1 I.C) THE DISTRIBUTION OF PRIMES I the last sectio we showed via a Euclid-ispired, algebraic argumet that there are ifiitely may primes of the form p = 4 i.e ). I fact, this is true for primes of the form 4 + as well, ad the ratio of primes of these two forms less tha N teds to as N. We say that the primes are distributed asymptotically equally betwee 4 + N} ad 4 N}. More geerally, takig P N to deote the primes, N a,b := a + b Z} N, ad P a,b := N a,b P, there is the famous theorem o primes i arithmetic progressios: Theorem Dirichlet, 837). Give a, b N such that a, b) =, the set P a,b is ifiite. Moreover, for each fixed b, the primes are distributed asymptotically equally betwee the P a,b } 0 < a < b. a, b) = I the early 0th cetury, people bega to otice that the N a,b cotaied cosecutive sequeces of primes, e.g. N 3,4 3, 7, } [legth 3] ) N 7,30 7, 37, 67, 97, 7, 57} [legth 6] 909) N 99,0 99, 409,...} [legth 0] 90) A sequece of legth was t foud util 999; the logest kow today has legth 6 ad begis with a 6-digit umber). I light of this, the theoretical result is impressive: Theorem Gree ad Tao, 004). Give ay k, there exist a ad b such that k cosecutive elemets of N a,b are prime.
2 MATH I.C) Oe questio which may bug you for istace, i relatio to the sequeces )) is: How do you kow if a umber N is prime? Naively, it s eough to check that o umber N divides N, but we will fid better methods later. A secod questio is: How ca oe costruct primes? There is o ice aswer here o kow fuctio which produces distict primes ad oly primes). There are may other logstadig riddles regardig the primes: for example, Cojecture 3 Goldbach). Ay eve umber is the sum of two primes. This is kow up to 8 digits but ot proved i geeral. A famous result of Viogradov from the 930s says that ay sufficietly large odd umber is the sum of 3 primes.) Alteratively, oe might try to go further tha Theorem ad ask whether, give ay k ad b with b eve), there exist ifiitely may sequeces m + b, m + b,..., m + kb} cosistig etirely of primes. Takig k = yields the veerable Cojecture 4 de Poligac, 849). Give ay eve b N, there exist ifiitely may pairs p, q P with p q = b. The case b = is kow as the twi prime cojecture. A spectacular ad uexpected recet advace is: Theorem 5 Zhag, 03). Cojecture 4 holds for some b < 70, 000, 000. Recet work has brought this upper boud dow to 46, but for the momet, the twi prime cojecture remais ope. I the exercises, you will verify that o polyomial fuctio ca possibly do this.
3 MATH I.C) 3 Dirichlet s L-fuctios. We ow tur to the idea behid the proof of Dirichlet s theorem i the special case b = 4), startig with Euler s aalytic approach to the ifiitude of primes. What follows is far from beig rigorous. Let s >, ad cosider the Euler product p s = + p s + p s + p 3s + where we have expaded each factor p s ) as a geometric series o the right. If we formally expad the right-had product, the by the Fudametal Theorem of Arithmetic, each s = p a pa k p a s p a ks k occurs exactly oce i the result, yieldig = + s + 3 s + 4 s + 5 s + 6 s + = s =: ζs), the Riema zeta fuctio. Remark 6. The series coverges for s > i R, by the itegral compariso test ˆ [ s < dx x x s = s+ ] = s + s, ) k ) s = ad more geerally for Res) > i the complex umbers C. Oe may aalytically cotiue it to get a aalytic fuctio o C\} with a simple pole at ). Now what could some aalytic fuctio have to do with the distributio of primes? Quite a bit: to begi with, formally takig the limit of the above as s + gives p = =, which proves the ifiitude of primes. 3 because there exists a uique prime factorizatio of each N 3 I mathematics, formally ofte meas maipulatig symbols, which is about as far from rigor as oe gets ad is great for producig or coveyig ideas but also
4 4 MATH I.C) The idea of Dirichlet s proof is to refie this observatio. Let 0, eve χ 0 ) :=, odd ad you should check that 0, eve χ ) :=, = 4k +, = 4k + 3 ; ) χ i m) = χ i m)χ i ) for m, N ad i = 0, ). We ow carry out a aalogue of the above argumet, but i reverse, startig with the Dirichlet series or L-fuctio) χ Lχ i, s) := i ) s which usig the Fudametal Theorem ad ) becomes = k 0 χ i p k ) p ks ) 3) = Takig log of 3) yields log Lχ i, s) = = χ. ip) log k 0 which usig log x) = k x k k becomes = k χ i p k ) χ kp ks = i p) χ ip) ) ) χp) k ), χ + i p k ) k kp ks. }} =: f i s) ca be terrifically misleadig). For a proof without the quote marks, see the ext subsectio.
5 We ca boud this last term for i = 0 or ) by f i s) which for s is Fially, usig ad we have 4) ad 5) k MATH I.C) 5 kp ks χ 0 ) + χ ) χ 0 ) χ ) k p s s 4. = = ) k =, = 4k + 0, otherwise, = 4k + 3 0, otherwise, log Lχ 0, s) + log Lχ, s)) = f 0 + f + p = 4k + p s p s } } A) log Lχ 0, s) log Lχ, s)) = f 0 f + p = 4k + 3. } } B) Sice Lχ, ) = χ ) coverges by the alteratig series test with ozero limit π 4 ), oly the log Lχ 0, s) ad p terms of 4) ad 5) diverge as s +. It follows that A) ad B) diverge at the same rate. This proves that there are ifiitely may primes of the form 4k +, ad suggests that they are distributed asymptotically equally to those of the form 4k + 3.
6 6 MATH I.C) The ifiitude of primes. Fially, we shall describe oe way of makig Euler s argumet above completely airtight, which has the added bous of puttig a lower boud o partial sums of iverse primes. Lemma 7. e x+x x for x [0, ]. I particular, e p + p p each prime p.) for Proof. It suffices to show that x)e x+x. The left-had side of this is at x = 0 ad has derivative x x)e x+x 0 for x [0, ]. Theorem 8. For ay real umber y >, p > loglog y). Corollary 9. p diverges. I particular, there are ifiitely may primes.) Proof of Theorem 8. Give y >, set N y := N = p a pa k k, all p i y },
7 MATH I.C) 7 ad deote the greatest iteger less tha or equal to y by y. Now usig the lemma together with the Fudametal Theorem, we fid e p + P Takig log of both sides, = p + p + p + p 3 + ) = N y y ˆ + y dx = test) x > logy). log log y < log = < p y prime = log + y ) e p + p ) p + ) p p + =, which by the itegral test is < ˆ p + dx x }. } < I the ext sectio, we will study the fuctio πx) := umber of primes less tha or equal to x
8 8 MATH I.C) o R + = 0, ). As a prelimiary step, we ca push Theorem 8 a bit further to get Corollary 0. For x >, πx) x + ˆ x πu) du > loglog x). u Proof. Write πu) = χ [p, ) u), where for ay subset S R,, u S χ S u) := 0, u / S is the characteristic fuctio. The we have 4 as desired. ˆ x πu) u du = = = = = ˆ x ˆ x p χ [p, ) u) du du u u [ ] x u p p p πx) x x > loglog x) πx) Thm.) x, 4 Note that oly fiitely may terms of the sum cotribute, so switchig with the itegral is permissible.