A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the preci...A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.展开更多
Let P（x） denote the greatest prime factor of ∏<sub>x【n≤x+x<sup>1/2</sup></sub>n. In this paper, we shall prove that P（x）】x<sup>0.728</sup>holds true for sufficiently large x.
Let π△ be the automorphic representation of GL（2, QA） associated with Ramanujan modular form A and L（s, π△） the global L-function attached to π△. We study Selberg＇s integral for the automorphic L-function L...Let π△ be the automorphic representation of GL（2, QA） associated with Ramanujan modular form A and L（s, π△） the global L-function attached to π△. We study Selberg＇s integral for the automorphic L-function L（s, π△） under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.展开更多
文摘A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.
基金Project supported by the Tian Yuan Item in the National Natural Science Foundation of China.
文摘Let P（x） denote the greatest prime factor of ∏<sub>x【n≤x+x<sup>1/2</sup></sub>n. In this paper, we shall prove that P（x）】x<sup>0.728</sup>holds true for sufficiently large x.
基金Supported by National Natural Science Foundation of China （Grant No. 10571107）Acknowledgements The author expresses her thanks to Professor Jianya Liu and Professor Yangbo Ye for encouragernent, and to Professor Xiumin Ren for valuable suggestions. This work was completed when the author visited The University of Iowa supported by CSC. The author would like to thank Department of Mathematics, The University of Iowa for hospitality and support.
文摘Let π△ be the automorphic representation of GL（2, QA） associated with Ramanujan modular form A and L（s, π△） the global L-function attached to π△. We study Selberg＇s integral for the automorphic L-function L（s, π△） under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.
