(1 q an+b ). n=0. n=0

Transcription

1 AN ELEMENTARY DERIVATION OF THE ASYMPTOTICS OF PARTITION FUNCTIONS Diel M Ke Abstrct Let S,b { + b : 0} where is iteger Let P,b deote the umber of prtitios of ito elemets of S,b I prticulr, we hve the geertig fuctio, P,b q q +b We obti symptotic results for P,b whe gcd, b Our methods deped o the combitoril properties of geertig fuctios, symptotic pproximtios such s Stirlig s formul, d i depth lysis of the umber of lttice poits iside certi simplicies Itroductio d Sttemet of Results For positive itegers d b, let S,b {+b : 0} where is iteger be the set of turl umbers, t lest b, tht re cogruet to mod b Let P,b deote the umber of prtitios of ito elemets of S,b I prticulr, we hve the geertig fuctio, P,b q q +b A fmous theorem of Hrdy d Rmuj is tht whe b P, 4 3 eπ /3 s Their proof which mrs the birth of the circle method depeds o properties of modulr forms A symptotic formul for P,b for ll pirs of, b reltively prime ws first ttied by Ighm see [] d his proof ws lter refied by Meirdus i [3] d [4] The correct expressio is: b P,b Γ π b/ 3/ b/ 3 b/ /+b/ +b exp π 3 99 Mthemtics Subject Clssifictio Primry P7, P68 Key words d phrses Asymptotic formuls, prtitio fuctios Typeset by AMS-TEX

2 DANIEL M KANE Lter Erdös ws ble to study the symptotics usig elemetry methods ivolvig recursive formuls for them I prticulr, he showed i [5] tht for some costt c, P, c eπ /3 This proof ws lter refied by Do Newm i [] to obti to full symptotics of P, Usig differet methods we obti similr results for P,b to withi costt multiple whe gcd, b Our methods deped o the combitoril properties of geertig fuctios, symptotic pproximtios such s Stirlig s formul, d i depth lysis of the umber of lttice poits iside certi simplicies I prticulr, we shll prove the followig theorem; Theorem If d b re turl umbers d gcd, b, the there re costts 0 < C,b < such tht for ll sufficietly lrge C +,b C,b e π +b 3 I fct, we shll compute such costts < P,b < C+,b e π +b 3 Defiitios d Prelimiry Lemm We will use the followig defiitios For coveiece, we let C { }{b }{z } F { }{b }{z } deote the sum of F over sequeces of turl umbers { i } i, {b i } i,, {z i } i so tht coditio C is stisfied For series fq d gq let fq q gq me tht the coefficiets of the Mcluri series i q of gq re greter th or equl to the correspodig coefficiets of the series for fq Defie fq q gq,fq < q gq, etc similrly Furthermore, let 3 σ,b m+b where the sum is over o-egtive itegers d turl umbers m so tht m + b Let N 0 deote the set of o-egtive itegers Let γ lim i i log 577 be Euler s costt Let! Γ + 0 t e t dt Recll Stirlig s formul, which sttes tht 4! e π m,

3 PARTITION ASYMPTOTICS 3 Lemm Suppose tht, : N 0 R R + d b, : N 0 R R + re mps for which the followig re true:, d b, coverge for ll R,, b, < C where lim C 0, 3 lim, +, 0 d lim The we hve lim b, b+, 0, b, Proof of Lemm For y ɛ > 0 fid N so tht > N implies C < ɛ 4 Fid N so tht, for < N d > N, +, < ɛ 4+ɛ For > N, N, N, ɛ 4, ɛ 4 Fid N b similrly Now, for > mxn, N b, b,,, N N N, b, N b, b, By looig t the bsolute vlues of the logs of these frctios, we fid tht the differece betwee this d is t most ± 4 ɛ 3 < ɛ This proves the lemm This result will be used lter i sectios 6 through 9 3 The Strtegy I order to prove Theorem, we shll wor to prove the followig theorem, Theorem If d b re turl umbers d if b, the there exist costts 0 < C,b < C+,b so tht for ll sufficietly lrge we hve C,b b e π 3 < P,b < C+,b e π b 3 To prove Theorem, we begi with some importt observtios Recll tht, by P,b q q +b

