Chapter 2 Elementary Prime Number Theory for

Transcription

1 Chapter 2 Elemetary Prime Number Theory for [5 lectures] I keepig with the elemetary theme of the title I will attempt to keep away from complex variables. Recall that i Chapter we proved the ifiitude of primes by relatig p /pσ to ζ(σ) for σ >. From the Euler Product, we formally get (so ot worryig about covergece), logζ(σ) = p log ( p ) σ for σ >. We will equate the derivatives of both sides, usig the logarithmic derivative alog with d dσ logf(σ) = f (σ) f(σ) = f f (σ), d dσp = d σ dσ e σlogp = logp p. σ The, because the resultig sum, () below, coverges uiformly for σ +δ, for ay δ > 0 (see Backgroud: Complex Aalysis II for a discussio of uiform covergece) we ca differetiate term-by term to get ζ ζ (σ) = d dσ logζ(σ) = p d ( dσ log p ) σ = p = p ( p σ ) d ( p ) dσ σ logp ( ) p. () σ p σ (I the Appedix we show this series coverges uiformly for σ + δ). Expad ( /p σ ) as a geometric series to get ζ ζ (σ) = p logp p σ k 0 p kσ = p r logp p rσ,

2 o relabellig k + as r. The right had side here is a double sum. See the Backgroud: Product of Series otes to see that whe the double sum is absolutely coverget, as it is i this case, the it ca be rearraged i ay way ad the resultig series will coverge to the same value. I particular, we ca write out the right had side startig as 0 s + log2 2 s + log3 3 s + log2 4 s + log5 5 s s + log7 7 s + log2 8 s + log3 9 s s +... This o-rigorous itroductio is simply to motivate the followig defiitio. Defiitio 2. vo Magoldt s fuctio is defied by { logp if = p r, Λ() = 0 otherwise. The the above argumet cocludes with ζ ζ (σ) = for σ +δ for ay δ > 0, i.e. σ >. = Λ() σ, NoteItheextchapterwewillshowthisequalityholdswithreal σ replaced by complex s for Res >. Defiitio 2.2 A Dirichlet Series is a sum of the form = a s, for some sequece {a } of complex umbers, where s C. Aside Give a sequece {a } the associated Dirichlet Series may ot coverge for ay s C. If it does coverge for some s the it will do so i some half-plae Res > c (which may be the whole of C, i.e. c = ). We ca also look at absolute covergece; agai if it coverges absolutely at some poit the it will do so i some half-plae Res > c a. Sice absolute covergece implies covergece we have c c a. It ca be show that 0 c a c. Ed of Aside 2

3 Example 2.3 ζ(s) ad ζ (s)/ζ(s) for Res > are Dirichlet Series. Notatio For a iteger > ad prime p, the p a meas that a is the largest power of p that divide p. For example, if = = the Note that , ad log = 2log2+4log5+2log3 = a geeral form of which will be see i the ext proof. p a alogp, This otatio allows a efficiet way of writig a iteger i terms of its prime divisors as = p a p a The basic ad importat property of vo Magoldt s Λ is Theorem 2.4 For Λ(d) = log. (2) d Proof If = both sides of (2) are zero. If > the Λ(d) = p Λ(p r ), r d which simply meas that we have excluded the terms with d ot a prime power, for i such cases Λ(d) = 0. Yet o prime powers Λ(p r ) = logp so logp. p r Λ(p r ) = p r Observe this is really a double sum, over p ad r. Write = p a p a, 3

4 asaproductofdistictprimes. Thep r if, adolyif, p a ad r a. I which case logp = ( ) logp = p r p a r a p alogp a = logp a = log p a p p a = log. a Combie all these steps to get the stated result. Notatio 2.5 The summatory fuctio of the vo Magoldt fuctio is deoted by ψ(x) = xλ(). Deote the sum over the logarithm of primes by θ(x) = p x logp, ad the uweighted sum by π(x) = p x. Ladau s big O-otatio If f(x), g(x) ad h(x) are fuctios the we write f(x) = O(h(x)) if there exists C > 0 such that f(x) < Ch(x). The costat C is referred to as the implied costat. The otatio is exteded so that f(x) = g(x)+o(h(x)) meas there exists C > 0 such that f(x) g(x) < Ch(x). It is further exteded so that f (x) g(x)+o(h(x)) meas there exists a fuctio k(x) satisfyig both f(x) g(x) + k(x) ad k(x) = O(h(x)). 4

5 Viogradov s (read as less tha less tha ) otatio. f(x) g(x) meas exactly the same as f(x) = O(g(x)). If g(x) f(x) g(x) we write f(x) g(x). Little o-otatio We write f(x) = o(g(x)) iff f(x) lim x g(x) = 0. Asymptotic We write f(x) g(x) iff f(x) lim x g(x) =. Example 2.6 x = logx+o(). (3) Proof i Chapter it was show that for iteger N log(n +) N logn +. Give real x > apply this with N = [x], the iteger part of x. The N x < N + ad we deduce logx x logx+. (4) This meas, o writig E(x) = x logx, that 0 E(x). Weake this to E(x), the defiitio of E(x) = O(). Hece result. 5

