SUMMARY OF THE ZETA REGULARIZATION METHOD APPLIED TO THE CALCULATION OF DIVERGENT SERIES

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1 SUMMARY OF THE ZETA REGULARIZATION METHOD APPLIED TO THE CALCULATION OF DIVERGENT SERIES AND DIVERGENT INTEGRALS s d s Jose Jver Grc Moret Grdute studet of Physcs t the UPV/EHU (Uversty of Bsque coutry I Sold Stte Physcs Addres: Prctctes Ad y Grjlb 6 G P.O Portuglete Vzcy (Sp Phoe: ( E-l: josegrc@yhoo.es MSC: 97M5, 4G99, 4G, 8T55 ABSTRACT: We study geerlzto of the zet regulrzto ethod ppled to the s cse of the regulrzto of dverget tegrls d for postve s, usg the Euler Mclur suto forul, we ge to epress dverget tegrl ter of ler cobto of dverget seres, these seres c be regulrzed usg the Re Zet fucto ζ ( s s >, the cse of the pole t s= we use property ' of the Fuctol detert to obt the regulrzto = (, wth ( the d of the Luret seres oe d severl vrbles we c eted zet regulrzto to the cses of tegrls f d, we beleve ths ethod c be of terest the regulrzto of the dverget UV tegrls Qutu Feld theory sce our ethod would ot hve the probles of the Alytc regulrzto or desol regulrzto Keywords: = Re Zet fucto, Fuctol detert, Zet regulrzto, dverget seres. ZETA REGULARIZATION FOR DIVERGENT INTEGRALS:

2 Soetes thetcs d physcs, we ust evlute dverget seres of the for, of course ths seres s dverget ules Re ( >, however cses le = or =3 pper severl clcultos of strg theory d Csr effect, for the cse of 3 Csr effect [3] the result = ppers to gve the correct result for the Fc cπ Csr force = h here A s the re d d the seprto betwee the 4 A 4 pltes, c d h re the speed of lgth d the Plc s costt. The de behd the Zet regulrzto ethod s to te for grted tht for every s the detty s = ζ ( s, follows lthough ths forul s vld just for Re (s >, to eted the defto of the Re Zet fucto to egtve rel ubers, oe eed to use the fuctol equto for the Re fucto π s ζ ( s = ( π ( scos ζ ( s π ( s ( s = ( s( π s Ths gves the epressos t s=, the Hroc seres fte vlue =, = d = due to the pole s NOT zet regulrzble, lthough t c be gve = γ = , ths vlue c be justfed by usg the theory of Zet-regulrzed fte products (deterts, s we shll see lter the pper o Zet regulrzto for dverget tegrls: s Let be f = wth Re(-s < -, the the Euler-Mclur suto forul for ths fucto reds s d = d ζ ( s s = r r ( r s d r= B ( s ( r! ( r s N ( Here forul ( ll the seres d tegrls re coverget, forul ( s usully worthless, sce t s trvl to prove tht d = for Re( >,d the Re zet fucto ζ = =, so othg ew c be obted fro (, the de s s to use the Fuctol equto ( for the Re d Zet fucto to eted the

3 defto of equto ( to the whole cople ple ecept s=, cse (-s s postve there wll be o pole t =, so we c put = d te the lt s B!( r ( r!( r! r r d = d ζ ( d (3 r= Forul (3 s the Alytc cotuto of forul ( wth = d c be used to obt fte defto for otherwse dverget tegrls,ppretly ths recurrece equto hs fte uber of ters but the G fucto hs pole t = d t beg soe egtve teger, soe eples of forul (3 I = ζ ( = = ζ ( = I d I d I B I = ζ ( I d = (4 3 B 3 I3 = ( I ζ ( I ζ ( 3 B3I = d So our ethod c provde fte regulrzto to dverget tegrls, wth the Ad of the zet regulrzto lgorth. Also our forule ( (3 d (4 re cosstet wth the usul suto propertes, fct f Λ d s fte for fte Λ d we use the property of the Re d Hurwtz Zet fucto [ ] to get the su of the -th powers of o the tervl [, Λ ] Λ = = ζ ( ζ (, Λ, ζ ( s, Λ = ( Λ s defed for Re(s > (of course for postve s s Λ the secod ter goes to B!( r ( r!( r! Λ Λ Λ r r d = d ζ ( ζ (, Λ d (5 r= B For teger (, ζ H = we fd the Beroull Polyols, the powers Λ Λ of Λ would ccel the tegrl d =, so the ed forul (5 we would B get the usul defto of Zet regulrzto ( ζ H = for teger. Of course oe could rgue tht spler regulrzto of the dverget tegrls should s s be I( s = d( = d I( = d( = log s, ths s just droppg out the ter proportol to log s or sde the tegrl to e t fte, however f w plugged ths result to the Euler-Mclur suto forule ( (3 or (5 the 3