4 4 DANIEL M KANE By tig the log of both sides we fid tht log P,b q log q +b log q +b q m+b m m by 3 σ,b q Expoetitig both sides of this d usig the series expsio for e x, we fid tht 0 Equtig coefficiets, we fid tht, P,b P,b q e 0 σ,bq σ,b q! 0 i x i {x } σ,b x i! i Summig over, the previous expressio implies i x i P,b m σ,b x i! m0 3 {x } i i α iβ i +b {α }{β } i α ib {α } α i i i α i!! {β, β,, β N 0 : α i β i + b } i I the ext few sectios we shll study the lst term of this expressio, {β, β,, β N 0 : α i β i + b } i {β, β,, β N 0 + : α i β i b 3 i i α i This is the umber of hlf-iteger lttice poits withi the -dimesiol simplex defied by the equtios x i 0 i d i α ix i b i α i I sectios 4 d 5 we will compute }

5 PARTITION ASYMPTOTICS 5 upper d lower bouds respectively for 3 I sectio 6 we will icorporte this ito 3 to remove the β depedece I sectios 7 d 8 we will clculte upper d lower bouds respectively o this ew expressio, removig the α depedece I sectio 9 we shll come up with symptotic bouds for the expressios tht cme out of sectios 7 d 8, provig Theorem i the process Filly, i sectio 0 we will use Theorem to prove Theorem 4 Upper Bouds For 3 I this sectio we derive upper bouds for the umber of hlf-iteger lttice poits withi certi right simplicies d hece obti upper bouds for 3 Lemm 4 For y sequece of positive rel umbers {S i } i if m, the {x,, x N 0 + : x i } S i i S i! i + S m We ote tht this is equivlet to the sttemet tht the umber of hlf-iteger lttice poits iside the right simplex with legs prllel to the xis, meetig t the origi, d of legths S i is t most the volume of the simplex plus the re of y oe of the fces Proof of Lemm 4 Let S be such simplex Let the ple x m 0 be N Let the fce of S defied by S N be F Defie lttice prism to be the set of poits defied by y i x i y i + for i d i m for some itegers y i Let the ceter of such lttice prism be the poit y +,, y m +, 0, y m+ +,, y + For lttice prism L, with ceter C defie P L to be the umber of hlf-iteger lttice poits i S L Defie V L to be the volume of S L Let the re of F L be AL We wish to show tht 4 P L AL + V L If P L 0, the the result follows trivilly becuse P L 0 AL + V L Otherwise, P L so C + e m S, where e m deotes the uit vector i the m directio This implies tht C S Whe hypercube is cut by ple, the hlf cotiig the ceter hs the lrger volume sice it d its reflectio bout the ceter cover the cube Therefore, AL The height of S bove the N is some lier fuctio, M, of the positio i the N ple Sice ll lttice poits i S L re i the sme colum, P L is the gretest iteger so tht C + e m S Therefore, P L is t most the height of S bove N t C plus oe hlf, or P L MC + Now, V L F L Mxdx Sice M is egtive outside of F, V L Mxdx Sice M is lier, L N this equls MC So, V L MC This mes tht P L MC + V L + AL, provig 4 By summig 4 over ll lttice prisms, we get tht P L L L V L + L AL

