Proof that the real part of all non-trivial zeros of Riemann zeta function is 1/2 6 KimichikaFukushima 1 ∗ 0 AdvancedReactor SystemEngineeringDepartment, 2 ToshibaNuclearEngineeringService Corporation, t 8, Shinsugita-cho,Isogo-ku,Yokohama235-8523,Japan c O 1 ThisarticleprovestheRiemannhypothesis,whichstatesthatallnon-trivialzerosofthezetafunctionhaveareal 3 partequalto1/2.Weinspectindetailtheintegralformofthe(symmetrized)completedzetafunction,whichisaproduct ] between the zeta and gamma functions. It is known that two integrallines, expressing the completed zeta function, M rotatedfromtherealaxisintheoppositedirections,canbeshiftedwithoutaffectingthecompletedzetafunctionowing G totheresiduetheorem.Thecompletedzetafunctionisregularintheregionofthecomplexplaneunderconsideration. . Forconvenienceinthesubsequentsingularityanalysisoftheaboveintegral,wefirsttransform(convert)thisintegralto h thatalongthelineparalleltotherealaxis. Wetheninvestigatethesingularitiesofthecompositeelements(causedby t a polynomialintegralsinoppositedirections),whichappearonlyinthecaseforwhichthedistancebetweenthecontours m andtheoriginofthecoordinatesapproacheszero. Therealpartofthezerosofthezetafunctionisdeterminedtobe [ 1/2alongasymmetrylinefromthesingularityremovalcondition.(Intheotherpoints,thesingularitiesareadequately cancelledasawholetoleadtoafinitevalue.) 6 v 3 5 5 1 Introduction 3 0 Byconnectingcomplexanalysiswithnumbertheory,Riemannobserved[1]that(denotingasetofrealnumbersbyR 2. andlettingx R)thefunctionp (x), whichdenotesthenumberofprimenumbersbelowagivennumberx, contains 0 thesummatio∈novernon-trivialzeros(pointsatwhichthefunctionvanishes)ofthezetafunction. Riemannexpected 6 (denotingasetofcomplexnumbersbyCandlettingz C)therealpartofthenon-trivialzerosofthezetafunction 1 z (z) to be 1/2, which is known as the Riemann hypot∈hesis. Furthermore, von Koch showed [2] that p (x) is well : v approximatedbytheoffsetlogarithmicintegralfunctionLi(x)as i X p (x)=Li(x)+O(x12logx), (1) r a which is equivalentto the Riemann hypothesis. We denote a set of naturalnumbersby N and let n N and z C, thenthe zeta functionz (z) is definedas a function,which is analyticallycontinuatedin the complex∈planefrom∈the expressiondefinedbelow[3-5] ¥ z (z):= (cid:229) 1, (2) nz n=1 forzthatsatisfiesRe(z)>1(wedenotetherealandimaginarypartsofzasRe(z)andIm(z), respectively.) Thezeta functionis also obtained with the help of the gammafunction G (z), and, lettingt R, then the gamma functionis ′ ∈ definedasafunctionthatisalsoanalyticallycontinuatedintoallpointsinthecomplexplanefrom[3,6-9] ¥ G (z):= dt′(t′)z−1exp( t′), (3) 0 − Z E-mail:[email protected];[email protected]:+81-90-4602-0490Phone/Fax:+81-45-831-8881 ∗ 1 forRe(z)>0. Concerningthezerosofthezetafunction,whichstatesz (z)=0,thereexisttrivialzeros,suchasnegativeintegers 2, 