黎曼猜想的一个推论

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iùߎ˜‡íؤAu*f‰Eúi/721006E-mail:feijinhuayoujian@msn.comÁ‡©¦^Š©ÙnØNevanlinna1Ľn§‰Ñiùߎ˜‡íØ'…cŠ©Ùnاiùzeta¼êMR(2000)ÌK©a30D3511M06¥ã©aO174AncorollaryofRiemannhypothesisJinHuaFeiChangLingCompanyofElectronicTechnologyBaoji721006P.R.ChinaE-mail:feijinhuayoujian@msn.comabstractThispaperuseNevanlinnasecondfundamentaltheoremofthevaluedistributiontheory,giveancorollaryofRiemannhypothesisKeywordvaluedistributiontheory;RiemannzetafunctionMR(2000)SubjectClassi cation30D35ChineseLibraryClassi cationO174Äk§‰ÑŠ©ÙnØ¥·‚‡^˜PÒ!½ÂÚ½n§ÙSN넩z[1]Ú[2]½Âµ1log+x=8:logx1x00x1´„§logxlog+xf(z)´«jzjR;0R1˜‡æX¼ê§Øðun(r;f)L«f(z)3 jzjr(0rR)þ4:‡ê§­?UÙ­êO§n(0;f)L«f(z)3:?4:­??¿Eêa6=1;n(r;1fa)L«f(z)a3 jzjr(0rR)þ:‡ê§­?UÙ­êO§n(0;1fa)L«f(z)a3:?:­?Pµm(r;f)=12Z20log+ f(rei') d'N(r;f)=Zr0n(t;f)n(0;f)tdt+n(0;f)logr½ÂµT(r;f)=m(r;f)+N(r;f);T(r;f)¡f(z)A¼êÚn1µef(z)3«jzjR(0R1)S)Û£=X¤§KkT(r;f)log+M(r;f)+rrT(;f)(0rR)d?M(r;f)=maxjzj=rjf(z)jy²„©z[1]57Ún2µf(z)3«jzjR(0R1)˜‡æX¼ê§Øðu0§Ę jzj(0R)Ú3d Sf(z):a(=1;2;:::;h)94:b(=1;2;:::;k)§Ù¥z˜:½4:Ñygê†Ù?ƒÓ§…z=0Ø´¼êf(z)˜‡:½4:§K3 jzjSkeúª2logjf(0)j=12Z20log f(ei') d'hX=1logjaj+kX=1logjbjdúª¡Jensenúªy²„©z[1]48.Ún3µ¼êf(z)ujzjRæX¿kf(0)6=0;1;1;f0(0)6=0Ku0rRkT(r;f)2N(R;1f)+N(R;f)+N(R;1f1)+4log+jf(0)j+2log+1Rjf0(0)j+24logRRr+2328ùÒ´Nevanlinna1Ľn˜«/ªy²„©z[1]75½n3.1¡I‡§·‚2Š˜OÚn4µxaž§f(x)´˜šK4~¼ê§K4limN!1NXn=af(n)ZNaf(x)dx!= 3§…0 f(a)§?˜Úóƒ§x!1ž§ef(x)!0§K Xanf(n)Zaf()d f(1);(a+1)y²„©z[3]91½n2.3Ps=+it´ECê§(s)´RiemannZeta¼ê§1ž§½Â(s)=1Xn=11nsŠâ©z[4]90§1žlog(s)=1Xn=2(n)nslognÙ¥(n)´Mangoldt¼êÚn5µé?¿¢êt§k£˜¤0:0426jlog(4+it)j0:0824£¤j(4+it)1j0:0426£n¤0:917j(4+it)j1:0824£o¤j0(4+it)j0:012y²µ£˜¤jlog(4+it)j1Xn=2(n)n4logn1Xn=21n4=49010:0824jlog(4+it)j1241Xn=31n4=1+2241Xn=11n4=984900:04264£¤j(4+it)1j= 1Xn=21n4+it 1241Xn=31n4=1+2241Xn=11n4=984900:0426(n)j(4+it)j= 1Xn=11n4+it 1Xn=11n4=4901:0824j(4+it)j= 1Xn=11n4+it 11Xn=21n4=21Xn=11n4=24900:917£o¤j0(4+it)j= 1Xn=2lognn4+it log2241Xn=3lognn4dÚn41Xn=3lognn4=Z13logxx4dx+ Ù¥0 log334Z13logxx4dx=13Z13logxdx3=log334+13Z13x4dx=log334132Z13dx3=log334+135¤±1Xn=3lognn4log334+135+log3345Ïd§j0(4+it)jlog2242log3341350:012y.01100,©±e§c1;c2;:::;þL«=†k'~êÚn6µ12;jtj2ž§kj(+it)jc1jtj12y²„©z[4]140½n29142½n4Ún7µf(z)3 jzz0jRþ)Û§Ké?