Primes in Arithmetic Progressions

Transcription

1 2 Primes i Arithmetic Progressios I 837 Dirichlet proved by a igeious aalytic method that there are ifiitely may primes i the arithmetic progressio a, a + q, a +2q, a +3q,... i which a ad q have o commo factor ad q is prime. The geeral case, for arbitrary q, was completed oly later by him, i 840, whe he had fiished provig his celebrated class umber formula. I fact, may are of the view that the subject of aalytic umber theory begis with these two papers. It is also accurate to say that character theory of fiite abelia groups begis here. I this chapter we will derive Dirichlet s theorem, ot exactly followig his approach, but at least iitially tracig his ispiratio. 2. Summatio Techiques A very useful result is the followig. Theorem 2.. Suppose {a } = is a sequece of complex umbers ad f(t is a cotiuously differetiable fuctio o [,x].set A(t = a. t

2 8 2. Primes i Arithmetic Progressios The a f( =A(xf(x x A(tf (tdt. Proof. First, suppose x is a atural umber. We write the left-had side as a f( = {A( A( }f( = A(f( A(f( + = A(xf(x + A( f (tdt = A(xf(x sice A(t is a step fuctio. Also, + A(tf (tdt = + x A(tf (tdt, A(tf (tdt, ad we have proved the result if x is a iteger. If x is ot a iteger, write [x] for the greatest iteger less tha or equal to x, ad observe that A(x{f(x f([x]} x [x] A(tf (tdt =0, which completes the proof. Remark. Theorem 2.. is ofte referred to as partial summatio. Exercise 2..2 Show that log = x log x x + O(log x. Exercise 2..3 Show that =logx + O(.

3 I fact, show that ( lim x log x 2. Summatio Techiques 9 exists. (The limit is deoted by γ ad called Euler s costat. Exercise 2..4 Let d( deote the umber of divisors of a atural umber. Show that d( =x log x + O(x. Exercise 2..5 Suppose A(x =O(x δ. Show that for s>δ, = a s = s A(t dt. ts+ Hece the Dirichlet series coverges for s>δ. Exercise 2..6 Show that for s>, ζ(s = s s s {x} dx, xs+ where {x} = x [x]. Deduce that lim s +(s ζ(s =. Cosider the sequece {b r (x} r=0 of polyomials defied recursively as follows: 0 b 0 (x =, b r(x = rb r (x (r, b r (xdx = 0 (r. Thus, from the peultimate equatio, b r (x is obtaied by itegratig rb r (x, ad the costat of itegratio is determied from the last coditio. Exercise 2..7 Prove that F (x, t = r=0 b r (x tr r! = text e t.

4 20 2. Primes i Arithmetic Progressios It is easy to see that b 0 (x =, b (x =x 2, b 2 (x =x 2 x 6, b 3 (x =x x2 + 2 x, b 4 (x =x 4 2x 3 + x 2 30, b 5 (x =x x x3 6 x. These are called the Beroulli polyomials. Oe defies the rth Beroulli fuctio B r (x as the periodic fuctio that coicides with b r (x o [0,. The umber B r := B r (0 is called the rth Beroulli umber. Notethatifwedeoteby{x} the quatity x [x], B r (x =b r ({x}. Exercise 2..8 Show that B 2r+ =0for r. The Beroulli polyomials are useful i derivig the Euler - Maclauri summatio formula (Theorem 2..9 below. Let a, b Z. We will use the Stieltjes itegral with respect to the measure d[t]. The a< b f( = b a f(td[t]. Notice that the iterval of summatio is a< b, so that a< b f( = b a f(tdt b a f(tdb (t because d[t] =dt d{t} ad B (t ={t} 2, by the theory of the Stieltjes itegral. We ca evaluate the last itegral by parts: b a f(tdb (t =(f(b f(ab b a B (tf (tdt,

