An arithmetic interpretation of generalized Li s criterion

Transcription

1 A riheic ierpreio o geerlized Li crierio Sergey K. Sekkii Lboroire de Phyique de l Mière Vive IPSB Ecole Polyechique Fédérle de Lue BSP H 5 Lue Swizerld E-il : Serguei.Sekki@epl.ch Recely we hve eblihed he geerlized Li crierio equivle o he Rie hypohei viz. deored h he u over ll o-rivil Rie ucio zeroe k Σ or y rel o equl o ½ re o-egive i d oly i he Rie hypohei hold rue; rxiv: Ukrii Mh. J A riheic ierpreio o hi geerlized Li crierio i give here.

2 . Iroducio I rece pper [ ] we hve eblihed he geerlized Li crierio equivle o he Rie hypohei d ir dicovered i [3] ee e.g. [4] or geerl dicuio o properie o he Rie zeucio well he cloely reled geerlized Bobieri Lgri heore cocerig he locio o zeroe o ceri cople uber ulie [5]. Nely we hve deored h he u k Σ ke over he o-rivil Rie ze-ucio zeroe kig io ccou heir ulipliciie or y rel o equl o ½ well he derivive d! dz z l ξ z z 3... or y </ re o-egive i d oly i he Rie hypohei hold rue correpodigly he e derivive whe >/ hould be o-poiive or hi; cople couge zeroe re o be pired whe uig or. We lo eblihed he relio bewee hee u d ceri derivive o he Rie i-ucio: d! dz z l ξ z z The i o hi echicl Noe i o eblih riheic ierpreio o hi e geerlized Li crierio iilr o hi h bee doe or Li crierio by Bobieri d Lgri [5]... A riheic ierpreio Our coiderio cloely ollow h o Bobieri d Lgri [5]. For uible ucio Melli ror i deied ˆ d

3 3 while ivere Melli ror orul i c d i Re ˆ π wih pproprie vlue o c. The ollowig i ore or le repeiio o Le ro [5] which i priculr ce correpodig o. Le. For 3 d rbirry cople uber he ivere Melli ror o he ucio k i P g i < < g i g i > where P! l ;!!! i bioil coeicie. Proo. We hve or Re>: d d d d! l! I i rbirry cople uber wih Re > or he ucio g we c pply he o clled Eplici Forul o Weil ee [5-7] which i give i [5]: Λ ~ l ~ ~ ˆ d d d γ π Here Λ i v Mgold ucio le u reid h or Re> Λ ' ς ς [4] d : ~ hu i our ce he ucio

4 ~ P l! hould be ued wheever pproprie. erily P L l d ~ P d d L l where L L! L c. [7].! d re geerlized Lguerre polyoil [6] Thi i ey o check h he ucio g do poe he ecery properie i ee o coiuiy d ypoic i priculr or oe δ poiive δ g O or eq. o be rue [5 8 9]. Such pplicio give {! lπ γ l { d l! d l Λ l d } Now i he ecod d hird iegrl i he r.h.. o 3 we ke vrible ror o / er wh hee iegrl ke he or I l d I d 3! { } 3. l d }. The ir wo iegrl re hdled by virue o eple o GR l ; Re > µ ν book []: / d Γ µ ν µ ce we ge l d! µ d Re ν >. Adopig or our l! d. 4

5 5 The ecod pr o he hird iegrl I 3 i by virue o eple o GR book [] equl o / 3 d I γ ψ ; here γ i Euler Mcheroi co d ψ i dig ucio. I he ir pr o hi iegrl we ke he vrible chge ep-: 3! l d I d e e!. Applyig Tylor epio e e e e we ge urher I 3 /!! ς where : ς i Hurwiz ze-ucio. Uig he relio d collecig everyhig ogeher we hve prove he ollowig heore. Theore. For 3 d rbirry cople wih Re > we hve Λ / l / l! ς π ψ 4. Thi reul w publihed i Ukrii Mheicl Jourl [].