4, [3]. Incontrast,Hardyshowedthatnumerousnon-trivialzerosofthezetafunctionexistalongthelinewith − − ···· the realpartequalto 1/2[10]; however,notall the realpartsof the non-trivialzerosare known. Thecomputational approach[11]stronglysuggeststhattherealpartofzerosofthezetafunctionis1/2. Ontheotherhand,lettingz,w C,forthecompletedzetafunctiondefinedby ∈ zˆ(z):=p −2zG (z)z (z), (4) 2 integralformofthe(completed)zetafunctionisexpressedas[12] p −1−2zG (1−z)z (1 z) 2 − =p −1−2zG (1−2 z) 0 1dwexpw(pz−i1we)xp(e−xpp(iw2p)iw) Z ց − − +p −2zG (2z) 0 1dwexp(wp −iwz)expe(pxpiw(2)p iw). (5) Z ւ − − The above integralis performedalong the integral lines 0 1 and 0 1 with the slopes 1 and +1, respectively, ց ւ − whichpassthroughanarbitrarypointintheregionbetween0and1oftherealaxis. Sincetheresiduetheoremexhibits theaboveequation,theintegralformisindependentoftheshiftofthisintersectionpointbetween0and1.Furthermore, intheoriginalform[12]oftheaboveequation,thefunctionG (z/2)inthesecondtermontherightisproportionalto theregularfunctionforRe(z)<1. ThefunctionG ((1 z)/2)ontheleftisregularintheregionRe(z)<1,whilethe rightside is also regularbecause ofthe existenceof th−e derivative[13, 14], andthe functionz (1 z) is analytically continuateduniquely[13]intotheregion0<Re(z)<1(therealpartofzerosofz (z)existsonlyin−thisregion). This papertakesintoaccounttheformmentionedabove. Sincethegammafunctionisregular,thenon-trivialzerosofthecompletedzetafunctionzˆ(z),whichistheproduct betweenthegammafunctionG (z)andzetafunctionz (z), coincidewiththoseofthezetafunctionz (z) intheregion being considered with the real part between 0 and 1. As is described in this paper, each of the two line integrals expressingthe completedzeta functionhas a singularitywhen the integrallines approachthe axis origin. However, thecompletedzetafunctionzˆ(z)doesnotdependonaspecificpointoftheintersectionpoint(shiftedbetween0and 1alongtherealaxis)betweentheaboveintegrallineandtherealaxisduetotheresiduetheorem,andzˆ(z)isregular intheconsideringregion. Then,thesesingularitiesmustexactlycanceleachotherforzˆ(z)=0,whichisexpectedto leadtothedeterminationoftherealpartofthezerosofthezetafunctionz (z). Consideringthestatusmentionedabove,thispaperisaimedatprovingtheRiemannhypothesis.Wefirsttransform (convert)theintegral(intheintegralformofthecompletedzetafunction)alongtheintegrallinerotatedfromthereal axis to the integralalong the line parallelto the coordinate(real/imaginary)axis, for conveniencein the subsequent analysisofthesingularityoftheintegralinacomplexplane.Bythisconversionoftheintegral,thesingularityanalysis canbeconcentratedonthecomponentoftheintegralalongthelineparalleltotherealaxis(ortheequivalentintegral component),droppingthecomponentoftheintegralinthedirectionoftheimaginaryaxis. Thisresearchthenaddressesthesingularitiesthatappearinthetwointegrallinesoftheintegralformofthecom- pletedzetafunction. Thesingularitiesoftheintegrandsforthecompositeelementsneartheoriginoftherealaxisare