4 ters volvg would ccel d we would flly fd tht ζ H ( = for every whch clerly s gst the defto of zet regulrzto of seres, for the cse of the logrhc dvergece, obted fro dfferetto wth respect to the eterl preter ths s result of tg the fte prt of the tegrl,whch ppretly wors. For the cse of the tegrls log d, we c sply dfferette - tes wth respect to regultor s order to obt fte vlues ters of ζ ( s d ζ '( s for egtve vlues of s uless =- (for other egtve vlues of we c e chge of vrble q =, ths s treted the et secto o Zet-regulrzed deterts d the Hroc seres: λ = Gve opertor A wth fte set of ozero Egevlues { } Zet fucto d Zet-regulrzed detert, Voros [] we c defe { } Tr A = ζ A( s = λ dζ A det( A ep ( = λ = ds (6 The proof of the secod forul sde (6 s pretty esy, the dervtve of the log λ Geerlzed zet fucto wll be ζ A '( s = s ow let s=, use the property of λ the logrth log(. b = log logb d te the epoetl o both sdes. For the cse of the Egevlues of sple Qutu Hroc osclltor oe deso [ ] λ =, the Zet fucto s just the Hurwtz Zet fucto, so we c defe zet-regulrzed fte product the for dζ H ep (, = ds dζ H (, log log π ds = (7 I cse we put = we fd the zet-regulrzed product of ll the turl ubers ( = π,see [5] f we te the dervtve wth respect to, we would fd the se regulrzed Vlue Ruj dd [] precsely ' = ( > ( d Hroc seres pper due to logrthc dvergece of the tegrl (, f we put = - sde forul (, usg regultor s, s we hve the Euler Mclur suto forul (8 r d B r = s s r s ( ( r= ( r! u ( = 4

5 Sce s > the tegrl d the seres sde (8 wll be coverget, ow we c tegrte s over sde (8 d use the defto of the logrth l log =, to s s d ' regulrze the tegrl ( s s s ters of the fucto plus soe fte correctos due to the Euler-Mclur suto forul. A fster ethod s just sple dfferette wth respect to sde the tegrl d di =, ow ths tegrl s coverget for every d equl to d, tegrto over g gves the vlue log c plus costt c tht wll ot deped o the vlue of sde the tegrl questo, the proof tht c s uque o tter wht s coes fro the fct tht the dfferece d log b =. b dζ H (, For the cse =, the dervtve of the Hurwtz Zet s = log ( π so f ds we pprote the dverget tegrl by seres, the we c get the regulrzed result d =. Appretly t sees tht usg two dfferet regulrztos we get soe dfferet results, the de s tht f we use the Strlg syptotc forul pproto for the logrth of the Zet fucto B z log ( z = z log z z log ( π r (9 r= r ( r If we te the dervtve wth respect to z sde (9, s ow ore ppret tht for d µ the logrthc dervtve log here c = log µ s costt obted fro dfferetto wth respect to to regulrze the dverget tegrl, ths costt c ust be relted to soe physcl costt or cse the qutty hs deso of Eergy the µ ust hve lso desos of eergy so the logrth s desoless, ths costt c would be the oly free djustble preter tht would pper sde our clcultos to regulrze tegrls. If s egtve there s etr ter due to the vlue log( = π, for ore cople logrthc tegrl oe c use the defto log ( d log µ wth the se eergy scle c = log µ o Regulrzto of dverget tegrls df ( : I geerl, the dverget tegrls tht pper Qutu Feld Theory [ ] re vrt 4 uder rottos, for eple 4 d p p or d p ( p q p, f we use 4-5

