ZETA REGULARIZATION APPLIED TO THE PROBLEM OF RIEMANN HYPOTHESIS AND THE CALCULATION OF DIVERGENT INTEGRALS

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1 ZETA REGULARIZATION APPLIED TO THE PROBLEM OF RIEMANN HYPOTHESIS AND THE CALCULATION OF DIVERGENT INTEGRALS Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EHU (Uiversity of Bsque coutry) I Solid Stte Physics Address: Address: Prctictes Ad y Grijlb 5 G P.O Portuglete Vizcy (Spi) Phoe: () E-il: josegrc@yhoo.es MSC: 45G5, 47H3, w,..Gh ABSTRACT: I this pper we review soe results of our previous ppers ivolvig Rie Hypothesis i the sese of Opertor theory (Hilbert-Poly pproch) d the pplictio of the egtive vlues of the Zet fuctio ( s) to the diverget itegrls s d d to the proble of defiig cosistet product of distributios of the for D ( ) D ( ), i this pper we preset ew results of how the sus over the o-trivil zeros of the zet fuctio h( ) c be relted to the Mgoldt fuctio ( ) ssuig Rie Hypothesis.Throughout the pper we will use the ottio ( s R ) ( s ) the zet regulriztio for the diverget series s s> or s= eig tht we use Keywords:Zet regulriztio, Urysoh equtio, epoetil olierity, Rie Hypothesis Hilbert-Poly opertor, diverget itegrl.spectrl Zet fuctio ( s ) d Rie Hypothesis : H I cse Rie Hypothesis (RH) is true, i previous pper [6] we give the physicl equivlece betwee the eplicit forul for the Chebyshev fuctio ( ) d the ˆ forul for the trce of the Uitry opertor ˆ iuh U e, where H is the Hiltoi opertor ˆ ih, tht is H is precisely the Hilbert-Poly opertor solutio to Rie Hypothesis, let be the itegrl represettio

2 (.) () s log( ci ( s) ) ( ) ds () i ci ( s) s Lettig u e, d differetittig with respect to u we fid the (trce) idetity u u / u / u / d( e ) e iue ˆ ˆ iuh 3u u du e e e e e Tr U e u > (.) Usig the seiclssicl represettio for the trce e iue i ters of itegrl over Phse Spce, we hve tht the potetil V() iside Hiltoi H c ot be rbitrry but ust stisfy id of olier Urysoh itegrl equtio ( r > ) log( r) d ( r) r d e r dr r r iv ( ) i / 4 3 r u e (.3) d( ) The derivtive of the Chebyshev fuctio is defied s ( p ) d log( ) p, p (su te over prie d prie powers ).However (.3) is too cople to hve ow lytic solutio, good ethod to solve would be to suppose tht the Opertor proposed by Berry d Ketig [] plus iterctio is the correct Hilbert-Poly opertor, i tht cse Hb p W ( ) d we c lierize (.3) t first order i the couplig costt s ˆ iuh ˆ ( ), Tr e iu dpf W u u iu( V ( ) ) Also, if we itroduce the fuctio Z u Z u de ˆ ( ), iup F W u de W ( ) (.4), with cotiuos prtil derivtives ( ), the solvig (.3) is equivlet to fidig solutio to the iitilvlue proble Zu ( ) Zu ( ) iud Zu ( ) ( iu) u u / / / ( ) i u u d e e u 4 e e e Z () 3u u u du e e (.5) Epressio (.8) tells us tht provig RH is equivlet to show tht the ODE give i d V ( ) (.5) with d R d d, V ( ) d usig (.5) together with! d

3 fiite power epsio for V(), usig (.5) we could obti the costts d to get pproite solutio for the potetil V(). If RH is true d ie, with E E beig the eigevlues of certi opertor H p V ( ), usig epressio (.) d the fuctiol equtio s s ( s) ( ) Cos ( s) ( s),the for we c defie spectrl Zet fuctio, ivolvig the otrivil zeros of Zet d pries d prie powers s Sec R t ithˆ s ( s) t / t d( e ) s dttr e dte e t s 3t t E ( s) ( s) d e e The vlue E Eigevlues E e E, let us itroduce d () ds would be the regulrized product of ll the positive (.6) this epressio c lso be used to obti Dirc esure for the E dt t t s E E s s s dtk( t) t s (.7) ( ) ( ) Usig the properties of the Melli trsfor pplied to solve lier itegrl opertors I[ f ] dtr( t) f ( t), if we cobie (.6) d (.7) we get the result / t / t / t dt / t e d( e ) e ( ) K( t) e 3/ t / t E E t t dt e e (.8) If we too the Melli trsfor d s iside (.8) together with the chge of vrible t=z we would recover equtio (.6), ote tht the Melli trsfor of the Kerel K ( t) does ot deped o the otrivil zeros it. Usig test fuctios i h iside (.8) obtied fro our Trce forul for ˆ iuh Tr e we c relte the coverget su h( ) to su over pries d prie powers 3