causedby polynomials, only in the case whenthe contour-origindistance approacheszero. These singularitiesade- quatelycanceleachotheryieldingafinitevalueindependentoftheintegralcontourasawhole. Incontrast,fromthe equationzˆ(z)=0forcompletedzetafunctionzˆ(z),therealpartofthezerosofzˆ(z)isdeterminedbyrequiringthese singularitiesto be an identical order power of the integral variable in the integrandsleading to the exact singularity cancellation(givenbyTheorem2). Thisrequirementresultsinavalueof1/2fortherealpartofzerosofthecompleted zetafunctionzˆ(z)(andtheoriginalzetafunctionz (z))duetothesymmetrywithrespecttothe1/2realpart,whichis the originalityof the presentstudy andprovesthe Riemann hypothesis. TheRiemann hypothesisis oneof the most 2 important unproved problems in mathematics, and has its equivalent and advanced (extended) conjectures in other relatedfields. ThepositiveproofoftheRiemannhypothesisadvancesmathematicsinotherrelatedfields[15,16]. Thecontentsofthispaperareasfollows. Section2providesthetransformationoftheintegralintheintegralform ofthecompletedzetafunctiontothelineparalleltotherealaxisforconvenienceinthesubsequentsingularityanalysis oftheintegral.Section3presentstheproofthattherealpartofallnon-trivialzerosofthezetafunctionisequalto1/2, aswasconjecturedbyRiemann,followedbytheconclusion. 2 Transformation of the integral to that along the line parallel to a coordi- nate axis for the singularity analysis in a complex plane Thissectionprovidesthetransformation(conversion)oftheintegral(intheintegralformofthecompletedzetafunc- tion)alongtheline rotatedfromthe realaxisto theintegralalongthe line parallelto the realaxis(orthe equivalent integral component)for conveniencein the subsequent analysis (in Section 3) of the singularity of the integral in a complexplane. Bythistransformation,theanalysisoftheintegralinacomplexplanebecomesindependentfromthe integralcomponentinthedirectionoftheimaginaryaxis. Notationsusedinthispaperareasfollows. Letz Cand x,y Randletibetheimaginaryunit,then ∈ ∈ (x,y):=z=x+iy. (6) Wedenotetherealandimaginarycomponentsas z :=Re(z)=x, (7) R z :=Im(z)=y. (8) I We first define the integrands, the main points in the complex plane, and integrals with contours for use in the subsequentlemmainthissection. Definition1. Letw,z C,thentheintegrandsI andI inw-planearedefinedby N P ∈ IN:=p −1−2zG (1−2 z)expw(pz−i1we)xp(e−xpp(iw2p)iw), (9) − − IP:=p −2zG (2z)exp(wp −iwz)expe(pxpiw(2)p iw). (10) − − Definition2. Leta ,a ,a Rbepositivefinitenumbersbetween0and1,witha =a forI inEq.(9)anda =a 0 0l 0r 0 0l N 0 0r forI inEq. (10),respective∈ly. Letb Rwithb >0beasmallnumber. Thenwedefinethepointsw ,w Cin P 0 0 lU lL ∈ ∈ thecomplexplane,indicatedinFig. 