6 desol polr coordtes we c reduce these tegrls to the cse d / π d drf ( r r d / the the UV dvergeces pper whe r, here d=4 s the deso of the spcete, depedg o the vlue of d we c hve severl types of dvergeces Λ d drf ( r r Λ blog Λ, f b = for = the UV dvergeces re qudrtc f = the dvergeces re ler, cse = d b = the dvergeces re 4 of logrthc type, for eple d p p hs oly logrthc dvergece deso 4, for lower vlue of the deso (d=3 ths tegrl ests. To study the rte of dvergece, we c epd the fucto to Luret seres vld for z, f = c ( s fte uber d es tht the fucto f ( hs power lw dvergece for bg, the the de to copute dverget tegrl would be ths, we dd d substrct Polyol plus ter proportol to to splt the tegrl to fte prt d other dverget tegrl, both cses we ust lso troduce regultor ( for turl uber so we e the tegrls coverget for soe Re(s > d b f b ( b ( d b s s ( = ( ( Also we c use the chge of vrble (, so the ew lts of tegrto would be (,, sce s turl uber, the the followg detty ( = = ζ ( s holds for every postve d the sese of zet regulrzed seres. Of course sde ( our substrcto we c clude o-teger powers of sce the recurso forul ( s stll vld for the. The uber of ters s chose so the frst tegrl s FINITE, ths frst tegrl c be coputed by Nuercl or ect ethods d yelds to fte vlue, the rest of the tegrls re just the logrthc d power-lw dvergeces, they c be regulrzed wth the d of forule ( (3 (4 (6 (8 to get fte vlue volvg ler ζ H (, cobto of ζ ( =,,,..., d other vlue proportol to s d µ or log for eple we c lyze ths sple dverget tegrl > d 6

7 d ζ ( = d B r ' r ζ r r= ( r! u = ( The frst tegrl ( s coverget d hve ect vlue of log, order ' to regulrze the logrthc tegrl we hve used the result = ( plus ( the Euler-Mclur suto forul. The thetcl justfcto of ths s the followg, gve dverget tegrl df we troduce regultor d F( s = f s so the tegrl F(s ests for soe bg s, f we dd d substrct powers of the for s for teger d s, we c splt F(s to coverget tegrl I (s vld for s d soe dverget tegrls of the for d ( d d s, usg forule ( (3 (4 d (8 we c epress these tegrls ters of the seres s d s, whch wll be coverget for Re( s > d Re( s >, ow usg the Fuctol equto for the Hurwtz d Re Zet fucto we c e the lytc cotuto of both seres to s vodg the pole t s= by the use of Re Zet fucto t egtve ' tegers ζ ( plus soe correctos volvg of course the rules for chge of vrble d stll vld so df ( = duf ( u ths c be used to vod soe IR dvergeces t = by splttg the tegrl to IR dverget prt d UV dverget prt du = du du. For other types of dverget tegrls le β α d log for postve α d β oe could dfferette wth respect to or s sde forul ( order to obt recurrece equto for the tegrls β α d log, ths recurrece equto s fte (pprotely sce for Re ( p > β log d p s fte d do ot eed to be regulrzed provded >. Other useful dettes c be ( / or the epso of the logrth vld for y.4 7

8 > log = to e logrths ore trctble, lso we could use Luret epsos to hdle coplcte o-polyol epressos le ( µ by epdg t for bg to syptotc (verse power seres. o Regulrzto of tegrls the for d d f d : ( ( b Utl ow, we hve oly cosdered the UV dverget tegrls, the tegrls whose tegrd f s, fro the defto of proper tegrl ε d ε = = F reg ( / ε ( ε d = = Freg ( ε = ( N As N, ths ply tht our regulrzto producedure Freg ( s = Freg ( s, for the cse s > we c use forule ( d (3 to regulrze the dverget tegrl, d by the forul reltg s d (-s oe could lso regulrze IR (frred dverget d tegrls slr wy we dd for d, ecept for the cse = (logrthc tegrl,for the cse of ths tegrl d d ow we e the chge of vrbles, to rewrte ths s d d / d oe could splt t to d, these re g logrthc dverget tegrls d c be regulrzed wth the d of the zet regulrzed product b = e ζ ' = reg ( plus the Euler-Mclur suto forul, d the detty ( ' H (, b. For the cse of ore geerl dverget tegrl le f d f c d > c > (3 ( f f ( c( c ( d ( c ( c c Frst tegrl sde (3 s fte d fter severl pultos the other dverget r d tegrls c be wrtte s ( for soe rel d postve preters c ε r c d, by ultplyg both uertor d deotor by ( c r 8

