iùßíؤAu*fEúi/721006E-mail:feijinhuayoujian@msn.comÁ©¦^©ÙnØNevanlinna1Ľn§ÑiùßíØ' c©Ùnاiùzeta¼êMR(2000)ÌK©a30D3511M06¥ã©aO174AncorollaryofRiemannhypothesisJinHuaFeiChangLingCompanyofElectronicTechnologyBaoji721006P.R.ChinaE-mail:feijinhuayoujian@msn.comabstractThispaperuseNevanlinnasecondfundamentaltheoremofthevaluedistributiontheory,giveancorollaryofRiemannhypothesisKeywordvaluedistributiontheory;RiemannzetafunctionMR(2000)SubjectClassication30D35ChineseLibraryClassicationO174Äk§Ñ©ÙnØ¥·^PÒ!½ÂÚ½n§ÙSNë©z[1]Ú[2]½Âµ1log+x=8:logx1x00x1´§logxlog+xf(z)´«jzjR;0R1æX¼ê§Øðun(r;f)L«f(z)3jzjr(0rR)þ4:ê§?UÙêO§n(0;f)L«f(z)3:?4:??¿Eêa6=1;n(r;1f a)L«f(z) a3jzjr(0rR)þ:ê§?UÙêO§n(0;1f a)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)+r rT(;f)(0rR)d?M(r;f)=maxjzj=rjf(z)jy²©z[1]57Ún2µf(z)3«jzjR(0R1)æX¼ê§Øðu0§Äjzj(0R)Ú3dSf(z):a(=1;2;:::;h)94:b(=1;2;:::;k)§Ù¥z:½4:ÑygêÙ?Ó§ z=0Ø´¼êf(z):½4:§K3jzjSkeúª2logjf(0)j=12Z20logf(ei')d' hX=1logjaj+kX=1logjbjdúª¡Jensenúªy²©z[1]48.Ún3µ¼êf(z)ujzjRæX¿kf(0)6=0;1;1;f0(0)6=0Ku0rRkT(r;f)2N(R;1f)+N(R;f)+N(R;1f 1)+4log+jf(0)j+2log+1Rjf0(0)j+24logRR r+2328ùÒ´Nevanlinna1Ľn«/ªy²©z[1]75½n3.1 ¡I§·2OÚn4µxa§f(x)´K4~¼ê§K4limN!1NXn=af(n) ZNaf(x)dx!=3§ 0f(a)§?Úó§x!1§ef(x)!0§KXanf(n) Zaf()d f( 1);(a+1)y²©z[3]91½n2.3Ps=+it´ECê§(s)´RiemannZeta¼ê§1§½Â(s)=1Xn=11nsâ©z[4]90§1log(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=490 10:0824jlog(4+it)j124 1Xn=31n4=1+224 1Xn=11n4=98 4900:04264£¤j(4+it) 1j=1Xn=21n4+it124 1Xn=31n4=1+224 1Xn=11n4=98 4900:0426(n)j(4+it)j=1Xn=11n4+it1Xn=11n4=4901:0824j(4+it)j=1Xn=11n4+it1 1Xn=21n4=2 1Xn=11n4=2 4900:917£o¤j0(4+it)j=1Xn=2lognn4+itlog224 1Xn=3lognn4dÚn41Xn=3lognn4=Z13logxx4dx+Ù¥0log334Z13logxx4dx= 13Z13logxdx 3=log334+13Z13x 4dx=log334 132Z13dx 3=log334+135¤±1Xn=3lognn4log334+135+log3345Ïd§j0(4+it)jlog224 2log334 1350:012y.01100,©±e§c1;c2;:::;þL«=k'~êÚn6µ12;jtj2§kj(+it)jc1jtj12y²©z[4]140½n29142½n4Ún7µf(z)3jz z0jRþ)Û§Ké?¿0rR,3jz z0jrþkjf(z) f(z0)j2rR r(A(R) Ref(z0))Ù¥A(R)=maxjz z0jRRef(z)y²©z[4]61½n2yb½iùb¤á§{¡RH¤á§=12§(+it)vk:P«12+;jtj1Ú«2;jtj1¿8«DÏ3«DS§(+it)QÃ:§Ã4:§d§3«DS§log(+it)´k½Âõ)Û¼ê§Ùzü©|2iêb½3«DS§3:s0§¦(s0)=1£XJØ3ù:§@oÚn9(JCN(;1 1)=0§lÒ±©½n(J¤§éuØÓü)Û©|§log(s0)=log1ØÓ§0;2ki;(k=1;2;::::::)6§·Àlog(s0)=log1=0ü)Û©|Ï«D´üëÏ«§¤±d)Ûmÿü5½n£ë©z[5]276½n2Ú©z[6]155½n1¤§3«DS§log(+it)´ü)ۼ꧱eþTü)Û©|´§¼ê(+it)=1K7klog(+it)=0§=¼ê(+it)1-:7´¼êlog(+it):Ún8µeRH¤á§01100§K12+2;jtj16§kjlog(+it)jc2logjtj+c3y²µ3Ún7¥§z0=0;f(z)=log(z+4+it);jtj16;R=72 ;r=72 2;§log(z+4+it)3jz z0jRþ)Û§Ïd3jz z0jrþkjlog(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)=maxjz z0jRlogj(z+4+it)j12logjtj+logc12dÚn5§jlog(z+4+it)jc2logjtj+c3dujtj16?¿5§12+2jlog(+it)jc2logjtj+c37y.Ún9µeRH¤á§01100§Kjtj16;=72 2§3jzjþ§kN;1(z+4+it) 1loglogjtj+c4y²µ3Ún2¥§f(z)=log(z+4+it);R=72 ;=72 2;a(=1;2;:::;h)´log(z+4+it)3jzjS:§?UÙêO§log(z+4+it)3dSÃ4:§ log(4+it)اÏklogjlog(4+it)j=12Z20loglog(4+it+ei')d' hX=1logjajdÚn59Ún8§hX=1logjajloglogjtj+c4Ïz=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;jtj16§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=72 2;r=72 3§Ï(z+4+it)3jzjRSX§QÃ:§Ã4:§¤±NR;1f=0;N(R;f)=02dÚn9§KkT(r;(z+4+it))2loglogjtj+c53Ún1¥§R=72 2;=72 3;r=72 49n§3jzjrþ§klog+j(z+4+it)jc6loglogjtj+c7dujtj16´?¿§¤±12+4klog+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