6 To proceed with our ivestigatio ito prime umbers we eed a versio of this with a smaller error term. This is achieved usig the followig importat result. But first a Subtle Poit. If f is differetiable o a iterval cotaiig [a,b] it may appear obvious that f(b) f(a) = b a f (t)dt, (5) but it may ot, i fact, be true. You eed to ivoke the Fudametal Theorem of Calculus that says that if f is cotiuous o [a,b] (alteratively that f has cotiuous derivative) the (5) holds. Theorem 2.7 Abel or Partial Summatio (Cotiuous Versio) Let g : N C ad set G(x) = x g(). Let f : R >0 C have a cotiuous derivative o x > 0. The g()f() = f(x)g(x) G(t)f (t)dt. x Note We sometimes write this as g()f() = f(x)g(x) x G(t)df(t). Proof The proof is a exercise i the iterchage of a fiite sum ad a fiite itegral. Start with the simple observatio that, sice f has a cotiuous derivative, f() = f (x) ( f (x) f() ) = f(x) f (t)dt. The, multiplyig by g() ad summig over x gives g()f() = ( ) g() f(x) f (t)dt x x = f(x)g(x) x The secod term here ca be writte as g()f (t)dt. x t 6 g() f (t)dt.

7 Fiite itegrals ad sums ca be iterchaged, with the restrictio o the itegral of t reiterpreted as a coditio o the sum of t. This gives x }{{} t g()f (t)dt = x }{{} t g()f (t)dt = f (t) t g()dt as required. = G(t)f (t)dt, A importat Special Case is whe g() = for all. Notatio For real x defie [x], the iteger part of x, to be the largest iteger x. Defie {x} = x [x], the fractioal part of x. This satisfies 0 {x} < for all real x. Thus, i the otatio of the previous Theorem, G(x) = x = [x]. We ca ow state a fudametal result o approximatig sums by itegrals. Propositio 2.8 Euler Summatio Let f have a cotiuous derivative o x > 0. The x f() = for all real x. f(t)dt+f () {x}f(x)+ {t}f (t)dt Notes i) If x = N is a iteger the the {N}f (N) term is zero. So we ca use the propositio to prove results valid for all real x ad improved results for itegral x. ii) We have f() o both sides of this result, so it could have bee writte as 2 x f() = f(t)dt {x}f(x)+ {t}f (t)dt, but there is a dager that if this was doe, you would ot have oticed that the left had side was a sum oly over 2 x, ot x. 7

8 Proof By the result above f() = f(x)[x] x = f(x)[x] = f(x)[x] [t]f (t)dt (t {t})f (t)dt tf (t)dt+ We itegrate the secod itegral by parts to get tf (t)dt = [tf(t)] x f(t)dt = f(x)x f() Substitutig back i we get f() = f(x)[x] f(x)x+f()+ {t}f (t)dt. f(t)dt. f(t)dt+ x {t}f (t)dt = f(t)dt+f() {x}f(x)+ {t}f (t)dt. To see the stregth of Propositio 2.8 we improve (3), Theorem 2.9 There exists a costat γ such that ( ) = logx+γ +O, x for real x >. x Note how the estimate o the error here is best possible (i.e. you could ot replace it by a faster dimiishig fuctio of x). This is because as x varies by a miuscule amout from just below a iteger to just above it, the left had chages by /, yet the mai terms logx+γ chage imperceptibly (beig cotiuous i x), while it is the error term O(/x) which exactly matches the chage i the left had side. 8

9 Proof From above with f(x) = /x we have x = dt {x} + t x x The secod itegral coverges absolutely sice {t} dt t 2 dt t 2. {t} t 2 dt. Thus we ca complete the itegral up to, the error i doig so is Combiig, x Hece the result follows with x {t} dt t 2 = logx++o x ( ) x dt t 2 x. {t} dt t 2. γ = {t} dt t 2. Fudametal Idea. This method of completig a coverget itegral up to ifiity ad the boudig the tail ed is ofte used ad should be remembered. The costat γ is the called Euler s costat or sometimes the Euler- Mascheroi costat. Reiterpreted, Theorem 2.9 says ( ) γ = lim x logx. x This ca be used to calculate γ though the speed of covergece is very slow. Numerically γ It is ot kow if γ is irratioal! If we assume more about the fuctio f we ca state a very useful versio of Euler s summatio. Useful i that it easily allows a sum to be replaced by a itegral. 9

10 Corollary 2.0 If f has a cotiuous derivative o x > 0, is o-egative ad mootoic the f() = x f(t)dt+o(max(f(),f(x))), (6) for all real x. Proof Sice f is mootoic its derivative f (x) is of costat sig. Thus {t}f (t)dt {t}f (t) dt = f (t) dt sice {t} f (t)dt sice f (t) is of costat sig = f(x) f(). Hece, by the triagle iequality applied twice f() {x}f(x)+ {t}f (t)dt f() + {x}f(x) + f()+f(x)+ f(x) f() f()+f(x)+f(x)+f() 4max(f(),f(x)). {t}f (t)dt I the last lies of this proof we have used a+b 2max(a,b) for a,b > 0. Make sure you believe this. 0

11 A immediate applicatio of Corollary 2.0 is Example 2. Choose f(x) = logx to deduce log = xlogx x+o(logx) for real x >. x Agai we have the best possible error term for real x. We ca though do better whe x = N a iteger ad i a umber of Problem Sheet Questios we look at improvig ad geeralisig this result o the sum of logarithms. The iterest comes from the fact that N log = logn! ad we thus get boud o N!.