9 Aother possblty s to vod the pole t cert pot = by usg the Alytc cotuto of the tegrl volvg severl preters d d( d = F (, = α α ( = (4 = ( α α The de s to clculte the tegrl (4 tht wll deped o two preters α ± =, d flly set α =, α = ± f F, ests, ths c be regrded s the regulrzed vlue of the tegrl, of course (4 y be dverget s so we y eed to dd d substrct ters to e t coverget slr wy we dd (, other for to regulrze (4 s defg F, α d the clculte ths tegrl for geerl vlue of α, the ed we would put α = ±. Aother useful detty to regulrze frred dvergeces wheever = s (tbles of tegrls by Abrowtz d Stegu [] r r d ( π ( = = I(,, r r ε (5 ( ( ε π π s ( r! 3 r Itegrl sde (4 wll be coverget wheever r- > f ths s ot the cse we could use the Euclde lgorth to splt ths tegrl to coverget ter defed s (4 d soe dverget tegrls d ( = -,,,,... the de to justfy why the frred dvergeces re eser to regulrze th ultrvolet oes, s tht for the frred, you could sert sll cople ter ε to regulrze t d e t coverget, so there re soe cople vlues of tht e (4 to be well-defed, however for the ultrvolet dvergeces ths s ot the cse sce there s o vlue of 4 d tht es coverget, uless we use soe d of regulrzto REGULARIZATION OF MULTIPLE INTEGRALS: Utl ow, we hve oly cosdered tegrls oe vrble (fter chge to polr coordtes, the t rses the questo f oe c pply our ethod of zet regulrzto to ore coplcte tegrls le I( s = d q d q... d q F( q, q,..., q R q, q,..., q ( ( (6 = ( q 9

10 Here we hve troduced regultor depedg o eterl preter s order the tegrl (6 to coverge for bg s d the use the lytc regulrzto to te the lt s, ths regultor ust be chose wth cre order ot to spol y syetres of the Physcl syste ths regultor y be of the for R q, q,..., q q = R ( q q q = ( q =,,..., (7 = Our frst stz would be to defe -desol polr coordtes so we c rewrte (6 s ultple tegrl depedg o r,,3,4,..., - the for q = r d severl gles = = Ω (, θ ( (, θ d dθ s ( θ Ω = I s d drg r r R r Ω = (8 θ = We y choose the frst regultor sde (7 so t does ot deped o the gulr coordtes, the de s tht cse (6 hs ultrvolet dvergece ths dvergece wll pper wheever r, so f we perfor the tegrl over the gulr vrbles Ω = s we re left wth tegrl I( s = dru ( r r ( r d dθ θ = order to regulrze ths we defe coverget tegrl (by substrcto plus soe dverget ters I( s = dr ( r U ( r r ( r ( r dr (9 = = U (r s the fucto obted fter tegrto over the gles, d s fte uber to perfor the l substrcto of ters order the frst tegrl to be coverget eve for s =, f the tegrl over the gles s too coplcte to hve ect for we could replce ths tegrl over the gles by pprote fte su dω (su over ll the gulr vrbles order to e the tegrl eser to clculte, ths c be usg Motercrlo ethods of tegrto., o Substrcto ethod: Oce we hve de the chge of vrble to sphercl coordtes sde our tegrl I( q, q,..., q oe could substrct soe ters to reder the tegrl fte I s d dr G r r f r f d dr r j j = Ω (, θ j ( θ ( j ( θ Ω ( ( Ω j= Ω j=

