e x x s 1 dx ( 1) n n!(n + s) + e s n n n=1 n!n s Γ(s) = lim

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1 Lecure 3 Impora Special FucioMATH-GA 45. Complex Variable The Euler gamma fucio The Euler gamma fucio i ofe ju called he gamma fucio. I i oe of he mo impora ad ubiquiou pecial fucio i mahemaic, wih applicaio i combiaoric, probabiliy, umber heory, differeial equaio, ec. Below, we will pree all he fudameal properie of hi fucio, ad prove ha hey all aurally follow from i iegral repreeaio.. Defiiio Theorem he Euler gamma fucio: There exi a uique fucio Γ o C uch ha: a Γ i meromorphic o C b N, Γ + =! c Γ = π d C uch ha R > e C excep for pole f C where Γ = Γ = + =! + + e x x dx e x x dx + Γ = eγ + e = γ = lim l i called he Euler coa g C excep for pole! Γ = lim h Γ ha o zero; i oher word, /Γ i a eire fucio i The pole of Γ are he opoiive ieger =,,,.... The pole of Γ a =, wih N i a imple pole, wih reidue Re = Γ =! j C excep for pole, Γ + = Γ C excep for pole, ΓΓ =. Provig he properie π iπ Le u ar wih he iegral repreeaio, which may be viewed a a defiiio for he fucio: C : R >, Γ = e x x dx For N, le f := e x x dx

2 f i aalyic, ad Γ f x R e x dx The righ had ide coverge uiformly i every half-plae R δ wih δ >, o Γ i aalyic o R >. For R >, iegraio by par immediaely yield Γ + = Γ Furhermore, Γ =, o by iducio oe readily fid which prove b. Γ + =! Γ/ = e x x / dx = e d = π, which prove c. We ca ue he fucioal equaio Γ + = Γ o aalyically coiue Γ o a meromorphic fucio o C. Ideed Γ + Γ := i a aalyic fucio o { C : R > } \ {} uch ha Γ = Γ for R >. Furhermore, = i a imple pole of Γ, wih Re = Γ = Γ = Liewie, for uch ha R > ad,, defie Γ = Γ + = Γ + + which i aalyic o { C : R > } \ {, } ad coicide wih Γ for R >. = i a imple pole of Γ, wih reidue -. By iducio, if we have Γ a he aalyic coiuaio of Γ o R >, / { +, + 3,...,,, }, he we defie Γ := Γ + = Γ which i a meromorphic fucio for R >, wih pole = +, +,...,, ad reidue! a =. Le u ow wrie Γ = e x x dx + e x x dx The ecod erm o he righ-had ide i aalyic for all C. We call he fir erm F. We have F := e x x dx = + = x + dx =! + =! x + dx where we have ierchaged he order of ummaio ad iegraio uig abolue covergece. We hu ge he fial expreio: + Γ =! + + e x x dx = a give i iem e, which ha he deirable propery of highlighig he pole of Γ. Coider lim x/ x dx for R >.

3 A, x x e x x poiwie. Furhermore, N, x [, ], x e x Hece, by he domiaed covergece heorem, lim x x dx = Now, for R >, oe ca how by iducio ha Thi i how i goe. The propery hold for = : Le u aume i hold for. The x x dx = = + = e x x dx = Γ x x! dx = xx dx = + } d = {[ ] + d d =! d where we have ued he iducio hypohei for he la ep. Hece, for R >,! Γ = lim We wa o exed hi reul o C, excludig he pole of Γ. Le u coider he fucio /Γ. For R >, Γ = lim ! = lim e l lim e = l = + e / = e γ = + e / From he Weierra facorizaio heorem, we ow ha hi repree a eire fucio wih zero a he opoiive ieger, which prove f ad h. I remai o prove. Away from he pole of Γ, oe ca wrie ΓΓ = ΓΓ = eγ = + e / e γ e / = = where he la equaliy follow from he example we reaed i cla i he la lecure. = = iπ π 3