1,as w :=(a ,b ), (11) lU 0 0 w :=(a , b ). (12) lL 0 0 − Lete¯ Cbedefinedby 3 ∈ 3 e¯ :=exp( p i), (13) 3 4 andletr Rwithr >0betheradiusofarcscenteredatw andw ,thenwedefinethefollowingw ,w ,w ,w 2 2 lU lL 11 12 21 22 CinFig.∈1 ∈ 3 3 w :=r e¯ +w =(r cos( p )+a ,r sin( p )+b ), (14) 11 2 3 lU 2 0 2 0 4 4 3 ✻ w C Im 11 1 ❅❅ ց ❅ ❅ I1f I ❅ 1 ց❅ I w12↑ 1← ❅wlU R✲e ← O w w lL❅ I ← 22 2 ❅ ← ↑ ❅❅I2 I2f ց ❅ ❅❅ ց w C 21 2 w : (a ,b ) lU 0 0 w : ( √2r +a ,√2r +b ) 11 − 2 2 0 2 2 0 w : ( r +a ,b ) 12 2 0 0 − w : (a , b ) lL 0 0 − w : (√2r +a , √2r b ) 21 2 2 0 − 2 2− 0 w : (r +a , b ) 22 2 0 0 − r :radiusofarcscenteredatw andw 2 lU lL a =a (between0and1)andr >0 0 0l 2 Note:b >0 0 Figure1: Locationofcomplexnumbers(indicatedatthebottomofthefigure)andcontoursinthecomplexw-plane. ContourC passes throughw , w and w as in the integraldirections , and , in order. ContourC passes 1 11 lU 12 2 ց ← ↑ throughwlL,w21 andw22 asinthedirections , and ,inorder. SymbolsI1 ,I1 andI1f (givenbyEq. (26))are integralsofIN (ofEq. (9))alongC1,whileI2ց,I↑2f and←I2 (givenbyEq. (31)ց)are←integralsofIP alongC2. (I1f and I2f areangularintegralsaroundwlU andwlL,rցespectively.)← w :=r ( 1,0)+w =( r +a ,b ), (15) 12 2 lU 2 0 0 − − 1 1 w := r e¯ +w =(r cos( p )+a ,r sin( p ) b ), (16) 21 2 3 lL 2 0 2 0 − −4 −4 − w :=r (1,0)+w =(r +a , b ). (17) 22 2 lL 2 0 0 − Inasimilarway,usinga (a =a forI inEq. (10))andb (withb >0)definedinthesentenceaboveEq. (11) 0 0 0r P 0 0 andlettingw ,w C,thenwedefinethepointsinFig.2as rU rL ∈ w :=(a ,b ), (18) rU 0 0 w :=(a , b ). (19) rL 0 0 − Additionally,lete¯ Cbedefinedby ∈ 1 e¯ :=exp( p i), (20) 4 4 Im ✻ C3 w31 (cid:0)(cid:0) (cid:0)ւ (cid:0) (cid:0) I3ւ I3f (cid:0) I OwrU(cid:0) 3→ ↑w32✲ w42 → I (cid:0)wrL → Re 4 ↑ → (cid:0) I4f I4 (cid:0) ւ(cid:0) (cid:0) (cid:0)(cid:0)ւ w 41 C 4 w : (a ,b ) rU 0 0 w : (√2r +a ,√2r +b ) 31 2 2 0 2 2 0 w : (r +a ,b ) 32 2 0 0 w : (a , b ) rL 0 0 − w : ( √2r +a , √2r b ) 41 − 2 2 0 − 2 2− 0 w : ( r +a , b ) 42 2 0 0 − − r :radiusofarcscenteredatw andw 2 rU rL a =a (between0and1)andr >0 0 0r 2 Note: b >0 0 Figure2: Locationofcomplexnumbers(indicatedatthebottomofthefigure)andcontoursinthecomplexw-plane. ContourC passes throughw , w andw as in the integraldirections , and , in order. ContourC passes 3 31 rU 32 4 ւ → ↑ throughwrL,w41 andw42 asinthedirections , and ,inorder. SymbolsI3 ,I3 andI3f (givenbyEq. (33))are integralsofIP(ofEq. (10))alongC3,whileI4ւ,I↑4f and→I4 (givenbyEq. (35)ւ)are→integralsofIPalongC4. (I3f and I4f areangularintegralsaroundwrU andwrL,rւespectively.)→ and let r R with r > 0 be the radius of arcs centered at w and w , then we further define the following 2 2 rU rL w ,w ,w∈,w CinFig. 2 31 32 41 42 ∈ 1 1 w :=r e¯+w =(r cos( p )+a ,r sin( p )+b ), (21) 31 2 rU 2 0 2 0 4 4 w :=r (1,0)+w =(r +a ,b ), (22) 32 2 rU 2 0 0 3 3 w := r e¯+w =(r cos( p )+a ,r sin( p ) b ), (23) 41 2 rL 2 0 2 0 − −4 −4 − w :=r ( 1,0)+w =( r +a , b ). (24) 42 2 lL 2 0 0 − − − Definition3. UsingthedefinitionsofI ,w ,w andw inEqs. (9),(11),(14)and(15),respectively,andwiththe N lU 11 12 helpofFig.1,wedefinetheintegralsandcontoursas dwIN:=I1 +I1 +I1f , (25) IC1 ց ← 5 wherecontourC passesthroughw ,w andw asintheintegraldirections , and ,inorderinFig. 