4 K t dh dt e c. c h( ) t t dt e e / t / t / t i / t e d( e ) e 3/ t / t (.9) Forul (.9) d its result c be copred with sus (eplicit forul for Chebyshev fuctio) d Z( ) N, tht c be clculted ectly. o The Trce ˆ iuh Tr e d the su h( ) Eve though we c ot solve equtio (.3) we c use the Trce epressio (.) to fid stites for sus h( ). First we defie couple of fuctio g() d h() with the followig properties Both g()=g(-) d h()=h(-) re eve fuctios g( ) li eists d it is fiite The fuctios h() d g() re relted by Fourier Cosie trsor dh( ) Cos( ) g( ) The fuctio h() c be defied by lytic cotiutio to the regio of cople ple defied by Re (s) =, i prticulr h( i / ) i If RH (Rie Hypothesis) is true, the the Trce (.3) is just su of cosies cos( u), the if we te g(u) s test fuctio ˆ iuh h( i / ) h( i / ) ( ) u / g( u) u e dug( u) Tr e g(log ) due h( ) (.) I order to obti (.) we hve used the represettio i ters of Dirc delts of the u u / d( e ) ( ) derivtive of Chebyshev fuctio to get dug( u) e g(log ), d du the Euler forul for cosie to represet the itegrl dug( u) e u / s the su ( / ) ( / ) h i h i. A specil cse is wheever we choose h( ) ( s ) ( s ) d g( u) cos( u) (.) 4

5 The we c use the fuctios i (.) d the forul (.) to get '(/ is) '(/ is) ( / ) ( s ) (/ is) (/ is) 4s / s (.) (.) is the desity of sttes i QM, d c be used to ow how y zeros of the T for ½+is re with igiry prt less tht give T sice N( T ) ds ( s ) A forl derivtio of (.) c be obtied cosiderig the followig idetities for diverget series ivolvig zet regulriztio or lytic cotiutio ( ) '(/ is) e (lierity) (.3) / is e (/ is) ( ) ( ) is ( ) cos( s log ) is (.4) Ad the Lplce trsfor of Cosie st dte cos( t) s( s ).The poles iside (.4) re of three ids s fro the No-trivil zeros of Zet fuctio, s i / due to the diverget vlue () s i / N for the trivil zeros of zet fuctio -,-4,-6,... d o Rie-Weyl forul d solutio for the iverse of potetil V(): A siilr forul to (.) hd bee previously itroduced by Weyl i 97 [9] ( ) ' ir h( ) h( i / ) h( i / ) g()log g(log ) h( r) dr 4 (.5) Weyl sutio forul c be used to solve equtio (.) if we e iside this itegrl equtio the chge of vrible V ( ), the (.) is siply proportiol to the iverse Fourier trsfor of proportiol to other su e iu i / 4iu o the itervl u, which is just ivolvig the igiry prts of the Rie zet zeros, to get rid off the su we c use (.5) to epress the iverse of potetil 5

6 A ( ) ( / 4) i V BCos c D ( )cos( log / 4) ir E / Fp. v dr / / log 4 r r (.6) Where A, B, c, D, E, F R re rel preters tht defie the potetil, fro forul (.6) the Hiltoi would be self djoit d its eergies (igiry prt of zeros) would be rel ubers. (If the ew liits of itegrtio fter tig the chge of vrible V ( r) were ot but rel ubers c,d the we could ultiply the left side of equtio by W d ( ) H ( c) H ( d), c, d V ( ) ) c The su ivolvig ( ) c be treted usig frctiol clculus d zet regulriztio / / ( )cos( log / 4) d '(/ i ) d '(/ i ) i i / / / log d (/ i ) d (/ i ) (.7) At the poits the iverse of potetil becoes, s we c epect fro / / / (.8) Although we hve ivestigted the trce ivolvig the Chebyshev fuctio ( ) ( ), with soe chges it c lso be pplied to fid Trce ivolvig + M ( ) ( ) Z the Mertes fuctio M ( ) ( ) M ( ) otherwise, if there re o ultiple zeros of Rie zet fuctio so '( ), the we c fid epressio for Mertes fuctio ivolvig su over Zet zeros s ( ) M ( ) '( ) ( )! ( ) > (.9) Sice the Mertes fuctio is just step fuctio its first derivtive it will be set of Dirc delt fuctio so dm ( e ) ( ) ( ) (log ) (.) du u u / due g u g Where if is ot squre-free. ( ) = ( ) if is squre-free with - distict prie fctor is Mobius fuctio 6