11 We chose the uber d the fuctos f j ( θ so the frst tegrl sde ( s coverget, for the secod tegrl we could perfor tegrto over the gulr vrbles d the use forule ( d (3 to regulrze o Iterted tegrto o severl vrbles: ( r dr. Aother ethod s to cosder the ultple tegrl s terte tegrl d the e the substrcto for every vrble for eple, = = q F( q q,..., q ( q,..., q ( q q ( q,..., q ( q The sybol q es tht the tegrl s de over the vrble q eepg the other vrbles costt, the uber s chose so the frst tegrl s fte, ths tegrl wll deped o I( q,..., q, the dverget tegrls (eve for the logrthc cse =- c be regulrzed. Now we hve regulrzed the frst tegrl, we hve reduced oe vrble the ultple tegrl, repetg the tertve process for the fuctos ( q, q,..., q ( q ( q, q..., q b j( q,..., q ( q q b( q,..., q ( q j= j= ( Usg ( d ( for every step we c reduce the deso of the tegrl utl we rech to the oe desol cse, whch s eser to hdle. As eple j dy d d y dy y y y y y y (3 ( y dy d y d dy = s s ( s I the lt s we c regulrze the dverget tegrls over vrble y, eepg costt, so fro (3 we get ( ( d f b b 3 dy f = (4 ( y y dyy Wth = d b = so for tl gve tegrl wth y reg reg overlppg dvergece s y we hve de substrcto to get fte tegrl over y (3 repetg the se process we c regulrze the tegrl over,

12 I order to tegrte the fte prt of the tegrl we c use severl uercl ethods. 3 dy For eple, the tegrl f = c be clculted uerclly to ( y y gve 3 f order to vod ters wth log(, r t( or ( y y j j j slr oes sde (3 d (4, other ethod to clculte (3 would be to troduce regultor the for y = R( s,, y, tht wll e (3 to coverge for cert vlues of s, f we use polr coordtes (,y ths becoes R(, y, s π / ( r s( u r ddy( y = dr du y (5 ( cos( u s( u r Itegrto over the gulr vrble u c be crred by uercl ethods, to produce soe ew dverget tegrls, tht wll oly deped o the vlue of r drg ( r ( r, ths tegrls c be regulrzed wth the se ethods we used to gve fte eg to -D tegrls, ule the desol regulrtzto, the regultor s ot the deso of spce-te, so we hve o proble wheever chgg to sphercl coordtes d=4 to overcoe the UV dvergeces. If the tegrd F( q, q,..., q hd o sgulrtes for every q j >, we y epd ths tegrd to ultple Luret seres of severl vrbles, d the perfor the substrcto C ( q b ( q b...( q b order to defe fte prt of the tegrl s, s,..., s,,...,,,..., d q d q d q F C q b q b q b (6 s, s,..., s ,,..., ( (...(,,..., Plus soe correctos due to dverget tegrls ( q b dq =-,,,.... I y cses lthough the tegrls gve ( d ( re fte they wll hve o ect epresso or the ect epresso wll be too coplcte, ths cse we c use the Guss-Lguerre Qudrture forul ( cse the tervl s [, to pprote the tegrl by su over the zeros of Lguerre Polyols w f ( q, q,..., q, wth the wegth epressed ters of Lguerre Polyols d ther roots w =, L ( = ( L ( =

13 FRACTIONAL DERIVATIVES FOR IR AND UV DIVERGENCES: Aother for to del wth IR dvergeces usg our Zet regulrzto lgorth ( d (3 s the followg, the ter ( p ε / ε c be regrded s the dervtve wth respect to of the UV dverget epresso p ε, so f we wsh to clculte the followg tegrl zet-regulrzto lgorth order to gve fte eg to we splt the tegrd to the secod tegrl y be wrtte s d = J (, we ust use our ε d ε, the frst tegrl s o loger probletc t = ε Bol theore to epd the squre root powers of d, ow we c use the ε > d use forule ( d (3 to regulrze the dverget tegrls.for other o-teger epoet ( ε frctol clculus ( 3/ ( α α d vld for p α we y sply use the forul of d = d / α / α wth ths ethod we tur UV dvergece to IR oe by forl dfferetto-tegrto, for the cse of teger ( p ε, we oly hve to study the cse = sce by dfferetto wth respect to other cses c be obted, we use prtl frcto descoposto d dvso of Polyols to set the followg dettes d ( p = p p p C± = F± ( p p ± p ± (7 wth F polyol o p d C costt, ow the sese of Prcpl vlue d µ = log π d µ = log (8 Here c = log µ (Eergy or UV scle s dverget costt obted the regulrzto of the logrthc dverget tegrl, due to the pole t s = we ust recll the defto of Fuctol detert to defe zet-regulrzed product ζ H '(, e = d te the logrthc dervtve wth respect to d pply 3