1,and 1 11 lU 12 ց ← ↑ wlU w12 w11 I1 := dwIN, I1 := dwIN, I1f := dwIN. (26) ց Zw11 ← ZwlU Zw12 In the last integral above, let f R be an angle of circular coordinates(centered at w ) of the integral variable, 1 lU ∈ wheretheangleismeasuredcounterclockwisefromtheaxisparalleltotherealaxis,denotedas w=r exp(if )+w =(r cos(f ),r sin(f ))+(a ,b ), (27) 2 1 lU 2 1 2 1 0 0 withitsderivative dw =ir exp(if ). (28) df 2 1 1 ThenthelastintegralinEq.(26)isdefinedby I1f =Zww1211dwIN:=Zp 34p df 1ddfw1IN. (29) In a similar way, let f ,f ,f R be anglesof circular coordinatescentered at w , w and w , respectively. 2 3 4 lL rU rL ∈ Using I , I (Eqs. (9), (10)), w , w (Eqs. (11), (12)), w , w (Eqs. (16), (17)), w , w (Eqs. (18), (19)) and N P lU lL 21 22 rU rL w -w (Eqs. (21)-(24)),andwiththehelpofFigs. 1-2,theotherintegralsandcontoursaredefinedby 31 42 dwIN:=I2 +I2f +I2 , (30) IC2 ց ← contourC passesthroughw ,w andw asinthedirections , and ,inorderinFig. 1,and 2 lL 21 22 ց ↑ ← , I2ց:= wwlL21dwIN, I2f := ww2212dwIN:= −041p df 2ddfw2INwithw=r2exp(if 2)+wlL, I2←:= ww2l2LdwIN (31) R R R R dwIP:=I3 +I3 +I3f , (32) IC3 ւ → contourC passesthroughw ,w andw asinthedirections , and ,inorderinFig. 2,and 3 31 rU 32 ւ → ↑ 1p , I3ւ:= ww3r1UdwIP, I3→:= wwr3U2dwIP, I3f := ww3321dwIP:= 04 df 3ddfw3IPwithw=r2exp(if 3)+wrU! (33) R R R R dwIP:=I4 +I4f +I4 , (34) IC4 ւ → contourC passesthroughw ,w andw asinthedirections , and ,inorderinFig. 2,and 4 rL 41 42 ւ ↑ → . I4ւ:= wwr4L1dwIP, I4f := ww4412dwIP:= −−34pp df 4ddfw4IPwithw=r2exp(if 4)+wrL, I4→:= ww4r2LdwIP (35) R R R R Wethenpresentthelemmaconcerningthetransformationmentionedatthebeginningofthissection. Lemma 1. Leta andb be numbersthatspecify thex andy coordinatesofw , w , w andw asexpressed in 0 0 lL lU rU rL Eqs. (11)-(12)and(18)-(19)with thehelpofFigs. 1-2. Letr betheradiusofthecontourforthecircularintegrals 2 indicatedinEqs. (27), (31), (33)and(35). UsingtheintegralsI , I ,I , I , I , I , I andI definedin 1 2 3 4 1 2 3 4 Eqs. (26),(31),(33)and(35)withthehelpofFigs. 1-2,wederiveց ց ւ ւ ← ← → → (I +I +I +I )= (I +I +I +I ) 1 2 3 4 1 2 3 4 ← ← → → − ց ց ւ ւ forapositivefinitevalueofa inthelimitb 0whichfollowsr ¥ (withtherelationsuchasb >2/r ). (36) 0 0 2 0 2 → → 6 Theaboveequationimpliesinthee d notationthat − ( e >0)( d >0, d >0)( b , r R) 0 0 2 ∀ ∃ ∃ ∀ ∀ ∈ 2 ( <d <b <d (I +I +I +I )+(I +I +I +I ) <e ). (37) 0 0 1 2 3 4 1 2 3 4 r2 ⇒| ← ← → → ց ց ւ ւ | The sumson theleft andrightofthe aboveequationtake a definitefinitevalue, statingthatthe sum ofthe integrals inthe e¯¥ direction(indicatedby )andinthe e¯¥ direction(indicatedby )aretransformed(converted)tothe 3 − ց − ւ sumoftheintegralsalongthelineparalleltotherealaxis(indicatedby and ). Theaboverelationholdsforthe ← → exchangez 1 z. ↔ − Proof. UsingEqs. (25),(26)and(30)-(35),eachclosedcontourofthefollowingintegralsdoesnotenclosethesingu- larityat the coordinatesoriginin the complexplane whena and b (indicatedin Eqs. (11)-(12) and(18)-(19) with 0 0 thehelpofFigs. 