5 2. Summatio Techiques 2 sice B (b =B (a =B (0. From B 2 (t =2B (t, we ca write b f(tdb (t =(f(b f(ab f (tdb 2 (t, a 2! a provided that f is differetiable o [a, b]. We ca iterate this procedure to deduce he followig theorem: Theorem 2..9 (Euler-Maclauri summatio formula Let k be a oegative iteger ad f be (k +times differetiable o [a, b] with a, b Z. The a< b f( = b a f(tdt + + ( k (k +! k r=0 b Example 2..0 For itegers x, 2 a b ( r+ (r +! (f (r (b f (r (ab r+ B k+ (tf (k+ (tdt. =logx + γ + 2x + 2x 2 + O ( x 3. Solutio. Put f(t =/t i Theorem 2..9, a =, b = x, adk =2. The =logx + ( 2 x + ( x 2 x 2 B 3 (t t 4 dt, so that Sice we must have Also, =logx + 2 x 2 B 3 (t t 4 dt + 2x 2x 2. γ = lim x log x, γ = 2 2 B 3 (tdt t 4. x B 3 (tdt t 4 so that the result is ow immediate. ( = O x 3,

6 22 2. Primes i Arithmetic Progressios Exercise 2.. Show that for some costat B, =2 ( x + B + O x. Exercise 2..2 For z C, ad arg z π δ, where δ>0, show that log(z + j = j=0 ( z + + log(z + 2 ( z B (xdx log z z + x. 2.2 Characters mod q Cosider the group (Z/qZ of coprime residue classes mod q. A homomorphism χ :(Z/qZ C ito the multiplicative group of complex umbers is called a character (mod q. Sice (Z/qZ has order ϕ(q, the by Euler s theorem we have a ϕ(q (mod q, ad so we must have χ ϕ(q (a =for all a (Z/qZ. Thus χ(a must be a ϕ(qth root of uity. We exted the defiitio of χ to all atural umbers by settig χ( = { χ( (mod q if (, q =, 0 otherwise. Exercise 2.2. Prove that χ is a completely multiplicative fuctio. We ow defie the L-series, L(s, χ = = χ( s. Sice χ(, the series is absolutely coverget for Re(s >.

7 Exercise Prove that for Re(s >, L(s, χ = p where the product is over prime umbers p. The character ( χ(p p s, χ 0 :(Z/qZ C 2.2 Characters mod q 23 satisfyig χ 0 (a =for all (a, q =is called the trivial character. Moreover, if χ ad ψ are characters, so is χψ, as well as χ defied by χ(a =χ(a, which is clearly a homomorphism of (Z/qZ. Thus, the set of characters forms a group. This is a fiite group, as the value of χ(a is a ϕ(qth root of uity for (a, q =. But more ca be said. If we write q = p α pα k k as the uique factorizatio of q as a product of prime powers, the by the Chiese remaider theorem, is a isomorphism of rigs. Thus, Z/qZ i Z/p α i i Z (Z/qZ i (Z/p α i i Z. Exercise Show that (Z/pZ is cyclic if p is a prime. A elemet g that geerates (Z/pZ is called a primitive root (mod p. Exercise Let p be a odd prime. Show that (Z/p a Z is cyclic for ay a. I the previous exercise it is crucial that p is odd. For istace, (Z/8Z is ot cyclic but rather isomorphic to the Klei four-group Z/2Z Z/2Z. However, oe ca show that (Z/2 α Z is isomorphic to a direct product of a cyclic group ad a group of order 2 for α 3.

8 24 2. Primes i Arithmetic Progressios Exercise Let a 3. Show that 5 (mod 2 a has order 2 a 2. Exercise Show that (Z/2 a Z is isomorphic to (Z/2Z (Z/2 a 2 Z,fora 3. Exercise Show that the group of characters (mod q has order ϕ(q. Exercise If χ χ 0, show that χ(a =0. a(mod q Exercise Show that { ϕ(q if (modq, χ( = 0 otherwise. χ(mod q 2.3 Dirichlet s Theorem The cetral idea of Dirichlet s argumet is to show that lim s + p a(mod q p s =+, where the summatio is over primes p a (mod q. If q =,thisisclear,because ζ(s = p ( p s ad log ζ(s = p log ( p s upo usig the expressio = p ( = p s log( x = = x.