7 u The differetitig respect to d settig e, ssuig RH is true so ll the o-trivil zeros re of the for / it, (with t rel ) d choosig two eve test fuctios (g, h) relted by Fourier trsfor g( ) h( u)cos( u) du oe gets ( ) h( ) ( ) ( ) (/ ) u g(log ) dug( u) e ( )! ( ) (.) ' i (.) is siilr epressio to Rie-Weyl eplicit forul for the su reltig e whe > pries d Rie Zet zeros. As eple let be g( ) with whe < ' ' rel d positive uber, if / oe should hve the Prie Nuber theore, ( ) ( ) so, eve stroger coclusio is tht if RH is true the ust be / coverget for every positive d equl to ( ) 4( ) ( ). ' / i i ( )! ( ) 4 (.) /. Zet regulriztio for diverget itegrls: Give the fuctio f ( ), we c use the Euler-Mcluri sutio forul to obti recurrece reltio betwee itegrl of the for with d d the series i, ref [7] i I(, ) p dp Z B I I r I r r (, ) ( / ) (, ) i r( ) (, ) i r ( r)! (.) B r!( r ) r d d d ( ) ( r)!( r )! r ( ) The coefficiets r vish whe r, hece the su iside ( r ) (.) is fiite if is iteger, i the physicl liit the cutoff, this es the 7

8 series i to be diverget for, i this cse we should use the Fuctiol i equtio for the Zet fuctio to obti the (Regulrized) vlue li 3... R( ) ( ) (.) (.) is the Zet-regulrized vlue for the diverget su evolved i (.), usig this ethod we c copute the diverget itegrls I (, ), for =,,3 I(, ) () / d I(, ) I(, ) ( ) B I(, ) I(, ) ( ) I(, ) d d (.3) 3 B 3 I(3, ) I(, ) ( ) I(, ) ( 3) B3I (, ) d The cse = is just equl to the diverget series tig the regulrized vlue -/ evluted fro () For rbitrry fuctio f() so its itegrl would diverge s power of the cutoff we could epd f() ito Luret series coverget for < d > so we fid N N i j df ( ) c (, ) ri r c I (,, ) O( ) d ci c j (.4) r i j c i, tig, d usig (.) (.) (.3) to regulrize the diverget itegrls I (, ) N we could obti regulrized (fiite) vlue for the itegrl df ( ), d however the logrithic diverget itegrl I (,, ) c ot regulrized by our forule, the solutio would be to use the Euler-Mcluri sutio to pproite the diverget itegrl by diverget Hroic su tht c be ttched Ruj su ( =Euler-Mscheroi costt) 8

9 o Zet regulrized product of distributios: Forule (.-.3) c be used to copute diverget itegrls of the for s d, but lso could give swer to the proble of ultiplictio of two distributios ivolvig Dirc delt d its derivtives D ( ), if we tried to defie the product of distributios ivolvig delt fuctios we could use the covolutio theore pplied to the Fourier trsfor ( A=orliztio costt) : ( ) i D ( ) D ( ) F AF dtt ( t) (.5) Ufortutely (.5) es o sese, the itegrl is diverget for every rel or cople vlue of, if d re positive itegers usig the Bioil epsio [ R] i D ( ) D ( ) = i AD ( )( ) i D () (.6) i D ( ) D ( ) = AD ( )( ) i ( ) d (.7) R stds for regulriztio (regulrized vlue), the diverget itegrls coe ow fro the dirc delt d its derivtives evluted t =, which re proportiol to d for =r+ (Odd) the itegrl cosidered i Pricipl Vlue is, for =r (eve iteger) the itegrl c be writte s r r i D () I( r, ) r, I(, r) d (r=iteger) d c be evluted usig (.) d (.). The epressio (.7) is rel,this is wht oe would epect sice the product of two distributios tig oly rel vlues ust be rel, however (.6 ) is ot still ivrit uder the chge d (this is iste we de i pper [7] ) so we should te ore syetricl product of distributios defied by (.8) R D D ( ) D ( ) D ( ) D ( ) D ( ) The siplest cse is == so ( ) A ( ) R For the cse of d ot beig iteger or we hve shifted dirc delt D ( ), we could use the ideties for the -th power of or the trsltio D d opertor e d D i the for d 9