¿0rR,3 jzz0jrþkjf(z)f(z0)j2rRr(A(R)Ref(z0))Ù¥A(R)=maxjzz0jRRef(z)y²„©z[4]61½n2yb½iùb¤á§{¡RH¤á§=12ž§(+it)vk:P«12+;jtj1Ú«2;jtj1¿8«DÏ3«DS§(+it)QÃ:§Ã4:§dž§3«DS§log(+it)´˜‡k½ÂõŠ)Û¼ê§Ùz‡üŠ©|ƒ 2iêb½3«DS§3:s0§¦(s0)=1£XJØ3ù:§@oÚn9(JCN(;11)=0§lҌ±†©½n(J¤§éuØÓüŠ)Û©|§log(s0)=log1ŠØÓ§Œ0;2ki;(k=1;2;::::::)6§·‚Àlog(s0)=log1=0üŠ)Û©|Ϗ«D´üëÏ«§¤±d)ÛmÿüŠ5½n£ë„©z[5]276½n2Ú©z[6]155½n1¤§3«DS§log(+it)´˜‡üŠ)ۼ꧱eþTüŠ)Û©|´„§¼ê(+it)=1K7klog(+it)=0§=¼ê(+it)1-Š:7´¼êlog(+it):Ún8µeRH¤á§01100§K12+2;jtj16ž§kjlog(+it)jc2logjtj+c3y²µ3Ún7¥§z0=0;f(z)=log(z+4+it);jtj16;R=72;r=722;§log(z+4+it)3 jzz0jRþ)Û§Ïd3 jzz0jrþkjlog(z+4+it)log(4+it)j7(A(R)Relog(4+it))=µjlog(z+4+it)j7(A(R)+jlog(4+it)j)+jlog(4+it)jdÚn6A(R)=maxjzz0jRlogj(z+4+it)j12logjtj+logc12dÚn5§jlog(z+4+it)jc2logjtj+c3dujtj16?¿5§Œ12+2žjlog(+it)jc2logjtj+c37y.Ún9µeRH¤á§01100§Kjtj16;=722ž§3 jzjþ§kN;1(z+4+it)1loglogjtj+c4y²µ3Ún2¥§f(z)=log(z+4+it);R=72;=722;a(=1;2;:::;h)´log(z+4+it)3 jzjS:§­?UÙ­êOŽ§log(z+4+it)3d SÃ4:§…log(4+it)؏§Ïklogjlog(4+it)j=12Z20log log(4+it+ei') d'hX=1logjajdÚn59Ún8§hX=1logjajloglogjtj+c4Ϗz=0QØ´log(z+4+it):§Ø´4:§¤±§er0˜·ê§KhX=1logjaj=Zr0logtdn(t;1f)=logtn(t;1f) r0+Zr0n(t;1f)tdt=Z0n(t;1f)tdt=N;1f=N;1log(z+4+it)N;1(z+4+it)18y.½nµeRH¤á§K12+4;01100;jtj16ž§kj(+it)jc8(logjtj)c9y²µ3Ún3¥§f(z)=(z+4+it);jtj16§dÚn5§f(0)=(4+it)6=0;1;1;f0(0)=0(4+it)6=0§…f0(0)=0(4+it)0:012;jf(0)j=j(4+it)j1:0824;R=722;r=723§Ï(z+4+it)3 jzjRSX§QÃ:§Ã4:§¤±NR;1f=0;N(R;f)=02dÚn9§KkT(r;(z+4+it))2loglogjtj+c53Ún1¥§R=722;=723;r=7249ŒŠn§3 jzjrþ§klog+j(z+4+it)jc6loglogjtj+c7dujtj16´?¿§¤±12+4žklog+j(+it)jc6loglogjtj+c7Klogj(+it)jc6loglogjtj+c7=9j(+it)jc8(logjtj)c9y.ë©z[1]ZhuangQ.T,Singulardirectionofmeromorphicfunction,BeiJing:Sci-encePress,1982.[2]YangL,Valuedistributiontheoryandnewresearch,BeiJing:SciencePress,1982.[3]HuaL.G,Introductionofnumbertheory,BeiJing:SciencePress,1979.[4]PanC.D,PanC.B,Fundamentalsofanalyticnumbertheory,BeiJing:SciencePress,1999[5]ZhuangQ.T,ZhangN.Y,Complexvariablesfunctions,BeiJing:PekingUniversitypress,1984[6]HuaL.G,Introductionofadvancedmathematics(BookOneofsecondvolume)BeiJing:SciencePress,198110

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