9 2.3 Dirichlet s Theorem 25 Observig that lim ζ(s =+ s + by virtue of the divergece of the harmoic series, we get Cosequetly, ( lim s + p lim log ζ(s =+. s + p s + p 2 I view of the fact for s, p s p p p p 2 2 p s =+. p(p <, we deduce lim s + p s =+. p Exercise 2.3. Let χ = χ 0 be the trivial character (mod q. Show that lim log L(s, χ 0=+. s + Exercise Show that for s>, log L(s, χ =ϕ(q χ(mod q p (mod q Exercise Show that for s> the Dirichlet series = a s := χ(mod q L(s, χ has the property that a =ad a 0 for 2. p s. Exercise For χ χ 0, a Dirichlet character (mod q, show that χ( q. Deduce that coverges for s>0. L(s, χ = = χ( s

10 26 2. Primes i Arithmetic Progressios Exercise If L(, χ 0, show that L(, χ 0, for ay character χ χ 0 mod q. Exercise Show that lim s +(s L(s, χ 0=ϕ(q/q. Exercise If L(,χ 0for every χ χ 0, deduce that ad hece lim s +(s p (mod q χ(mod q L(s, χ 0 p =+. Coclude that there are ifiitely may primes p (mod q. This exercise shows that the essetial step i establishig the ifiitude of primes cogruet to (mod q is the ovaishig of L(, χ. The exercise below establishes the same for other progressios (mod q. Exercise Fix (a, q =. Show that { ϕ(q if a (mod q, χ(aχ( = 0 otherwise. χ(mod q Exercise Fix (a, q =. If L(,χ 0, show that Deduce that lim s +(s χ(mod q p a(mod q L(s, χ χ(a 0. p =+. The essetial thig ow is to show that L(,χ 0 for χ χ 0. Historically, this was a difficult step to surmout. Now, there are may ways to establish this. We will take the most expediet route. We will exploit the fact that F (s := L(s, χ χ(mod q

11 2.4 Dirichlet s Hyperbola Method 27 is a Dirichlet series = a s with a =ad a 0. If for some χ, L(,χ =0, we wat to establish a cotradictio. Exercise Suppose χ χ (that is, χ is ot real-valued. Show that L(,χ 0by cosiderig F (s. It remais to show that L(,χ 0whe χ is real ad ot equal to χ 0. We will establish this i the ext sectio by developig a iterestig techique discovered by Dirichlet that was first developed by him ot to tackle this questio, but rather aother problem, amely the Dirichlet divisor problem. 2.4 Dirichlet s Hyperbola Method Suppose we have a arithmetical fuctio f = g h. That is, f( = g(dh(/d d for two arithmetical fuctios g ad h.defie G(x = g(, H(x = h(. Theorem 2.4. For ay y>0, ( x f( = g(dh + ( x h(dg d d d y d x y Proof. We have g(dh(e f( = de x = g(dh(e+ g(dh(e de x de x d y d>y = ( x g(dh + d d y e x y = ( x g(dh d d y + e x y ( x G(yH. y { ( x } h(e G G(y e ( x ( x h(eg G(yH. e y

12 28 2. Primes i Arithmetic Progressios The method derives its ame from the fact that the iequality de x is the area udereath a hyperbola. Historically, this method was first applied to the problem of estimatig the error term E(x defied as E(x = σ 0 ( {x(log x+(2γ x}, where σ 0 ( is the umber of divisors of ad γ is Euler s costat. Exercise Prove that σ 0 ( =x log x +(2γ x + O( x. Exercise Let χ be a real character (mod q. Defie f( = χ(d. d Show that f( = ad f( 0. I additio, show that f( wheever is a perfect square. Exercise Usig Dirichlet s hyperbola method, show that where f( = d χ(d ad χ χ 0. f( =2L(,χ x + O(, Exercise If χ χ 0 is a real character, deduce from the previous exercise that L(,χ 0. Exercise Prove that >x χ( = O ( x wheever χ is a otrivial character (mod q.