10 j D r j r j e D ( ) ( ) D ( ) ( ) j! r D D j r (.9) I cse of itegrls o d R df( ) d R, if the fuctio F, is ivrit uder Loretz trsfortios, the ig Wic rottio to igiry tie t it,the etric becoes ds d dy dz dt which is ivrit uder rottios, tig 4- diesiol polr coordites our itegrl c be evluted s d / d drf ( r) r ( d / ), if ot we could replce the itegrl over the cross sectio (gles) d by discrete su i drf ( r, ) r d i, with d equl to the diesio of spce-tie Eple: d with i this cse the itegrl hs power-lw (qudrtic) divergece, > d iteger (this is ot relevt sice the itegrl diverges oly for big ), the Luret series for bigger th is ( ) j j j3 of / by the diverget series, if we pproite the logrithic diverget itegrl (fter chge of vrible =t+) the, the pproite Zet regulrized vlue of the itegrl would be j () '( ) ( ) d ( ) ( ) j3 j R j (.) Aother eple without logrithic divergece, would be d 4 ( ) i this cse ( ) 3 the regulrized fiite vlue is just (), the logrithic derivtive 3 of G fuctio iside (.) is just the Ruj resutio of the Hurwitz series (, ) voidig the pole t s= H Aother ethod to evlute these id of diverget itegrls is, substrct su of the N N i i for ci, so the itegrl defied by d f ( ) ci d eists i the Rie i i sese, i cse we hd Fourier trsfor, we would dd d substrct ters i the for N i iu ci e i which re proportiol to the derivtives of ( u).

11 Eple 4 d d d d ( ) the first itegrl o the right of = is coverget, the logrithic itegrl c be (pproitely) regulrized by es of '(3) Ruj sutio to the fiite vlue d the other itegrl is just the (3) regulriez vlue () 3. O R siply use the -diesiol polr coordites d try usig substrctio to i obti itegrl of the for dr f ( r, r U diverget itegrls of the for r dr N i Coclusios d fil rer i, the ide here is isolte the I this pper we hve used the ethod of Zet regulriztio of series pplied to the proble of fidig fiite results for diverget itegrls d Hilbert-Poly opertor i order to solve Rie Hypothesis. d to give dequte O the first prt of the pper we show tht if RH is true the the Chebyshev fuctio evluted t =ep(u) is just the trce of the epoetil of Hiltoi H p V ( ) whose eigevlues re precisely the igiry prt of the otrivil zeros, we eted this ide d defie the distributio iu Z( u) e cos( u) which c be clculted for every u bigger th d whose vlue is relted to the derivtive of Chebyshev fuctio ( u d e ). We lso discuss the pplictios of this du iu trce forul for Z( u) e d how c be used to obti the vlues of the su f ( ) i siilr wy to Rie-Weyl eplicit forul, we lso obti ethod to clculte IT N( T ), usig zet regulriztio we obti ect epressio for the oscilltig ter i N(T) s rg it which coes fro cos( s log ) itegrti the regulrized vlue of the series ( ) It y see t first sight tht there is o reltio betwee Rie-Weyl forul for the su f ( ) d the oe we obtied i (.), however we c prove the followig idetity (i the sese of distributios)

