(I.D) THE PRIME NUMBER THEOREM

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1 (I.D) THE PRIME NUMBER THEOREM So far, i our discussio of the distributio of the primes, we have ot directly addressed the questio of how their desity i the atural umbers chages as oe keeps coutig. But we did at least defie the fuctio π(), which couts the umber of primes, ad you might woder or maybe how fast does it grow? is the aswer good for aythig? Though we certaily should t epect ay fireworks from obserivig that π() <, we ca at least put it together with the iequality π() + ˆ from Corollary I.C. to get that F() := ˆ It follows that the derivative 2 2 π(u) du > log(log ) u2 π(u) du log(log ) + 2 >. u2 F () = π() 2 log must have oegative lim sup. Multiplyig by log, we fid that hece lim sup lim sup π() / log π() / log, This meas that give ay egative costat, ad ay M, there eists > M such that F () eceeds this costat. (Otherwise F() would go egative.)

2 2 MATH (I.D) which is to say that there are a lot of primes. A better result is the Prime Number Theorem: Theorem (de la Vallée Poussi/Hadamard, 896). We have π() log(), i.e. lim π() / log = eactly. This was already cojectured by Gauss ad Legedre i the 79s based o umerical evidece, perhaps of the sort i the followig table: 4 5 π() π() Notice that the differeces betwee the bottom etries stabilize to roughly 2.3 log, which suggests k π( k ) k log() = log(k ), which the suggests (if you are Gauss or Legedre) Theorem. Idea of the proof. This uses comple aalysis, ad is based o the study of three fuctios: () ϕ() := p (2) Φ(s) := s ad the Riema zeta fuctio (3) ζ(s) = s = log(p) ; ϕ() d ; s+ p s } {{ } Euler product What ca (3) possibly have to do with () ad (2)?.

3 MATH (I.D) 3 We start by epressig (2) as a series: Φ(s) = [ ] (log p)d log p s p s+ = s p s s p log p = p s, which apparetly makes sese (like (3)) for comple umbers s with Re(s) >. Now use d ds p s = (log p)p s to write ζ (s) ζ(s) = d ds log ζ(s) = d ds ( log( p s ) ) = log p p s ( p s ), ad otice that by epadig p s this becomes Φ(s) + H(s) where H (which ivolves +higher) is aalytic for Re(s) > p 2s 2. The reaso why this is importat, is that ζ(s) s = ˆ s ˆ + = s d ( s ) }{{ s d } s Re(s)+ eteds to a aalytic fuctio o Re(s) >. Moreover, the Euler product coverges o Re(s) >, from which we see that ζ(s) has o zeroes there, ad a deeper aalysis shows that ζ(s) has o zeroes o (or accumulatig to) Re(s) = just the pole at s =. The upshot of this discussio is that Φ(s) eteds to a aalytic fuctio o Re(s) > 2, ecept for poles at s = ad poles at zeroes of ζ(s) (with Re(s) < ɛ). I particular, we coclude from this that the fuctio g(s) = ϕ() s+ d = Φ(s) s s is aalytic i a eighborhood of s =. The itegral oly coverges a priori for Re(s) >, but a deep Tauberia theorem i comple aalysis shows that sice (amog other thigs) g eteds through, the itegral actually does coverge there: i.e., we have ϕ() 2 d <,

4 4 MATH (I.D) which implies ϕ() as, hece (4) lim ϕ() =. To fiish off the Prime Number Theorem, write (5) ϕ() = p log(p) p log() = π() log() ad (for < y < ) (6) π() = π(y) + p (y, ] π(y) + log p log < y + ϕ() log y. p (y,] Takig y = ϕ() log, multiplyig (6) by log 2 ad (5) by, gives ( < π() log < log 2 + ) ϕ() log log 2 log log = log + ϕ() log log 2 log log. By (4), the ed terms of this iequality limit to as ; therefore π() log also so that the beast is, i the ed, tamed by the squeeze theorem from Calculus. I fact we kow more about ζ(s): it eteds to a aalytic fuctio o all of C (ecept for the pole at ), with zeroes: at egative eve itegers; ad i the critical strip < Re(s) < these are called the critical zeroes. For a easy $,,, you should prove Cojecture 2 (The Riema Hypothesis). The critical zeroes are all o Re(s) = 2.

5 MATH (I.D) 5 The PNT was proved usig some of what we kow about ζ(s). Oe might epect that a eve better result would follow from Cojecture 2. I fact, the fuctio Li() := is kow to do a better job tha ˆ log() the better result would use this istead: dt log t at approimatig π(), ad Theorem 3 (Schoefield, 976). If the Riema Hypothesis holds, the for all π() Li() log() 8π There are may other cosequeces: to metio just oe more, recall that Propositio I.B.9 says that there eist gaps of arbitrary legth betwee the primes. O the other had, oe may have to look at very large umbers just to get a small gap. If the RH holds, the (accordig to a result of Cramér) we ca make this last statemet very precise: the gap betwee prime p ad the et prime is bouded by a costat times p log(p). A applicatio of the Prime Number Theorem. Here s the what is it good for part: we ll use the PNT to prove that ζ(3) = 3 is irratioal. Set d := lcm{, 2, 3,..., }(=product of prime powers ). Lemma 4. d < 3 for sufficietly large. Proof. To begi with, d = p p log p < p p log p = π().

6 6 MATH (I.D) Now let ɛ > be such that e +ɛ < 3. By the PNT, there eists N N such that N = π() < ( + ɛ) log Lemma 5. For s > r N, (a) (b) ˆ ˆ ˆ ˆ = π() log < ( + ɛ) = ep π() < e (+ɛ) = (e +ɛ ) < 3. log y y r y r d dy = 2ζ(3) log y y r y s d dy d 3 Z. s Proof of (a) ((b) is similar). Write ˆ ˆ t+r y t+r y d dy = k ˆ ˆ ( r 3 ) }{{} d 3 Z r t+r+k y t+r+k d dy = k Differetiate with respect to t (usig d dt t = t log ), set t = (t+r+k+) 2. = ˆ ˆ log y y d dy = 2 k (r + k + ) 3. Now defie the Legedre polyomials P () := ( ) d ( ).! d Lemma 6. For N, ˆ ˆ log y y P ()P (y)d dy ( 2 ) 4 2ζ(3). Sketch. Notice that P ()P (y) is a sum of terms of the form r y s. By Lemma 5, the itegral equals (A d 3 + ζ(3)b ) for some A, B Z. The rest is complicated itegratio by parts ad boudig.

7 Theorem 7 (Apéry, 978). ζ(3) / Q. MATH (I.D) 7 Proof. Sice the itegral i Lemma 6 is ozero, we have for each N Therefore < A + ζ(3)b d 3 < 2ζ(3)( 2 ) 4. < A + ζ(3)b < 2ζ(3)d 3 ( 2 ) 4 < Lemma 4 2ζ(3)(33 ( 2 ) 4 }{{} < 2ζ(3)(.9). <.9 Suppose ζ(3) = Q P, for some ozero P, Q Z. The the above yields hece for large eough < A + P Q B < 2 P Q (.9) < A Q + PB < 2P(.9) <, ) which is impossible sice A Q + PB is a iteger. That 3 3 ( 2 ) 4 is less tha is a miracle. May similar wouldbe proofs (e.g., for Catala s costat) fail solely because the correspodig umber eceeds!