13 2.5 Supplemetary Problems 29 Exercise Let a = d χ(d where χ is a opricipal character (mod q. Show that a = xl(,χ+o( x. Exercise Deduce from the previous exercise that L(,χ 0for χ real. Thus, we have proved the followig Theorem: Theorem (Dirichlet For ay atural umber q, ad a coprime residue class a (mod q, there are ifiitely may primes p a (mod q. 2.5 Supplemetary Problems Exercise 2.5. Let d k ( be the umber of ways of writig as a product of k umbers. Show that d k ( = for every atural umber k 2. x(log xk (k! + O(x(log x k 2 Exercise Show that log x = x + O(log x. Exercise Let A(x = a. Show that for x a positive iteger, a log x x = A(tdt. t Exercise Let {x} deote the fractioal part of x. Show that { x =( γx + O(x } /2, where γ is Euler s costat.

14 30 2. Primes i Arithmetic Progressios Exercise Prove that log k x = O(x for ay k>0. 3 Exercise Show that for x 3, ( log =loglogx + B + O. x log x Exercise Let χ be a opricipal character (mod q. Show that χ( ( = O x. x Exercise For ay iteger k 0, show that log k = logk+ x k + + O(. Exercise Letd( be the umber of divisors of. Show that for some costat c, for x. d( = ( 2 log2 x +2γlog x + c + O x Exercise Let α 0 ad suppose a = O( α ad A(x := a = O(x δ for some fixed δ<. Defie b = a d. d Prove that for some costat c. ( b = cx + O x ( δ(+α/(2 δ,

15 2.5 Supplemetary Problems 3 Exercise 2.5. Let χ be a otrivial character (mod q ad set f( = χ(d. d Show that f( =xl(,χ+o(q x, where the costat implied is idepedet of q. Exercise Suppose that a 0 ad that for some δ>0, we have a x (log x δ. Let b be defied by the formal Dirichlet series = b ( s = a 2. s = Show that b x(log x 2δ. Exercise Let {a } be a sequece of oegative umbers. Show that there exists σ 0 R (possibly ifiite such that f(s = coverges for Re(s >σ 0 ad diverges for Re(s <σ 0. Moreover, show that the series coverges uiformly i Re(s σ 0 + δ for ay δ>0ad that f (k (s =( k a (log k for Re(s > σ 0 (σ 0 is called the abscissa of covergece of the Dirichlet series = a / s. = = a s s

16 32 2. Primes i Arithmetic Progressios Exercise (Ladau s theorem Let a 0 be a sequece of oegative umbers. Let σ 0 be the abscissa of covergece of f(s = = a s. Show that s = σ 0 is a sigular poit of f(s (that is, f(s caot be exteded to defie a aalytic fuctio at s = s 0. Exercise Let χ be a otrivial character (mod q ad defie σ a,χ = d χ(dd a. If χ,χ 2 are two characters (mod q, prove that for a, b C, σ a,χ (σ b,χ2 ( s = = ζ(sl(s a, χ L(s b, χ 2 L(s a b, χ χ 2. L(2s a b, χ χ 2 as formal Dirichlet series. Exercise Let χ be a otrivial character (mod q.seta = b, χ = χ ad χ 2 = χ i the previous exercise to deduce that = σ a,χ ( 2 s = ζ(sl(s a, χl(s a, χl(s a a, χ 0 L(2s a a, χ 0 Exercise Usig Ladau s theorem ad the previous exercise, show that L(,χ 0for ay o-trivial real character (mod q. Exercise Show that ζ(s 0for Re(s >. Exercise (Ladau s theorem for itegrals Let A(x be right cotiuous for x ad of bouded fiite variatio o each fiite iterval. Suppose that A(x f(s = dx, xs+ with A(x 0. Let σ 0 be the ifimum of all real s for which the itegral coverges. Show that f(s has a sigularity at s = σ 0.

17 2.5 Supplemetary Problems 33 Exercise Let λ deote Liouville s fuctio ad set S(x = λ(. Show that if S(x is of costat sig for all x sufficietly large, the ζ(s 0 for Re(s > 2. (The hypothesis is a old cojecture of Pólya. It was show by Haselgrove i 958 that S(x chages sig ifiitely ofte. Exercise Prove that b (x = k=0 ( B k x k, k where b (x is the th Beroulli polyomial ad B deotes the th Beroulli umber. Exercise Prove that b ( x =( b (x, where b (x deotes the th Beroulli polyomial. Exercise Let s k ( = k +2 k +3 k + +( k. Prove that for k, (k +s k ( = k ( k + B i k+i i. i i=0

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