14 ' the epso of the Dg fucto reltes seres d tegrls. plus the Euler-Mclur forul tht If we wshed to clculte tegrl volvg the ter p we would hve 3/ ε ( ε ε proble sce p = 3/ ( p ε ε becoe sgulr s (3/ ( ε, f we epd the G fucto = ( er ts poles the for ( ε γ O( γ d troduce IR cut-off ε = ε IR, the we wll be ble to ε 3/ γ d defe regulrzed frctol dervtve s = 3/ reg so we c tur UV π d dverget quttes such s p ε to IR dverget quttes t p= p ε, ths would epl cert reltoshp we cojectured before volvg IR dverget tegrl wth other UV dverget tegrl by usg forl frctol dfferetto wth respect to eterl preters. o Multple tegrls: For the cse of ultple tegrls, we c e use of Polr coordtes y teger deso d the replce the tegrl over the gles dω by ultple su,..,, Ω ( Ω (, ( Ω Ω = ϕ dv F q q r dr r r b f r c (9 The fuctos b d c deped oly o the ANGLES but ot o the vrble r, for ultple tegrls we c ot lwys obt ect results for the tegrto over the gles so we should settle for pprotos. For the cse of the logrthc dverget tegrl, we c stll pply other trc we use the dettes =. d ( ( = order to obt P Rtol pproto (Pdé pprot for the squre root of wth Q ( P( d Q( Polyols of degree d respectvely so the logrthc tegrl becoes ore trctble tegrl tht c be regulrzed by usg ( d d P d j j c / s j c / s j Q c (3 c j= c j= 4

15 Due to the fctor, the Re the zet fucto ters ζ j wth j=-,,,... wll ot clude the pole t s= whe we use the forule ( d (3 to regulrze the dverget tegrls / d, the set of { j } c c s chose so the frst tegrl sde (3 s fte for every Re (s >, order to regulrze the secod dverget tegrl we wll eed forul (, c s chose order the Polyol Q( hs o roots o [ c,. For other d of logrthc dverget tegrls le c log ( p dp, dfferetto p wth respect wll e t coverget t the epese of troducg ew preter (costt u, see [] Zedler for further detls o the clculto of UV tegrls d Zet regulrzto. CONCLUSIONS AND FINAL REMARKS: We hve eteded the defto of the zet regulrzto of seres to pply t to the Zet regulrzto of dverget tegrl d > by usg the Zet regulrzto techque cobed wth the Euler Mclur suto forul. For good troducto to the Zet regulrzto techques, there s the boo by Elzlde [4] or the Boo by Bredt bsed o the thetcl dscoveres of Ruj d ts ethod of suto equvlet to the Zet regulrzto lgorth [], other good referece (but bt ore dvced s Zedler [], for the cse of Zet-regulrzed deterts [7] s good ole referece descrbg lso the process of Zet regulrzto v lytc cotuto d how t c be ppled to prove the detty ( = log π. Appretly there s cotrdcto, sce the Re Zet fucto hs pole t s= so the Hroc seres could ot be regulrzed, however E usg the defto of fuctol detert E = oe gets the fte µ ' result for the Hroc (geerlzed seres =, wth the d of the Euler-clur suto forul ths result for the Hroc seres c be used to gve pprote regulrzed vlue of the logrthc tegrl d, for the cse of other types of dverget tegrls d ( we c use g Euler- Mclur suto forul to epress ths dverget tegrls ters of the egtve vlues of the Hurwtz or Re Zet fucto ζ ( s H, = ζ ( s ζ H (, (UV =,,,3,4,...d the vlue of the dervtve of Hurwtz zet fucto log s = ζ (, (logrthc UV, these vlues ecode the UV dvergeces []. For the s H 5