6 6 DANIEL M KANE Ech term i the first sum is the umber of hlf-iteger lttice poits i S L for y lttice prism L Sice the lttice prisms tessellte spce d do ot overlp over y hlf-iteger lttice poit, this is merely {x,, x N 0 + : x,, x S}, which equls i S i! {x,, x N 0 + : x i } S i The secod term is the sum of the volumes of the itersectios of S with ll lttice prisms Sice lttice prisms tessellte spce d overlp oly o surfces of o volume, this is the volume of S, which is The lst term is the sums of the res of the itersectios of F with lttice prisms Agi sice lttice prisms tessellte spce d sice the itersectio of their overlps with F hs smller dimesio, this is just the re of F, which is Substitutig i these vlues, we get tht {x,, x N 0 + : i i i m i S i! x i } S i i S i +! i S i! i m i S i! + S m 5 Lower Bouds For 3 I this sectio, we derive lower bouds o the umber of hlf-iteger lttice poits withi simplex, d hece lower bouds o 3 We shll prove Lemm 5 For positive rel umbers S i, where i, we hve {x,, x N 0 N 0 + x i : } S i i S i! i S i i This is to sy tht the umber of hlf-iteger except for the first two coordites lttice poits withi simplex with sides log the xis tht meet t the origi of legth S i, is t lest the volume of the simplex times some correctio term Cll such poits l-poits Proof of Lemm 5 First ote tht i two dimesios, the umber of lttice poits iside the trigle with vertices 0, 0, x, 0, d 0, y is t lest x y This is true becuse ll the uit squres to the upper left of such lttice poits, cover the trigle Let K S S Let S be such simplex Let N be the hyperple defied by x x 0 Let F be the -dimesiol fce of S defied by S N Let lttice prism be the set of poits defied by y i x i y i + 3 i for some itegers y i Let the ceter of such lttice prism be 0, 0, y 3 +, y 4+,, y + For lttice prism L with ceter C, let P L be the umber of l-poits V L P L i L S Let V L be the volume of L S Let AL be the re of L F Let EL AL Defie the lier fuctio M s Mx 3,, x x i i3 S i Note tht the re of the sectio of S of the form, b, x 3,, x is KM We ow wish to put upper bouds o EL

7 PARTITION ASYMPTOTICS 7 First we will cosider the cse whe C F I this cse, AL By our previous discussio, P L KM C becuse tht is the re i the trigle bove C So, 5 EL V L P L KM xdx KM C F L KM xdx KM C N L N L KM x + KM C x KM Cdx K N L M x + MC Mx M Cdx K M x MxMC + M Cdx N L K Mx MC dx N L Notice tht this vlue is idepedet of the choice of L Now, whe C F, EL V L AL, which is the verge re bove poits i L F Notice tht the closer C is to the boudry of F, the lrger this is Therefore, this is t most twice the volume bove uit squre cetered o the boudry of F This is less th the qutity i 5, so therefore, 5 is upper boud o EL Let Q be the set of poits where x x 0 d x i for 3 i We hve tht EL K Q i3 S i i3 x i S i 4K! dx i3 x i S i +/ / +/ / +/ /, where +/ / +/ / +/ / is the differece of the evlutios of the expressio betwee plus d mius oe hlf over ll of the x i The lst fctor here is polyomil of degree i the S i Furthermore, it is odd i every S i Therefore, ll terms but those of the form Sh i3 S i ccel out These terms hve cotributio of 5 Let K S S i S i,,,, 3 EL S i 4K! K 6 i3 S i3 i S S from ech of the terms Therefore, S i i! 6 Si i3 S j3 i

8 8 DANIEL M KANE By 5 we hve EL K Therefore, P L V L K AL Summig over lttice squres L, we get P L L L V L K L AL By our discussio i the previous sectio, these terms re the umber of l-poits i S, the volume of S, d K times the re of F respectively Therefore, {x,, x N 0 N 0 + : x i } S i i i S i! K i3 S i! i i S i! S i! S i i S i i 6 Simplifictio of 3 We shll ow use the results from the lst two sectios to simplify 3 Let C be the coditio tht b i α i Let C be C with the further coditio tht t most oe α i equls oe Let C be C d tht α i for ll i Let C be the coditio C d ot C or, i other words, C d t lest two of the α i re equl to Let S α deote i α i Recll, tht expressio 3 is C {α } α i i! {β, β,, β N 0 : α i β i + b } i