1-2)arepositiveandfinite. Then,theassociatedintegralsalongtheclosedcontourvanish,duetothe residuetheorem,asexpressedby dwI = dwI = dwI = dwI =0 forb >0 (seealsonext6lines). (38) N N P P 0 IC1 IC2 IC3 IC4 (Wenoteinadvancethatbecauseofb >0theaboveequationholdsforapositivefinitevalueofa inthelimitb 0 0 0 0 thatfollowsr ¥ withtherelationsuchasb >2/r ,whichimpliesthat → 2 0 2 → ( e >0)( d >0, d >0)( b , r R) 0 0 2 ∀ ∃ ∃ ∀ ∀ ∈ 2 ( <d <b <d dwI <e , dwI <e , dwI <e , dwI <e ). (39) 0 0 N N P P r2 ⇒ |IC1 | |IC2 | |IC3 | |IC4 | Namely,becauseofb >0,nopolesexistwithintheregionsenclosedbytheintegralcontoursC ,C ,C ,C including 0 1 2 3 4 contours,andtheresiduetheoremworks.) Accordingly,withtheuseofEqs. (25),(30),(32)and(34),wehave I = dwI + dwI + dwI + dwI S N N P P IC1 IC2 IC3 IC4 =(I1 +I1 +I1f )+(I2 +I2f +I2 )+(I3 +I3 +I3f )+(I4 +I4f +I4 )=0. (40) ց ← ց ← ւ → ւ → Eachoftheaboveintegralterms(suchasI1 ,I1 andI1f inEqs. (26),(31),(33)and(35))hasadefinitefinitevalue withinafiniteregionthatdoesnotcontainanցysin←gularitiesoftheintegrandwhena andb (suchasthoseinEq.(11)) 0 0 arepositiveandfinite,andtheorderinthesumischangedtoproduce (I1 +I2 +I3 +I4 )+(I1 +I2 +I3 +I4 )+(I1f +I2f +I3f +I4f )=0. (41) ց ց ւ ւ ← ← → → For each of the last four terms I1f , I2f , I3f and I4f above, the real part of term p iw2 in the leading function exp( p iw2) (in the integrand IN given by Eq. (9) for I1f and I2f , such as that in∓(28)-(29), and the integrand IP given∓byEq. (10)forI3f andI4f )yields 2p Re(w) Im(w) inthelimitofr2 ¥ (thatis,forthelarge w)andthe functionexp( p iw2)vanishesforthelarg−e w|. Then|,|weder|ive → | | ∓ | | I1f =I2f =I3f =I4f =0 inthelimitofr2 ¥ (thatis,forthelarge w)forb0>0. (42) → | | Thisimpliesthatlettingb >0 0 (∀e >0)(∃d >0)(∀r2∈R)(r22 <d ⇒ |I1f |<e , |I2f |<e , |I3f |<e , |I4f |<e ).! (43) In contrast, with the use of Eqs. (4) and (5), the sum of the first four terms I , I , I and I in Eq. (41) 1 2 3 4 amountsto ց ց ւ ւ (I +I +I +I )=zˆ(1 z) 1 2 3 4 ց ց ւ ւ − forapositivefinitevalueofa inthelimitb 0whichfollowsr ¥ (suchasb >2/r ). (44) 0 0 2 0 2 → → 7 Thisimpliesthat (∀e >0)(∃d >0,∃d 0>0)(∀b0,∀r2∈R)(r22 <d <b0<d 0 ⇒ |(I1ց+I2ց+I3ւ+I4ւ)−zˆ(1−z)|<e ).!(45) Thus,Eqs. (41)-(44)leadto (I +I +I +I )= (I +I +I +I )= zˆ(1 z) 1 2 3 4 1 2 3 4 ← ← → → − ց ց ւ ւ − − forapositivefinitevalueofa inthelimitb 0whichfollowsr ¥ (suchasb >2/r ). (46) 0 0 2 0 2 → → Thismeansthat ( e >0)( d >0, d >0)( b , r R) (2 <d <b <d∀ (I ∃+I +∃I0 +I∀)0+∀(I2∈+I +I +I ) r2 0 0⇒|=1(←I1 +2←I2 +3→I3 +4→I4 )+1zցˆ(1 2ցz) <3eւ). 