16 d cse of the IR (frred dvergeces the for oe could e chge of vrble to re-terprette these tegrls s q q dq for the cse = we hve logrthc dvergece both t = s so we ust splt the tegrl / d d d to IR d UV dverget prt = ths tegrl wll be equl to d IR regultor so fter few sple clcultos / d = log µ, sce we c sply troduce forl UV Λ d l log UV Λ = Λ Λ, UV regultor s troduced to esure tht the tegrl wll be coverget.we lso beleve tht slr procedure c be ppled to eted our Zet regulrzto lgorth to ultple (ult-loop tegrls d q d q... d q F( q, q,..., q, oe of the dvtges of ths lgorth s tht the deso of the spce does ot pper eplctly so our ethod does ot hve the se probles s desol regulrzto, d c be used whe the Drc trces γ 5 = γ γγ γ 3 pper. The posto forul ( tht ust be turl uber s order to vod oddtes the process of Zet regulrzto wth the Zet d Hurwtz Zet fucto, sce uless s postve teger the equlty ζ (, = ( = does ot hold APPENDIX A: REGULARIZATION METHODS AND EXPANSIONS Λ I order to clculte dverget tegrl df Λ, for y rel d postve we ust epd the tegrd to Luret seres of postve d egtve ters c f = c s j s j s j= ( dz f ( z π (A. ( z c = For ths purpose the Bol theore d the epso of the logrth should be useful, order to fd ths Luret epso ( geerl for our dverget tegrls there wll be oly fte uber of ters sce the tegrls wll dverge t ost s power of the regultor Λ. r = u= u r u r u ( log p log p (A. p Oce we hve epded the fucto f( we y use forul ( to regulrze the dverget tegrls C d for teger or rel, the cse =- s just the 6

17 logrthc tegrl, d c be regulrze by usg the defto of Fuctol ' detert (fte products of ll the postve tegers so = (, y Λ cse f the tegrl s oly logrthc dverget df (, log Λ, forl dfferetto wth respect to eterl preter wll e t coverget, ths cse we ust troduce ew preter sde our theory c. Ths regulrzto odel c be ppled to lost y dverget tegrl tht ppers Qutu Feld theory, usg the trc of Wc rotto for eple wth the tegrl d p d p d q de 4 de = 4 (A R ( E p c c R ( E p c c ( q The de of ths Wc rotto s to chge to gry eerges by replcg the rel s by the gry oe d the e the substtuto E = q d cp = q wth =,,3,( for ore eplto see Zedler [] the lst tegrl (A.3 s dverget but usg sple chge to polr coordtes, t c be tured to the oe 3 d desol (dverget tegrl, ths s stll dverget, but c be regulrzed usg the techque of zet regulrzto for tegrls troduced ths pper. For ultple tegrls F( q, q,..., q we c e chge of vrble to - desol polr (sphercl coordtes, d replce the tegrl over the gulr vrbles dω by su dr r F r θ θ θ Ω (,,,...,, ths su wll clerly deped oly the vrble r, epdg ech ter to Luret seres o powers of (r for soe postve we c regulrze ultple tegrls pprote wy, (ote tht replcg tegrl by su s legtte d there re y Nuercl Qudrture ethods for ths purpose eve the cse tht the tegrd s ot vrt uder rottos (ule the desol regulrzto ethod or other ethods. For the cse of tegrls tht hve lso IR dvergece, we c defe severl preters µ > α >, ths cse the epressos, p α p p α ( ( p p µ re free of IR dvergeces, fter regulrzg the tegrl to cure the UV dvergeces, we c e the lytc cotuto our preters to the cses µ = ε d α = ε for sll preter ε, so the IR dvergeces s dp π pper. Aother useful forul s = s s ( s ( p ε, for d 7

