i?Au*fEi ?/721006E-mail: feijinhuayoujianmsn.com?n?Nevanlinna1?n?i?“cn izetaMR(2000)Ka30D3511M06aO174An corollary of Riemann hypothesisJin Hua FeiChangLing Company of Electronic TechnologyBaoji721006P.R.ChinaE-mail: feijinhuayoujianmsn.comabstractThis paper use Nevanlinna second fundamental theorem of the valuedistribution theory , give an corollary of Riemann hypothesisKeywordvalue distribution theory ; Riemann zeta functionMR(2000) Subject Classification30D35Chinese Library ClassificationO174kn?P!nSNz121log+x =logx1 x00 x 1(s) =Xn=11 nsz4 90 ? 1log(s) =Xn=2(n) nslogn(n)Mangoldt“n5?t k0.0426 | log(4 + it) | 0.0824?| (4 + it) 1 | 0.0426n0.917 | (4 + it) | 1.0824o| 0(4 + it) | 0.012y| log(4 + it) | Xn=2(n) n4lognXn=21 n4=4 90 1 0.0824| log(4 + it) | 1 24Xn=31 n4= 1 +2 24Xn=11 n4=9 84 90 0.04264?| (4 + it) 1 | =?Xn=21 n4+it?1 24Xn=31 n4= 1 +2 24Xn=11 n4=9 84 90 0.0426(n)| (4 + it) | =?Xn=11 n4+it?Xn=11 n4=4 90 1.0824| (4 + it) | =?Xn=11 n4+it? 1 Xn=21 n4= 2 Xn=11 n4= 2 4 90 0.917o| 0(4 + it) | =?Xn=2logn n4+it?log2 24Xn=3logn n4dn4Xn=3logn n4=Z3logx x4dx + 0 log 3 34Z3logx x4dx = 1 3Z3logx dx3=log3 34+1 3Z3x4dx=log3 341 32Z3dx3=log3 34+1 35Xn=3logn n4log3 34+1 35+log3 345d| 0(4 + it) | log2 242log3 341 35 0.012y.“?0 1 2( + it)vk“:“P 1 2+ ,|t| 1 2,|t| 1?8D“?3DS( + it)Q“:4:d3DSlog( + it)k?)z|?2i?“b3DS3:s0(s0) = 1XJ3?:on9?(JCN(,1 1) = 0 l?n?(J u?)|log(s0) = log1? ?0, 2ki,(k = 1,2,.)6?log(s0) = log1 = 0?)|“D d)m?5n z5 276n2z6155n1 3DSlog( + it)eT)|“( + it) = 1K7klog( + it) = 0=( + it)?1 -:7log( + it)?“:“n8eRH0 1 100 K? 1 2+ 2 , |t| 16 k| log( + it) | c2log|t| + c3y3n7?z0= 0, f(z) = log(z + 4 + it), |t| 16, R =7 2 , r =7 2 2,log(z + 4 + it)3?|z z0| R)d3?|z z0| rk| log(z + 4 + it) log(4 + it) | 7 ( A(R) Relog(4 + it) )=| log(z + 4 + it) | 7 ( A(R) + | log(4 + it) | ) + | log(4 + it) |dn6A(R) =max |zz0|Rlog| (z + 4 + it) | 1 2log|t| + logc12dn5| log(z + 4 + it) | c2log|t| + c3du|t| 16?5 ? 1 2+ 2| log( + it) | c2log|t| + c37y.“n9 eRH0 1 100K?|t| 16, =7 2 23?|z| kN? ,1 (z + 4 + it) 1? loglog|t| + c4y3n2?f(z) = log(z + 4 + it), R =7 2 , =7 22, a( = 1,2,.,h)log(z + 4 + it)3?|z| S?“:-?U-Olog(z + 4 + it)3d?S4: log(4 + it)“ ?klog|log (4 + it)| =1 2Z20log?log (4 + it + ei)?d hX=1log |a|dn59n8hX=1log |a| loglog|t| + c4z = 0Qlog(z + 4 + it)?“: 4: er0? KhX=1log |a|=Zr0? log t? dn(t,1 f) =? log t? n(t,1 f)?r0+Zr0n(t,1 f) tdt =Z0n(t,1 f) tdt = N? ,1 f?= N? ,1 log(z + 4 + it)? N? ,1 (z + 4 + it) 1?8y.“neRH K? 1 2+ 4 , 0 1 100, |t| 16 k|( + it)| c8(log|t|)c9y3n3?f(z) = (z + 4 + it), |t| 16dn5f(0) =(4 + it) 6= 0, , 1,f0(0) = 0(4 + it) 6= 0f0(0) = 0(4 + it) 0.012,|f(0)| = |(4 + it)| 1.0824,?R =7 2 2, r =7 2 3 (z+4 + it)3?|z| RS?X Q“: 4: N? R,1 f? = 0 ,N (R, f) = 02dn9 KkT (r,(z + 4 + it) 2loglog|t| + c53n1 ?R =7 22 , =7 2 3, r =7 2 49?n 3?|z| r klog+|(z + 4 + it)| c6loglog|t| + c7du|t| 16? ? 1 2+ 4klog+|( + it)| c6loglog|t| + c7Klog |( + it)| c6loglog|t| + c7=9|( + it)| c8(log|t|)c9y.“z1 Zhuang Q.T, Singular direction of meromorphic function , BeiJing: Sci-ence Press,1982.2 Yang L , Value distribution theory and new research , BeiJing: SciencePress,1982.3 Hua L.G , Introduction of number theory , BeiJing: Science Press,1979.4 Pan C.D, Pan C.B, Fundamentals of analytic number theory, BeiJing:Science Press, 19995 Zhuang Q.T , Zhang N.Y , Complex variables functions , BeiJing:Peking University press , 19846 Hua L.G , Introduction of advanced mathematics ( Book One of secondvolume ) BeiJing: Science Press, 198110