4ւ | (47) | ← ← → → − | The above equation implies that the sum of the integrals in the e¯¥ direction (indicated by ) and in the e¯¥ 3 − ց − direction(indicatedby )istransformed(converted)tothesumoftheintegralsalongthelineparalleltotherealaxis ւ (indicatedby and ),owingtothatthecompletedzetafunctionisregular. Namely,theintegralofthecompleted ← → zetafunctionalongthelinerotatedfromtherealaxiscanbetransformedtotheintegralalongthelineparalleltothe realaxisasawholeforconvenience.Wenotethattheaboverelationshipsholdfortheexchangez 1 z,keepingthe ↔ − region0<Re(z)<1underthisexchange. 3 ProofoftheRiemann’sconjecturethattherealpartofallnon-trivialzeros of the zeta function is 1/2 ThissectionfirstconvertstheintegralformofthecompletedzetafunctionexpressedbyEq. (5)totheusualformto giveTheorem1. Then,inTheorem2, weprovethattherealpartofallnon-trivialzerosofthezetafunctionmustbe 1/2.Asitisknownthatallnon-trivialzerosofthezetafunctionexistintheregion0<Re(z)<1inliterature[17,18], weconcentrateonthisregion.Furthermore,itisalsoknownthatthenumberofzeros(ofthezetafunction)withareal partof1/2isinfinite[10]. Theusualintegralformofthezetafunctionisasfollows. Theorem1. (the(third)integralform ofthe(completed)zetafunction)Letz,w C. Letzˆ(z)bethecompletedzeta ∈ functiondefinedbyEq. (4). Let zˆl(z):=p −2zG (2z) 0 1dwexpw(p−izwe)xp(e−xpp(iw2p)iw), (48) Z ց − − andlet zˆr(z):=p −1−2zG (1−2 z) 0 1dwexp(wpzi−w1)expe(xppi(w2p)iw), (49) Z ւ − − intermsofthegammafunctionG (z). Thenzˆ(z)isexpressedby zˆ(z)=zˆ(z)+zˆ(z), (50) l r whichiscalledthe(third)integralformofthe(completed)zetafunction. Proof. FromEq. (5),theabovecompletedzetafunctionzˆ(z)isobtainedbytheexchangez 1 z,wheretheregion ↔ − 0<Re(z)<1iskeptunderthisexchange.Theaboveregion0<Re(z)<1isconsistentwiththeregion0<Re(z)<1 weareconsideringinthispaper. 8 ✻ Im C1c w14 ← C C w16❤❤❤1n❤w❤15 ✭w1✭3 ✭1✭p✭wlU ← ↓ ↑ ← R✲e O w : (a ,b ) lU 0 0 w : (a /2,b /2) w : (0,r ) 13 0 0 14 1 w : ( a /2,b /2) w : ( a ,b ) 15 0 0 16 0 0 − − C : contourfromw tow 1p lU 13 C :arcfromw tow withradiusr centeredat(0,0) 1c 13 15 1 C : contourfromw tow 1n 15 16 a =a (between0and1)andr =[(a /2)2+(b /2)2]1/2 0 0l 1 0 0 Note:0<b <<a 0 0 Figure 3: A part of the componentparallel to the real axis in the contour C in Fig. 1, deformedinto the contour 1 composedofC ,C andC (withoutdisconnectionsfromotherparts)tobypasstheoriginofthecomplexw-plane. 1p 1c 1n Locationsofcomplexnumbersinthecomplexplaneareindicatedatthebottomofthefigure. Theanglef ofcircular c coordinatescenteredattheorigin(0,0)ismeasuredcounterclockwisefromtherealaxiswiththepositivesign. In addition, the completed zeta function satisfies the following known symmetry relation [1, 16, 15] for the ex- changez 1 z ↔ − zˆ(z)=zˆ(1 z). (51) − To derive the real part of non-trivial zeros of the zeta function, the present approach uses (in addition to the above symmetrygivenbyEq.