12 u / u / e si( u) ( ) du e si( log ) u e u log i Ilog i log log 4 (.) This forul c be ieditly obtied fro the defiitio of the fuctio tht gives us the uber of zeros with igiry prt less th give N(),the property of the Chebyshev fuctio ( u ) e d the properties of Fourier iverse sie d cosie trsfors for our Trce Z( u) cos( u ) with Z( u) u dn( )si( u) d i N( ) Ilog i log log 4 (.) Tig the iverse sie trsfor d the defiitio of Z(u) for u > we recover the bove forul reltig Fourier sie iverse trsfor d the igiry prts of the logrith of i d i, if we differetite respect to d use eve 4 ir test fuctios g() d h() with drh( r) e g( ),d tig ito ccout tht ' we c epd the Rel prt of is ito the diverget (but regulrized) su ( ) co s( s log ), the the oscilltig prt of the N() gives us the ter ( ) proportiol to g (log ), itegrtig over o (.) the right ters o the derivtive of (.) is precisely the Rie-Weyl cotributio to the su h( ) d ' i h( ) g() log the ter o the left would be 4 is just the epressio for the su g( u) e du e u u /, which h( ) obtied fro our trce Z(u), reeber tht for u < fro the defiitio of Chebyshev fuctios i ters of the eplicit forul, there is fctor sice the Fourier cosie d Fourier epoetil trsfors for eve F g( ) F g( ) fuctios re relted by c e O the secod prt of the pper, we use the Euler-Mcluri su forul together with the lytic cotiutio of the Zet fuctio ( s) to egtive vlues, ote tht E-M (Euler Mcluri) forul is correct for itegrls of the for d wheever

13 sice the series ( ) is coverget, the ide of our ethod is to use Euler- Mcluri forul d the perfor lytic cotiutio to vlues with, i order to clculte the itegrl d, so ( ) d we c defie recurrece equtio for every itegrl such s forul (.) beig the iitil ter i the recurrece d (), this id of zet regulriztio of series would llow to clculte diverget itegrls d to defie regulrized product of distributios D ( ) D ( ) by pplyig Covolutio theore to the weird fuctio dtt ( t), Z which c ot be defied for y uless we ow how to clculte diverget itegrls. Although we c ot use (.7) i order to regulrize d we c pproite this diverget itegrl by the Hurwitz series H (, ) which is still diverget but c be ssiged fiite vlue i the sese of Ruj '( ) resutio other ltertive would be differetitig respecto to to get ( ) coverget itegrl so we hve the result log( ) C with C ifiite costt. Appedi A: itegrl Trce for the Gree fuctio A forul for the su ( E E ) i ters of the Trce of the Resolvet (gree fuctio ) of Qutu Hiltoi Hˆ E c be defied s: (, ', ) 4 (,, ) ( ) R 4 Tr G E d G E E E G(, ', E) E i Hˆ (A.) Oe of the esiest ethod to prove this, is to cosider tht give coverget series t with su S d its Borel trsfor B(s) defied by B( S, ) dt e! the S=B(S), S i this cse if we te the series E ( ) ( i H ) t ( i Hˆ ) E dte ( i H )! E i H E E (A.) Where is sll uber so, the usig the forul for the Pricipl vlue 3

14 PV. i ( ), i this cse tig the trce of the opertor iside (A.) we i c give proof to (A.) usig the techique of Borel resutio. Aother eple of the ethod of Borel resutio, let be the geertig fuctio of the coefficiets ( ) P( ) ( ) ( ), let be the fuctio f(t) defied by dtf t t the usig gi the Borel-geerlized resutio ( s ) ( ) s P( ) dt ( ) ( t) f ( t) ( ) ( ) or P( ) f ( t) dt t (A.3) If we too the Melli trsfor o both sides d s Pˆ ( s) Kˆ ( s) Fˆ ( s), or i ters of iproper itegrls oe would fid ( s) dt ( )( ) sice si( s) s dtf ( t) t ( s) (A.5) This lst forul is ow s Ruj Mster theore, ote tht we hve proved this oly usig the fct tht for coverget series its sus d Borel trsfor ust be equl S=B(S). Accordig to tihs forul our K fuctio defied previously o (.7) si( ) ( ) would be equl to K( ). ( ) ( ) Appedi B: A Rie-Weyl sutio forul Rie-Weyl forul, is good tool to clculte sus over the igiry prts of the Rie zeros f ( ), usig Zet regulriztio d the i property of Dirc delt distributio d ( ) g( ) g( ) together with the Hdrd product z / z represettio ( z).. we c give proof of it, we will ( z ) z ( z / ) lso eed the followig forule li i ( ) P i d '(/ i is) ( ) log( ) /. is i e (B.) (/ i is) The first is Sohotsy s forul for delt distributio d the secod is just the usul zet regulriztio procedure for the Dirichlet su ssocited to Mgoldt fuctio, 4

