirichlet characters are comletely multilicative fuctios. So for σ >, we may aly Corollary 3.3 the Euler Product to write the Dirichlet L-series as Ls, χ χ s. rime Let us ow tur our attetio to the L-series for the ricial character χ : Theorem 6.3. Ls, χ is a aalytic fuctio for σ > 0, excet for a simle ole at s with residue ϕ/. Proof. Recallig that χ if gcd, ad χ 0 otherwise, Ls, χ χ s rime χ s s χ s s s ζs s. The secod factor above has fiitely may terms ad each term is aalytic, so the whole factor is aalytic. By Theorem 5., ζs has a simle ole at s with residue, so Ls, χ has a simle ole at s with residue, which is equal to ϕ by Theorem.8.

22 We are ow ready to rove the Dirichlet theorem. We follow Euler s idea for the divergece of the sum of the recirocals of rimes see roof of 3.5. For us, this ivolves the logarithm of the L-series, which is comlex-valued i geeral. We choose the brach such that for s real ad σ >, log Ls, χ is real. For σ >, usig the Euler roduct for Ls, χ, log Ls, χ log χ s log χ s χ s sice χ s < χ s + Rs where Rs χ. Now s Rs sice ζσ is bouded at σ. σ σ σ σ σ σ ζσ < as s + σ We retur to cosiderig the first sum i. We isolate the terms i the first sum where a mod. Sice a ad are corime, we ca fid b such that ab mod. Now multily by χb ad sum over all characters to get χb log Ls, χ χ χ χb χ s + χ s χb + R s χ : b mod ϕ : amod χbrs s ϕ + R s s + R s where R s : χ χbrs I the above, R s is bouded as s + sice Rs is. Each summad where χ χ o the left is bouded as s +, sice by Theorem 6., Ls, χ is aalytic as s +. The summad where χ χ goes to ifiity as s + by Theorem 6.3. Thus if each of L, χ 0 for χ χ, we would ot have a situatio o the left had side The the left had side would go to ifiity as

23 s +, so also would the right had side. Sice R s would remai bouded, we would obtai as s +, i.e. the series : amod : amod s diverges. This would show that there are ifiitely may rimes such that a mod, i.e. there are ifiitely may rimes i the arithmetic rogressio a, a +, a +,.... So let us show that L, χ 0 for χ χ : we show that this is the case searately for comlex ad real characters. Theorem 6.4. L, χ 0 for comlex characters χ χ. Proof. Let the characters cosidered be modulo. Cosider P σ χ Lσ, χ, which for σ > gives log P σ χ χ log Lσ, χ log χ σ usig the Euler roduct for Ls, χ χ 0, χ σ σ χ mod χ ϕ σ by 4.3 vi so lim if σ P σ. Suose that there is a χ such that L, χ 0. The L, χ L, χ 0, where χ is the cojugate character χχ χ, ot equal to χ sice χ χ. Also, Ls, χ has a ole at s by 6.3. So the roduct P of all the characters at s is zero, sice the oe sigle ole does ot cacel the two zeroes or ay other ossible zeroes. This cotradicts. Hece L, χ 0. Theorem 6.5. L, χ 0 for real characters χ χ. 3

24 Proof. Let the characters cosidered be modulo. Cosider the fuctio f d χd, which is multilicative as it is the Dirichlet covolutio of two multilicative fuctios, χ ad. Write α... α ad cosider the values of the fuctio at owers of these rimes. Set If i, the If i ad α i is odd, the If i ad α i is eve, the ψi α i m0 χ m i. ψi + χ i + χ i + + χ i α i α i ψi l+ 0 ψi l Thus f 0 for all. If m is a square, the each α i is eve, so fm. Thus F σ : f σ m ζσ mσ so F σ diverges at σ /. Hece the abscissa of covergece σ c /. Sice f 0 we may aly Ladau s theorem Theorem 5.8 to coclude that F s must have a sigularity at s σ c /. But for σ >, Ls, χ ad ζs coverge absolutely, so by Theorem 5., we have Ls, χζs m χm m s χd F s. Both Ls, χ ad ζs are aalytic for σ > 0 excet for the simle ole at s for ζs. If L, χ 0, the F s would be aalytic for σ > 0, ad thus caot have a sigularity at s σ c /. This is a cotradictio, so L, χ 0. f Havig established that L, χ 0 for χ χ, we have ow roved Dirichlet s theorem. d 4

25 7 Fial words After Dirichlet s theorem, more geeralisatios about rimes i arithmetic rogressios were made. We state some results obtaied by later mathematicias: Theorem 7. Lii 944. Suose that a ad d are corime, ad that a < d. Let a, d be the smallest rime i the arithmetic rogressio a + d. The there exist ositive costats c ad L such that a, d < cd L. Aother atural questio to as is: if each such arithmetic rogressio cotais ifiitely may rimes, what about cosecutive terms i the rogressio? How may cosecutive oes ca be rime? As a examle, it was foud that is rime for 0,,...,. It has aaretly bee roved i 004 by Gree ad Tao that there ca be arbitrarily cosecutively may, which has widely bee believed to be true for a log time. Theorem 7. Gree & Tao 004. The rime umbers cotai ifiitely may arithmetic rogressios of legth for all. A eve more rigid tye is cosecutive rimes i a arithmetic rogressio, that is, every umber i betwee the rimes is comosite. For examle, is such a sequece, for 0,,..., These geeralisatios show that eve though the umber of rimes thi out as the umbers grow larger, rime umbers are quite dese i the itegers. The famous rime umber theorem, roved i 896, describes the aroximate asymtotic distributio of rime umbers. It states that πx x log x, where πx is the umber of rimes ot exceedig x, ad ax bx meas lim x ax/bx. 5