(51))theproperty(withmerits)thataquantityinoneterminahighly(attainable)symmetrized integralformgeneratesacorresponding(paired)quantityinanotherterm. Here, for the Riemann-Siegelintegralform, each partof the contour(for the integralvariablew C) parallelto ∈ therealaxisin thecontoursC andC in Figs. 1-2 isfurtherdeformedto bypasstheorigininthe complexw-plane 1 4 as in Figs. 3-4 according to Definition 4 below. Each contour part deformedas in Figs. 3-4 is connected (without disconnections)withtheotherremainingcontoursinFigs1-2. Theintegrandshavepolynomialsofwwiththepower of z C, and the absolute value of the integrandis expressed using w and the argument(angle) arg(w) in circular ∈ | | coordinates. Then, bymakingarg(w) independentof w near the originof the w-plane, singularityanalysisbecomes simple. In Figs. 3-4, each linear contour employed has a constant arg(w), while arg(w) along each arc (of circle) contourisintegratedtothecircularintegral. Therefore,singularityanalysisfocusesonthepowerforthepolynomials of w. (Wenotethatnopoleexistsaroundthepolelocatedattheoriginofthecomplexw-plane,whentheconcerned | | regionissmall.) 9 ✻ Im O R✲e w43✭✭C→✭4n✭w✭44 ↓ ↑ ❤w4❤6 C❤→4❤p❤wrL C→4c w45 w :(a , b ) rL 0 0 − w : ( a , b ) w : ( a /2, b /2) 43 0 0 44 0 0 − − − − w : (0, r ) w : (a /2, b /2) 45 1 46 0 0 − − C : contourfromw tow 4n 43 44 C :arcfromw tow withradiusr centeredat(0,0) 4c 44 46 1 C : contourfromw tow 4p 46 rL a =a (between0and1)andr =[(a /2)2+(b /2)2]1/2 0 0r 1 0 0 Note:0<b <<a 0 0 Figure 4: A part of the componentparallel to the real axis in the contour C in Fig. 2, deformedinto the contour 4 composedofC ,C andC (withoutdisconnectionsfromotherparts)tobypasstheoriginofthecomplexw-plane. 4n 4c 4p Locationsofcomplexnumbersinthecomplexplaneareindicatedatthebottomofthefigure. Theanglef ofcircular c coordinatescenteredattheorigin(0,0)ismeasuredcounterclockwisefromtherealaxiswiththepositivesign. Definition4. Letb ,a Rwithb anda beingsmall(b >0asindicatedsucasinEq.(45);b <<a asdescribed 0 0 0 0 0 0 0 inNote belowEq. (65))∈. Letw C asdefinedin Definition2 with Figs. 1-2. Letting f R, we definethe angle lU c f ofcircularcoordinatescentered∈attheorigin(0,0)ofthecomplexw-plane,thatismeasure∈dcounterclockwisefrom c therealaxiswiththepositivesign. Letr R(radiusofthearc(ofcircle)contoursdefinedbelowandinFigs. 3-4) 1 suchthatr =a /2. Letw ,w ,w ,w ∈ C, then,asindicatedinFig. 3, wedefinethesenumbersinthecomplex 1 0 13 14 15 16 ∈ w-planeby a b 0 0 w :=( , ), w :=(0,r ), 13 14 1 2 2 a b 0 0 w :=( , ), w :=( a ,b ), (52) 15 16 0 0 − 2 2 − where a 2 b 2 1/2 0 0 r := + . (53) 1 2 2 "(cid:16) (cid:17) (cid:18) (cid:19) # UsingthequantitiesgiveninFig. 1,apartofthecomponent,paralleltotherealaxis,ofthecontourC inFig. 1is 1 deformedintothecontourcomposedofC ,C andC tobypassthecoordinateoriginasinFig. 3. Thecontoursin 1p 1c 1n Fig.3aredefinedasfollows. 10