9 Notice tht {α } C {α } {α } C By Lemm 4 α i i C α i i! α i i PARTITION ASYMPTOTICS 9!! {β, β,, β C {α } C Becuse the sums over α i seprte Becuse 6 Let 4 {α } {β, β,, β N 0 : {β, β,, β N 0 : N 0 + : α i i α i i 6 ɛ e Now, let 63 G By our prior discussio, C 3 {α } α i i! C {α } α i i By Lemm 4 C b /S 64 {α i } α i 3G + O ɛ 3 i α i β i + b } i α i β i + b } i α i β i b S }! +! + O O π π 6 3 {β, β,, β N 0 : 6 S b /S /b b //b π 6! π 6! 4 π 6! b /S!! + O e π 6 3 α i β i + b } + O ɛ i + O ɛ b /S

10 0 DANIEL M KANE Now let us cosider 3 uder the coditio tht t lest two of the α i WLOG α d α re equl to 3 {β, β,, β N 0 : α i β i + b } i {β, β,, β N 0 N 0 + b /S : α i β i } By Lemm 5 b /S α i! Usig this we fid tht 3 C 65 {α } {α } α i i i α i i i + G + O ɛ C! + G + O ɛ b / + G + O ɛ 5 6 G + O ɛ C {α } Combiig 64 d 65 we get 66 3G + O ɛ Let The summd i G is C {α } α i i α i i! i α i b /S i b /S b /S α i! P,b i 5 6 G + O ɛ i0 y π 6 b /S C {α } α i i!! b /S y!

11 PARTITION ASYMPTOTICS If we sum this over N where < y y 3/8 or > y + y 3/8 we get t most y y y y! y + y 7/8 + y 3/4 i0 i+y 3/8 y y 7/8 + y 3/4 i+y 3/8 + y By 4 e y y /8 3/8 O e y y /4 Let ɛ e y y /4 The we hve tht 67 G C b /S! + O ɛ α y <y 3/8 {α i } i i 7 Upper bouds o G We hve from 67 tht G C b /S! + O ɛ α y <y 3/8 {α i } i i We will exmie the summd for such tht y < y 3/8 This is 7 C {α i } α i i b /S! C {α i } α i i C {α i } α i i C {α i } i α i i e b /i i e b /α i e b /S b /S!!!! Let hx x

12 DANIEL M KANE This coverges bsolutely for x Furthermore we hve tht Becuse h ζ π 6 hx x 0 π 6 π 6 π 6 log t dt t x x log t dt t log tdt + x log x + x Let m b Usig this we fid tht π 7 6 b / + e b / y + + m b / y log e m y b / log +m y log e b / + e b / b / b / log + + m y y! y!!! Now sice y + O y /8, we hve tht m+oy /8 b / 7 e m+oy /8 y! m m 6eb / y π y! m 6eb / m y π! So it turs out tht 7 G N m y! where b 6eb / N π

13 We hve from 67 tht G y <y 3/8 PARTITION ASYMPTOTICS 3 8 Lower bouds o G C {α i } α i i b /S! + O ɛ We will exmie the summd for such tht y < y 3/8 This is C α i b /S b α {α } i i! b /S α {α } i i By the Arithmetic-Geometric Me Iequlity 8 Notice tht b i b /i i Rememberig tht m b Rememberig tht y π 6 Substitutig this ito 8 we get b Now, sice b y + m y log b + m y b i α i b {α } b i b /α i i b / i π 6 b y b y y m y γ O log m log b b y m y α i b /i i b i i m log + γ b log mγ b log my b γ y! my γ d sice y + O y /8, 8 e +Oy /8 b +m log m logb+mγ+o m e b e mγ b y! m m e b e mγ 6 y bπ y! y N m!, log y!!!!