6 Our ecod reul i he ollowig ior Theore. For 3 d rbirry cople i i rel we hve i i i li! i i N N i ψ / i / lπ Λ l i i N l i d i ς / i / 5. Proo. lerly or he ce i which ke plce here he δ ucio g do o hve ypoic g O ecery o pply he Weil eplici orul eq. direcly o ollowig gi Bobieri Lgri pper [5] we eed o iroduce ruced ucio g < < : g g i < g g i g i <. Their pplicio give er pproprie vrible ror i he {} brcke i eq. 3 epreio Λ l { li l d} ied o / Λ l { }. A ew vrible ror ro o / uder he iegrl ig give he epreio preeed i 5 hu o iih he proo i re o how h he relio ˆ li g g ˆ hold or he ce hd. Thi i doe by 6

7 verbi repeiio o he eril give i pp o [5]; ee lo Secio 3 o he pree pper. Equliie 4 d 5 re he riheic ierpreio we hve erched or. Now ew rerk re plce. Rerk. Reid h hould be rel or he poiiviy o u queio i equivle o he Rie hypohei. Rerk. The ce o he Theore give well kow equliy Λ ee e.g. [4] ψ / lπ. Rerk 3. For he ce we o coure recover riheic ierpreio o Li crierio give by Bobieri d Lgri Theore l. We eed u o chge d l N relio ς / ς. N d ue he quie kow Rerk 4. Ariheicl ierpreio o geerlized Li crieri or uerou oher ze-ucio ee dicuio i [ ] c be eblihed log iilr lie; c. he riheic ierpreio o Li crierio or Selberg cl i []. 3. ocludig rerk I ee iereig o lyze he poibiliy o he ue o he e pproch ivolvig he ruced ucio g o hdle he ce wih ller vlue o Re viz. / < Re <. O coure hi require oe hypohei cocerig zeroe locio. For eple we c eblih he ollowig 7

8 Theore 3. Aue h he Rie ucio ς i o-vihig or > where rel < < /. The or 3 d rbirry Re / cople wih > Re / δ where δ i rbirry ll ied poiive uber we hve:! ψ / lπ li N N Λ l N l ς / d 6 Proo. Le u ke uch h Re δ. Repeiio o he clculio preeed bove urihe eq. 6 o i re o how h g ˆ li g ˆ. We hve gˆ ˆ g Tl d < l d where T d i pproprie co. Furher!! l d l l.... I codiio o he heore we lwy hve Re δ > hu he cor δ ed o zero le h i er h y egive power o l. Thi ogeher wih he circuce h i he codiio o he heore he u i iie or ll [4] iihe he proo. Tkig io ccou he well-kow relio ς ' lπ ψ ς [4] bove heore how h i we ue he codiio o Theore 3 he or rbirry pproprie 8

9 ς ' Λ N li N ς N. Diereiio o hi equliy wih repec o redily give uber o equliie ivolvig higher order derivive d d ς ' ς d u N Λ l. Thi queio ogeher wih deiled uericl clculio will be ully coidered i epre publicio. REFERENES. Sekkii S. K. Geerlized Bobieri Lgri heore d geerlized Li crierio rxiv: Sekkii S. K. Geerlized Bobieri Lgri heore d geerlized Li crierio wih i riheic ierpreio Ukrii Mh. J Li X.-E. The poiiviy o equece o uber d he Rie hypohei J. Nub. Theor Tichrh E.. d Heh-Brow E. R. The heory o he Rie Ze-ucio lredo Pre Oord Bobieri E. Lgri J.. oplee o Li crierio or he Rie hypohei J. Nub. Theor G. Szegö Orhogol polyoil AMS Providece R.I M. W. oey Towrd veriicio o he Rie Hypohei: pplicio o he Li crierio Mh. Phy. Al. Geo Weil A. Sur le orule eplicie de l héorie de obre preier i Meddelde Fr Lud Uiv. Mh. Se. dedié M. Riez 95 pp Bobieri E. Rerk o Weil qudric uciol i he heory o prie uber I Red. M. Acc. Licei

10 . Grdhei I. S. Ryzhik I. M. Tble o iegrl erie d produc. N.- Y.: Acdeic 99.. Slovic L. O Li crierio or he Rie hypohei or he Selberg cl J. Nub. Theor