18 p dp eple by dvso of Polyols tegrls of the for c be reduced to p the bove tegrls plus so UV dverget tegrls le ltertve s to cosder the geerl propgtor dp. Aother brute force p ( p p λ, wheever lbd tes the vlue - ths tegrl hs IR dvergece t p = p ± but for lbd postve ths tegrl hs o IR dvergeces, the se would hod for ( p ε IR dverget for egtve vlues of preter lbd, we c use our zet regulrzto lgorth, by epdg the quttes ( p ε λ tht s λ ( p p λ c ( λ wth λ (, to coverget Luret seres of powers d the set = ( λ = fter hvg regulrzed the dverget tegrls by es of the zet fucto ζ ( s λ order to obt fte UV results. For the cse of frctol dervtves d IR dvergeces, we c lwys defe 3/ frctol dervtve reltg UV d IR dvergeces the for 3/ ε d λ γ 3/ ε λ = = wth ( q ε, q,..., q fucto of - π ε dλ vrbles whose zeros re precsely the IR dvergeces our theory. For ore coplcte IR dverget tegrl, f we could use the sector ethod so we c solte the dvergeces the for f (,... g( y,... y d... d dy... dy y... y α βε ( (,... α βε K ( C P( y,... y ( b ε ( b ε (A.4 Wth K d P polyols of severl vrbles C s costt d f d g re sooth fuctos er the org, the coeffcets { bε } so we c fctorze the IR dvergeces s ε, wth sple chge of vrble y = for =,,3,4,.. q the lst tegrl (A.4 becoes UV dverget tegrl g(( q,...( q ( b ε ( bε dq... dq( q...( q C P q q α βε (A.5 ( (,... Although we hve oly studed ultple tegrls, the se c be ppled to Fourer trsfor, for eple f the fucto sde the Fourer trsfor depeds oly o the odulus of posto vector f ( r the we c e chge of vrble to polr r r r. r cosθ coordtes to get dr f e drf ( r r g( θ e, here the gle s R 8

19 r r troduced by the sclr product. =. r.cosθ, the fucto g wll deped oly o the gle θ, so we y replce ultple fourer trsfor by pprote sus of Fourer trsfor oe deso for the fucto f ( r r APPENDIX B: HOW TO OVERCOME THE POLE ζ ( I ths pper we hve see how due to the pole of the Re zet t the pot s = we d could ot regulrze the tegrl uless we use the result for the Hroc seres = reg ( fro > d fte, the f we troduce ths result sde the ' ( Euler-Mclur suto forul we c get fte results for Aother ltertve s to use the detty = d. ( d ( α d = e = log!! α = (B. I ths cse we c evlute the tegrls sde (B. by d log log r log B r log ( = ζ ( α α α α r s = r= ( r! u ( = (B. ( Here α s sll o teger so the zet fucto d ts dervtves ζ ( α re FINITE Aother ltertve s to loo for Pde or Rtol pproto for the squre root of for eple. P Q (! = > (B.3 ( (! 4 d P d I ths cse (B.3 we hve the pproto 3/ Q forul d P c d 3/ 3/ c c d c 5/ Q (B.4, ow f we pply the Isde (B.4 ow there re o logrthc-dverget tegrls, so the pole ζ ( wll ot ow pper Refereces 9

20 [] Abrowtz, M. d Stegu, I. A. (Eds.. "Re Zet Fucto d Other Sus of Recprocl Powers." 3. Hdboo of Mthetcl Fuctos. New Yor: Dover, pp , 97. [] Berdt B. Ruj's Noteboos: Prt II Sprger; edto (993 ISBN-: [3] Dowlg Joth P, "The Mthetcs of the Csr Effect", Mth. Mg. 6, (989. [4] Elzlde E. ; Zet-fucto regulrzto s well-defed, Jourl of Physcs 7 (994, L [5] Elzlde E. ; Zet regulrzto techques wth pplctos Ed. World Scetfc Press ISBN [6] Mzuo, Y. "Geerlzed Lerch Foruls: Eples of Zet-Regulrzed Products." J. Nuber Th. 8, 55-7, 6. [7] Muñoz Grcí, E. d Pérez Mrco, R. "The Product Over All Pres s." Preprt IHES/M/3/34. My 3. [8] Pvel E "Note o desol regulrzto", Qutu felds d strgs: course for thetcs, Vol.,(Prceto, NJ, 996/997, Provdece, R.I.: Aer. Mth. Soc., pp , ISBN (999 [9] Prellberg Thos, The thetcs of the Csr Effect tl vlble t e-prt t [] Voros A. Zet fuctos over zeros of zet fuctos Ed. SPRINGER ( ISBN: [] Zee, A. (3. Qutu Feld Theory Nutshell. Prceto Uversty ISBN ISBN [] Zedler E. Qutu Feld Theory I : Bss Physcs d Mthetcs Ed : Sprger Verlg ISBN: