Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riem...Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the “critical line” . In this paper, we use Nevanlinna’s Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis. In this paper, we gave out the second proof, please read the reference.展开更多
Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference...Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.展开更多
This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective spacePn（C） under the assumption limsupT（r,f）/log 1/...This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective spacePn（C） under the assumption limsupT（r,f）/log 1/1-r = ＋∞ for hypersurfaces in general position. A heuristic principle concerning the existence of Julia directions of holomorphic mappings from the unit disk into Pn（C） is given also.展开更多
We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose dif...We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.展开更多
文摘Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the “critical line” . In this paper, we use Nevanlinna’s Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis. In this paper, we gave out the second proof, please read the reference.
基金supported by the National Natural Science Foundation of China(10471067)NSF of Guangdong Province(04010474)
文摘Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.
基金project supported in part by the National Natural Science Foundation of China(10971156)
文摘This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective spacePn（C） under the assumption limsupT（r,f）/log 1/1-r = ＋∞ for hypersurfaces in general position. A heuristic principle concerning the existence of Julia directions of holomorphic mappings from the unit disk into Pn（C） is given also.
基金Acknowledgements This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the first and third authors worked as visiting scholars. The authors wish to thank the anonymous referees for their very helpful comments and useful suggestions. This work was also supported by the National Natural Science Foundation of China （Grant No. 11271090）, the Tianyuan Youth Fund of the National Natural Science Foundation of China （Grant No. 11326083）, the Shanghai University Young Teacher Training Program （ZZSDJ12020）, the Innovation Program of Shanghai Municipal Education Commission （14YZ164）, the Natural Science Foundation of Guangdong Province （S2012010010121）, and the Projects （13XKJC01） from the Leading Academic Discipline Project of Shanghai Dianji University.
文摘We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.