14 4 DANIEL M KANE where Therefore, by 67 G N N e b e mγ y <y 3/8 m m 6 bπ y! + O ɛ But sice m y! < y!, by previous rgumets we hve tht 8 G N m y! + O ɛ 9 Fil lysis of G d proof of Theorem Let F y By 7 d 8 we hve m y! 9 N F y + O ɛ G N F y We ow use Lemm d 4 to fid tht 9 F y m y! m y e π +m+ + m + / + m + / m y e π + m + / m +m+ + m + +m+ y e 4 π + m + / m e +m+/ π / m+ π / 4y π y m/ /4 y +m+/ + m + /! If the 9 implies tht Hz z +m+/ + m + /!, F y π / y b/ H y

15 PARTITION ASYMPTOTICS 5 This leves us to fid symptotic formul for Hz Notice tht Hz is uiquely defied by the differetil equtio with iitil coditios H z Hz Ad H0 H 0 0 The solutio to this equtio is Hz ez z 0 ez z ez 0 zm+/ m + /! t m+/ e t m + /! dt e z z t m+/ e t m + /! dt + O z m+3/ 93 Proof of Theorem Combiig 9 d 93 we get tht F y e y πy b Substitutig y π 6, it turs out tht F y N 3 b e π 3, where N 3 π π 6 b 0 t m+/ e t m + /! dt Note tht for lrge tht this is much bigger th either ɛ or ɛ Combiig this with 9 we get tht N N 3 b e π 3 G N N 3 b e π 3 Now sice F π π 6 F 6, by 66 we hve tht 5 6 N N 3 b e π This proves the theorem for y costts C,b < 5 π 6 π 6 d C +,b > 3 π 3 i P,b i 3N N 3 b e π 3 b e b e b/ /γ π 6 b 6eb / π b/ / 6 bπ b Remr: Note tht whe d b, i P,bi P, So we hve just proved tht 0036 e π /3 P, 54 eπ /3 While Hrdy d Rmuj s symptotic formul gives 4 eπ /3 P,

16 6 DANIEL M KANE 0 Proof of Theorem Notice tht sice for d b reltively prime, y turl umber lrger th b + b c be writte uiquely s the sum of multiple of b less th + b plus multiple of + b less th b + b plus multiple of b + b, q +b q q + q b + q b + + q b+b + q +b + q +b + + q +b q +b q q Recll tht q sys tht the coefficiets of oe series re bigger th the correspodig coefficiets of the other This lst expressio implies tht We lso hve tht q b+b q q b+b q q b+b b + b q q q b q +b q q q b+b q q q b+b b + b q Ad sice q P lq q +b, where P lq is polyomil with P l + b, we hve tht 0 Q q q q +b q q b q +b q Q q q q +b, where Q q d Q q re polyomils i q the sum of whose coefficiets re b with highest degree t most 3b + b Proof of Theorem By 0 we hve bq 3b+b q q +b q or tht 3b+b b i q +b q P,+b i P,b b b q P,+b i By Theorem, we hve for y positive costts stisfyig C,b < b5 π +b 6 e +b e b/+/γ 6 π 6 + bπ d tht for ll sufficietly lrge C +,b > 3b π π C,b e π +b 3 6 +b i 6eb + / π < P,b < C+,b e π +b 3 b + q +b, b/+/ Acowledgemet I would lie to th Professor Ke Oo for dvisig me with the problem tht led to some of the importt ides i this pper, developig these ides, d writig d editig this pper

17 PARTITION ASYMPTOTICS 7 Refereces [] A E Ighm, A Tuberi theorem for prtitios, Als of Mthemtics 4 94, [] D Newm, A Simplified Proof of the Prtitio Formul, Michig Mth J 9, [3] G Meirdus, Asymptotische Aussge über Prtitioe, Mth Z 6, [4] G Meirdus, Über Prtitioe mit Differezebediguge, Mth Z 6 b, [5] P Erdös, O Elemetry Proof of some Asymptotic formuls i the Theory of Prtitios, Als of Mthemtics 43 94, Diel M Ke, 84 Reget Street, Mdiso, WI E-mil ddress: de@mitedu