Notes on Dirichlet L-functions

Transcription

1 Notes o Dirichlet L-fuctios Joth Siegel Mrch 29, 24 Cotets Beroulli Numbers d Beroulli Polyomils 2 L-fuctios 5 2. Chrcters Diriclet Series Alytic Cotiutio of Prtil Zet Fuctios The Vlues of L-fuctios t No-positive Itegers Beroulli Numbers d Beroulli Polyomils Our ivestigtio of the Beroulli umbers begis with the followig problem. Give iteger, we wish to fid polyomil B (x) such tht for ll rel umbers = + B (x)dx It is mybe ot immeditely cler tht such polyomils exist but i fct they re ot tht difficult to costruct. We begi by differetitig both sides of the bove equtio with respect to d obti Assumig tht > we see tht + = B ( + ) B () B (x)dx = B ( + ) B () This implies tht B (x) = B (x). Usig i dditio the fct tht for >, = = B (x)dx we see tht give B (x), B (x) is uiquely determied by the coditios B (x) = B (x) d B (x)dx =. Combiig this with the observtio tht B (x) = we c iductively costruct the Beroulli polyomils, which re the uique polyomils stisfig the desired coditio.

2 The first few Beroulli polyomils re B (x) =, B (x) = x /2, B 2 (x) = x 2 x + /6, etc. The umertor d deomitor of the coefficiets of the Beroulli polyomils grow quite rpidly. Noetheless, the recursive reltio stified by the Beroulli polyomils mkes them reltively esy to compute. Oe iterestig property of the Beroulli polyomils is the followig. Note tht +(+) +...+(+k) = Pluggig i = we see tht +k+ B (x)dx = + (B +(+k+) B + ()) k = + (B +(k + ) B + ()) Lettig =, we obti, for exmple, the well-kow formul tht k = 2 ((k + )2 (k + ) ) = (k2 + k) 2 We c compute B 3 (x) = x 3 (3/2)x 2 + (/2)x to obti the less obvious formul k 2 = 3 ((k + )3 3 2 (k + )2 + (k(2k + )(k + )) (k + )) = 2 6 d this process c clerly be geerlized. We ow defie B () = B to be the -th Beroulli umber. Notice tht ( ) B (x) = B i x i i i= i= This c be see by iductio s follows. It is clerly true for =. Now ssume it is true for, the B +(x) = ( + )B (x) = i= ( + )( i) Bi x i. Additiolly, B + is by defiitio the costt term of B + (x). So we compute ( ) + B + (x) = B i x i+ + B + i + i + Noticig tht ) ( i+( i = + ) i we hve + ( ) + B + (x) = B i x + i i i= s desired. Now ssume tht >. The otice tht sice the itegrl of B (x) o the itervl [, ] is d B (x) = B (x) we must hve B = B () = B (). I light of the previous formul for B (x) we obti the followig reltio betwee the Beroulli umbers. ( ) B = B i i i= 2

3 or ( ) = B i i i= for >. If =, the the correspodig reltio is simply B =. This could lso be used to compute the Beroulli umbers. Our ext gol is to derive the expoetil geertig fuctio for the Beroulli umbers d i the process to prove tht B = for odd >. We proceed s follows. Defie f(t, x) = = B (t) x! Tkig derivtives with respect to t we see tht df dt = B (t) ( )! x = = = B (t) x + = xf(t, x)! By solvig this differetil equtio we see tht f(t, x) = g(x)e xt fuctio g(x). Now we hve dditiolly tht for some f(t, x)dt = = B (t) x =! for ll x, sice ll of the itegrls i bove sum re except for the first oe by the recursive reltio tht the Beroulli polyomils stisfy. This permits us to solve for g(x). Nmely we hve = g(x)e xt dt = g(x) from which we obti tht g(x) = e xt = g(x) ex x x e x. Combiig this we see tht d lettig t = we hve f(t, x) = = xetx e x B! x = x e x Now we wish to show tht B = for odd >. Sice we kow tht B! = 2 we must simply show tht f(, x)+ x 2 is eve fuctio of x. But this is simple Replcig x by x we obti x e x + x 2 = 2x + xex x 2e x = x(ex + ) 2 2(e x ) x(e x + ) 2(e x ) = x( + ex ) 2( e x ) 3 = x(ex + ) 2(e x )

4 so f(, x) + x 2 is eve fuctio d thus B = for odd > s desired. Oe lst remrk i this sectio is the followig. If we isted decided to fid polyomils such tht = +h P (x)dx for y h we could proceed s follows. Cosider (h) = h(+) h P (x)dx Now mke the chge of vribles hy = x. The we see tht so tht (h) = = + + P (hy)hdy h P (hy)dy Cosequetly, by the uiqueess of the Beroulli polyomils we must hve tht B (x) = h P (hx) or P (x) = h B ( x h ). So these polyomils c be expressed simply i terms of the Beroulli polyomils. If we let h = k for some turl umber k, the we could hve solved the precedig problem differetly, by settig P (x) = k j= B (x + j k ). Oe c esily see tht + k for ech so we must hve or P (x)dx = + B (x)dx ( k k ) B (kx) = B (x + j k ) j= k B (kx) = k B (x + j k ) To coclude the sectio o Beroulli polyomils, I will sy tht ech of the properties derived here is iterestig d importt for the further pplictio of Beroulli polyomils. Also ote tht much of this sectio c be reformulted i the lguge of lier lgebr. We cosider the vectorspce R[x] of ll polyomils with rel coefficiets d lyse the lier mp T : R[x] R[x] which p(x)dx. It is simple mtter to verify tht this is lier mp such tht deg(p(x)) = deg(t (p(x))). This implies tht T is bijectio d so hs iverse T. The B (x) = T (x ). j= seds polyomil p(x) to the polyomil h(y) = y+ y 4

5 2 L-fuctios 2. Chrcters Let A be fiite beli group d cosider the group Hom(A, C ) = A of group homomorphisms from A to the multiplictive group of complex umbers. This is clled the group of chrcters. Lemm 2. Hom(A, C ) = A Proof Sice A is fiite beli group, we kow tht A is product of cyclic groups. Sice Hom(X Y, Z) = Hom(X, Z) Hom(Y, Z) we must oly show tht the lemm holds for fiite cyclic group. So ssume tht A is geerted by x d x =. The y χ Hom(A, C ) is uiquely determied by χ(x) d χ(x) must be -th root of uity. Hece Hom(A, C ) is isomorphic to the group of -th roots of uity which is cyclic of order. This completes the proof. The ext two lemms relte the chrcters to the dditive structure of C. Lemm 2.2 χ(x) = x A { χ A χ = Proof If χ =, the sum is x A = A. Now ssume tht χ d let y A such tht χ(y). The χ(x) x A χ(x) = x A χ(yx) = χ(y) x A so ( χ(y)) x A χ(x) =. As χ(y) we must hve x A χ(x) =. Lemm 2.3 χ(x) = { x A x = Proof If x =, the sum is = A. Now ssume tht x. If χ(x) = for ll χ Hom(A, C ), the Hom(A, C ) = Hom(A/(x), C ) (here (x) is the cyclic subgroup geerted by x). But the by the first lemm of this sectio A = A/(x) which is impossible sice the groups hve differet orders. So pick ψ Hom(A, C ) such tht ψ(x). The χ(x) = ψχ(x) = ψ(x) χ(x) so ( ψ(x)) χ(x) = d sice ψ(x) we coclude tht χ(x) =. I the followig we cosider the cse where A = (Z/Z) for some positive iteger. Give chrcter χ of A we exted χ : Z C by settig χ() = if (, ). The χ stisfies χ(b) = χ()χ(b) for y two itegers, b. 5

6 2.2 Diriclet Series I this sectio we defie the mi object of study, the Diriclet L-series. Defiitio Let χ be the extesio of chrcter (Z/Z) to Z by settig χ() = if (, ) >. Defie the Dirichlet L-fuctio of the chrcter to be L(χ, s) = = χ() s The gol of the remider of this chpter will be to derive s my properties of L-fuctios s possible. We begi with very simple lemm. Lemm 2.4 The bove sum coverges bsolutely for Re(s) >. Proof This sttemet follows sice χ/ s / Re(s) d thus the bove series coverges bsolutely by compriso with = /Re(s) for Re(s) >. The bove proof lso shows tht for y δ >, the bove series coverges bsolutely d uiformly for Re(s) + δ which implies tht the series coverges to lytic fuctio i the hlf-ple Re(s) >. We will proceed to derive formul for the lytic cotiutio of L(χ, s) to the etire complex ple. I order to do this we will itroduce the prtil zet fuctios ζ(s, ; r), lyticlly cotiue them d write our origil L-series i terms of the prtil zet fuctios. 2.3 Alytic Cotiutio of Prtil Zet Fuctios I this sectio we defie d compute the lytic cotiutio of the prtil zet fuctios. Defiitio Let N d r. The we defie. ζ(s, ; r) = k= (k r )s Notice tht if = r, we obti the fmilir Riem zet fuctio. The reso why the prtil zet fuctios re useful is tht we c lyticlly cotiue them usig the Euler-Mclure summtio formul. To obti the Euler-Mclure formul, cosider smooth fuctio f : R C such tht f(x) dx < d = f() <. We wish to relte = f() to f(x)dx. To this ed, rewrite f(x)dx = = + B (x )f(x)dx d itegrte ech of the terms i the sum by prts to obti f(x)dx = B ()f( + ) B ()f() = = + B (x )f (x)dx 6

7 Now the first sum = B ()f( + ) B ()f() c be simplified sice B () = 2 d B () = 2 so tht = B ()f( + ) B ()f() = f() 2 We del with the secod term prts 2l times to obti = + = B (x )f (x)dx = = + + f() = + B (x )f (x)dx by itegrtig by l k= B 2k (2k)! f (2k ) () B 2l+ (2l + )! (x )f (2l) (x)dx This works out sice B k (x) = kb k (x) d B k () = B k () = B k for k > so the sum which shows up whe itegrtig by prts telescopes except for the first term. The odd Beroulli umbers re so these terms re omitted from the bove sum. Rerrgig the bove we obti f() = = + = f(x)dx + f() 2 + l k= B 2k (2k)! f (2k ) () B 2l+ (2l + )! (x )f (2l) (x)dx which holds for y l N. This looks like complicted formul, but it is very useful for lyticlly cotiuig fuctios which re expressed s coverget series. This is especilly true if higher derivtives of f re more well-behved th f. How this works will become cler whe pplyig this to the fuctio f(x) = to lyticlly cotiue ζ(s, ; r). We see tht for Re(s) >, )s (x r + l k= ζ(s, ; r) = +s(s + )(s + 2)...(s + 2l ) s (x r dx + )s 2r s B 2k s(s + )(s + 2)...(s + 2k 2) s+(2k ) (2k)! r s+(2k ) = + We evlute the first itegrl explicitly to obti ζ(s, ; r) = B 2l+ (2l + )! (x ) (x r )s+2l dx s l r s ( s) + s 2r s + B 2k s(s + )(s + 2)...(s + 2k 2) s+(2k ) (2k)! r s+(2k ) +s(s + )(s + 2)...(s + 2l ) k= = + 7 B 2l+ (2l + )! (x ) (x r )s+2l dx

8 The exct detils of this formul re ot importt. Wht is importt is tht the lst sum is the oly ifiite prt of the formul, i.e. everythig else c be evluted explicitly usig fiite umber of opertios. Thus everythig except the lst sum represets meromorphic fuctio (defied everywhere except for pole t s = (due to the first term). Additiolly, sice B 2l+ (x) is bouded o [, ] (it is cotiuous, beig polyomil) we hve tht the lst sum is bouded by (sup B 2l+ (x)) [,] (x r )s+2l dx This is fiite s log s Re(s) > 2l (sice q x t dx < if q > d t > ). Thus by stdrd rgumet the fil sum coverges uiformly to lytic fuctio s log s Re(s) > 2l. Sice l ws rbitrry, this formul gives the lytic cotiutio of ζ(s, ; r). 2.4 The Vlues of L-fuctios t No-positive Itegers I this sectio we use the result of the previous sectio to lyticlly cotiue Dirichlet L-series to the etire complex ple d to evlute the correspodig L-fuctios t o-positive itegers (by L-fuctio I me the lytic cotiutio of L-series). Our method for doig this will be to ote tht the L-series c be writte i terms of the prtil zet fuctios s follows L(χ, s) = k= χ(k) k s = r= ζ(s, ; r) χ(r) s where χ is chrcter of coductor, i.e. the vlue of χ depeds oly o the cogruece clss mod. The bove formul follows sice ζ(s, ; r) χ(r) s = χ(r) ((k ) + r) s = χ(k) k s k= k r() Now we see tht the lytic cotiutio of the prtil zet fuctios derived i the precedig sectio provide the lytic cotiutio of L-series. Moreover, the vlues of the L-fuctios t o-positive itegers c be determied from the formul for the cotiutio of the prtil zet fuctios. We recll from lst sectio tht ζ(s, ; r) = s l r s ( s) + s 2r s + B 2k s(s + )(s + 2)...(s + 2k 2) s+(2k ) (2k)! r s+(2k ) +s(s + )(s + 2)...(s + 2l ) k= = + B 2l+ (2l + )! (x ) (x r )s+2l dx d tht if s is o-positive iteger d l is lrge eough, the the lst term i the sum is so we obti fiite expressio for ζ(s, ; r). I prticulr, we hve ζ( m, ; r) = ( ) (m+)+ ( ) m+ m + r 2 r m + 8 l k= B 2k ( m + 2k ) ( r ) m+2k

9 Where l is lrge eough so tht 2l m. Noticig tht B =, B = 2, d odd Beroulli umbers vish, we rewrite the bove s ζ( m, ; r) = m+ ( ) m + ( r B k m + k k= ) m+ k Recllig the reltioship betwee the Beroulli umbers d Beroulli polyomils we see tht ζ( m, ; r) = ( r ) m + B m+ I prticulr, if r =, we see tht Filly we ote tht sice we hve tht ζ( m) = m + B m+() = B m+ m + L(χ, s) = r= L(χ, m) = m + Now we mke the followig defiitio ζ(s, ; r) χ(r) s ( r ) χ(r) m B m+ r= Defiitio Let χ be Dirichlet chrcter of coductor. geerlized Beroulli umbers B χ,k s B χ,k = k r= B k ( r ) The defie the Usig this ew ottio we hve tht L(χ, m) = B χ,m+ m + Notice tht sice the Beroulli polyomils hve rtiol coefficiets, we see tht L(χ, m) Q(ζ ) (here is the coductor of m). Additiolly, the bove formuls give us explicit wy of represetig L(χ, m) i Q(ζ ). 9