15 tig logriths iside the Hdrd product d differtitig t the poit z is, where epsilo is sll qutity so we c igore cudrtic ters, we c ow obti the followig forul ' is ' s i is log( ) ( s ). i ip. V s 4 s 4 (B.) To obti (B.) we hve used both forule i (B.) so i ( s ) ip s i s with d the derivtive of log is i c be clculted usig Zet regulriztio i ters of su ivolvig the Mgoldt fuctio ( ) so (B.3) reds ow is ' s i ( ) islog( ) ( s ). i log( ). e ip. V / i 4 s s 4 (B.3) If we use iside (B.3) the test fuctios g() d h() relted by Fourier trsfor so d iu g( u) h( ) e with g() d h() beig eve fuctios stisfyig certi d h( )si( b) h( ) properties, i Cuchy s pricipl vlue h ( ) = d d 4 4 d if h() c be cotiued lyticlly to the criticl strip we fid g i ( ) /. d d e. d g( ) e. h( i / ) h( i / ). 4 (B.4) I order to chge the order of itegrtio iside (B.4), we ssue h() d g() re regulr eough d pply Fubii s theore. Appedi C: Geerliztio of Urysoh No-lier equtio The proble with itegrl equtio (.3) is the fct tht this iplies solvig itegrl equtio with distributio ( u d e ), ths id of itegrl equtio c be geerlized du to iclude test fuctios (g,h) with the followig properties 5

16 iu g( r, ) duh( r, u) e for rel preter r with g d h beig eve fuctios o the vrible g( r, ) g( r, ) g( r, ) is fiite i the liit The itegrl of g r e / (, ) iur o,, d the su ( ) g( r,log ) log du dr h( r, u), d the se holds for g(r,) so the olier Kerel re o L g( r, ) d h( r, ) re regulr eough so we c use Fubii s theore to iterchge the order of itegrtio The itroducig this tests fuctios o (.3) d tig the itegrl over u iterchgig the order of itegrtio d usig the Urysoh itegrl equtio (depedig o the choose of g ) h( r, ) dug( r, u) e iuv i ( ) / 4 we hve u / u / e ( ) g( r,log ) u e 4 (C.) du ug( r, u) e dh r, V ( ) log (C.) is the geerliztio of (.3) with the dvtge tht we re delig with fuctios isted of distributios, V() depeds o Mgoldt fuctio ( ) this is ot csul sice the pries d Rie Zet zeros re relted. We hve used the seiclssicl iue pproitio for the series (i the sese of distributio) Z( u) e, E R (if RH correct), this Z(u) will deped o the derivtive of Chebyshev fuctio ( u d e ) du I siilr er we c use the properties of the Lplce trsfor of si( t) d t use the lytic cotiutio to epress (.) ( ) t ( ) t si( log ) 4 log (C.) i I log i log log 4 As poited before, tig derivtives o both sides of (C.) we could ssig fiite ( ) vlue to the diverget su cos( log ), usig test fuctio we could copute 6

17 df the sigulr itegrl d I log i with Poles beig the o-trivil zeros d ( ) over the lie Re(s)=/, is relted to series f (log ). Refereces: [] Apostol, T. M. Itroductio to Alytic Nuber Theory. New Yor: Spriger- Verlg, 976. [] Berry, M. V. d Ketig, J. P. " d the Rie Zeros. I Supersyetry d Trce Forule: Chos d Disorder (Ed. I. V. Lerer, J. P Ketig, d D. E. Khelitsii). New Yor: Kluwer, pp , 999. [3] Corey, J. B. "The Rie Hypothesis." Not. Aer. Mth. Soc. 5, , 3. vilble t [4] Delbere E., Ruj's Sutio, Algoriths Seir, F. Chyz (ed.), INRIA, (3), pp [5] Elizlde E. ; Zet-fuctio regulriztio is well-defied, Jourl of Physics A 7 (994), L [6] Grci J.J Chebyshev Sttisticl Prtitio fuctio : A coectio betwee Sttisticl Mechics d Rie Hypothesis Ed. Geerl Sciece Jourl (GSJ) 7 (ISSN ) [7] Grci J.J A ew pproch to reorliztio of UV divergeces usig Zet regulriztio techiques Ed. Geerl Sciece Jourl (GSJ) 8 (ISSN ) [8] Polyi, A. D. d Mzhirov, A. V., Hdboo of Itegrl Equtios, CRC Press, Boc Rto, 998. [9] Weyl, A. "Sur les forules eplicites de l théorie des obres", Izv. Mt. Nu (ser. Mt.) 36 (97) 7