4 .3 Volume of a -dimeioal ball Coider he fucio of real variable We ca evaluae fx, x,..., x = exp R fdx = = = x e x dx = π Now, ice f i roaioally ymmeric, oe ca ue geeralized pherical coordiae o rewrie he iegral a follow: fdx = e r dadr = e r A rdr R S r where S r i he -phere of radiu r, da i he area eleme, ad A r i he urface area of he phere S r. Now, A r = r A, o R fdx = A r e r dr = A Comparig Eq. ad, we obai he equaliy: A r = π/ Γ r Hece, V r, he volume of he -ball of radiu r i give by V r = r A d = π/ Γ r e d = A Γ d = π/ Γ r = π/ Γ + 3 r We oberve omehig remarable: he volume of he ui ball icreae for 5, bu he decreae o a. If oe doe o reric o be a ieger, oe ca compue he max of V by eig dv /d =, ad obai max The Riema zea fucio Ju lie he gamma fucio, he Riema zea fucio play a ey role i may field of mahemaic. I i however much le well uderood ad characerized ha he zea fucio. There remai everal ope problem aociaed wih i, icludig THE ope problem of mahemaic: he Riema hypohei.. Defiiio Theorem he Riema zea fucio: There exi a uique fucio ζ o C uch ha: a ζ i meromorphic o C b For R >, ζ = = c For R >, ζ alo ha he ifiie produc repreeaio ζ = p where, a idicaed, he produc rage over he prime umber. 4

5 d ζ ha o zero i he regio R > e ζ ha o zero o he lie R = f The zero of ζ i he regio R are a =, N g ζ ha a uique pole, a =, wih reidue. h The value of ζ a eve poiive ieger are give by Euler formula: ζ = π B!, N where he B are he Beroulli umber, defied by he followig Taylor expaio: z e z = i ζ ae he followig value for egaive ieger: j ζ aifie he fucioal equaio m= B m m! zm ζ = B + +, N ζ = ζ where ζ i he ymmerized zea fucio defied by ζ = π / Γ ζ C \ {, } π / Γ ζ = + where he fucio θ i oe of he Jacobi hea erie, defied a l C \ {}, θ = = ζ = e π = + Γ πi C + + θ d = z dz e z z e π where C i he eyhole coour how i Figure, wih ɛ arbirary a log a he circle doe o ecloe a ieger muliple of πi. The brach of he logarihm i he iegrad i o be choe uch ha π < Arg z < π. m Coecio o prime umber eumeraio Defie ψx = p x l p, wih umber. ψ i called he Vo Magold weighed prime couig fucio. The, for ay oieger x >, ψx = x ρ x ρ ρ l π where he um i over he zero ρ of he Riema zea fucio. The formula above ha impora coequece for prime umber eumeraio, provided oe ca locae he zero ρ of ζ i he complex plae. For example, he fac ha ζ ha o zero uch ha R lead, afer ome wor, o he prime umber heorem give below. 5

6 Figure : Coour C ued for he iegral repreeaio of ζ i propery l. Theorem Prime umber heorem: Le πx deoe he umber of prime umber le ha or equal o x. We have πx = lim x x l x If here i eough ime a he ed of he coure, we will wor ou he deail of he proof of hi heorem baed o he properie of he Riema zea fucio. Of coure, he exac locaio of he orivial zero of he Riema zea fucio remai a ey ope problem. I i uually decribed a he Riema hypohei, which cojecure ha all he orivial zero of ζ are o he lie R =, called he criical lie.. Provig he properie We ar wih he adard defiio of ζ: ζ := I i clear ha hi erie coverge aboluely for R >, ad he covergece i uiform o ay half-plae R > δ wih δ >. Hece ζ i aalyic o R >. Liewie = p coverge aboluely iff p = p R coverge, which happe for R >. Hece F := p i aalyic ad ozero i R >. I remai o how ha ζ = F o hi e. For R >, le ζ N := p N, prime p = p N = p = =p c p c...p m m p,p,...,p m N p i prime where he la equaliy wa obaied by reorgaizig erm i he aboluely coverge erie. Hece, by he fudameal heorem of arihmeic, ζ ζ N >N N 6

7 which prove ha for R >, ζ = p which i called he Euler produc formula, ad prove poi c. Propery d follow immediaely from he Euler produc formula We ow ur o he Melli raform repreeaio. We ar by howig ha θ aifie >, θ = θ We ar wih fx = exp πx, whoe Fourier raform i ˆf := The Poio ummaio formula he ell u We oberve ha for >, θ = =+ = fxe πix dx = exp e π = = = e π π = θ θ = e π e π = e π e π Thu, θ = + Oe π for. So uig he equaliy θ = θ, we coclude ha θ = + Oe π/ =, + θ = O We are ow ready o ur o a repreeaio for ζ. For R >, we wrie Γ = e x x / dx π / Γ = Summig over o boh ide of he equaliy, we obai ζ = e π / d = = = e π / d =, + e π / d where we have ued he eimae for θ o exchage he um ad iegral ig. Now, le g aifie he equaliy Thu, ζ = g / d + = + g = g g / d = g := θ + g θ / / + / d which i he deired Melli raform repreeaio give i + + θ / d g / d 7

8 The iegral defie a eire fucio o C, howig ha ζ ca ideed be coiued o a meromorphic fucio o C. The Melli raform repreeaio immediaely yield ζ = ζ, which i propery j, ad which ca be rewrie a π ζ = π i Γ ζ 4 ζ ha imple pole a = ad a =, wih reidue ad. Therefore, ζ = Γ π/ ζ ha a pole π a = wih reidue = ad a pole a = wih reidue /Γ =. We ee ha he igulariy a Γ i i fac a removable igulariy, which prove g. 4 how ha he zero of ζ for R < are preciely =, N, which i propery f. Le u ow prove e: ζ ha o zero o he lie R =. Le σ > ad R ad coider he quaiy µ = l ζσ 3 ζσ + i 4 ζσ + i = 3 l ζσ + 4 l ζσ + i + l ζσ + i = 3 l p σ + 4 l p σ i + l p σ i = 3 l p σ 4 l p σ i l p σ i = [ 3RL p σ 4RL p σ i RL p σ i ] where, a alway, L i he pricipal brach of he logarihm. Our ex ep will be o ue power erie for L, which we ca ice p σ <, p σ i <, p σ i < For = a + ib uch ha R >, o ha Therefore, µ = = + p σ L p = RL p = + p = + p a = [3 + 4 co l p + co l p] = We ca herefore ay ha e µ, which mea ha cob l p = + p σ [ + co l p] ζσ 3 ζσ + i 4 ζσ + i 5 All we ow have o how i ha hi iequaliy preve ζ from havig a zero o he lie R =. Le u aume he corary: R uch ha ζ + i =. We he loo a he aympoic behavior of each erm i 5 a σ + : ζσ ζσ + i K σ ζσ + i K a σ +, K, K C where he fir aympoic eimae i igh, ad he oher wo are coervaive, i he ee ha ζσ + i could go o zero faer, ad ζσ + i could alo go o zero a σ +. We he obai he followig coervaive aympoic eimae a σ + : ζσ 3 ζσ + i 4 ζσ + i K 3 σ a σ + K 3 C 8

9 Thi coradic he reul ζσ 3 ζσ + i 4 ζσ + i σ > ad R. ζ doe o have ay zero wih real par equal o. Noe ha he miraculouly imple ric ued a he hear of our proof i ofe aribued o he Polih mahemaicia Fraz Mere. We coclude hi lecure wih a derivaio of propery l. The derivaio ar wih aoher ueful ideiy. Le C uch ha R >. e d = + e d = = + = e d = + = u e u du = ζγ where we have ued abolue covergece o ierchage he order of iegraio ad ummaio. Now, by Cauchy heorem, he value of he iegral z dz e z z C doe o deped o he hape of he curve C, provided C doe o ecloe a pole of he iegrad, i.e. a muliple of πi. We are herefore free o chooe he ey hole coour how i Figure, ad o ae he limi ɛ for ha coour. I i he raighforward o verify ha he coribuio from he circle of radiu ɛ ed o zero. Whe ɛ, he oly coribuio o he iegral hu come from he wo exeded brache of he coour C, wad e have + ρ e iπ e ρ dρ + ρ e iπ e ρ dρ = e iπ e iπ ρ e ρ dρ = i iπζγ I i a imple exercie o verify ha he iegral over he mall circle ed o zero a ɛ ed o whe R >. Hece, for R >, C z dz e z z = i iπζγ ζ = i iπγ C z dz Γ e z = z πi C z dz e z z We oberve ha he iegral i 6 i a eire fucio of, o 6 ca be viewed a a way o aalyically exed ζ o a meromorphic fucio i C which i equivale o he Melli raform repreeaio. Noe ha whe R, i i o rue aymore ha he coribuio from he circle of radiu ɛ ed o a ɛ. Propery i follow from 6. Thi i lef a a raighforward exercie, a well a propery h which follow from propery i ad he fucioal equaio. 6 9