Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+...Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+pαis at least c（g）x,where p is an odd prime and the constant c（g）depends only on g.展开更多
We compute the n-level correlation of normalized nontrivial zeros of a product of Lfunctions: L(s, π1) … L(s, π k ), where πj, j = 1, …, k, are automorphic cuspidal representations of GL mj (?A). Here the sizes o...We compute the n-level correlation of normalized nontrivial zeros of a product of Lfunctions: L(s, π1) … L(s, π k ), where πj, j = 1, …, k, are automorphic cuspidal representations of GL mj (?A). Here the sizes of the groups GL mj (?A) are not necessarily the same. When these L(s, π j ) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s, π j ) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m 1, …, m k ? 4, but are under Hypothesis H in other cases.展开更多
基金supported by the National Natural Science Foundation of China(Nos.10771103,10801075)the Natural Science Foundation of Huaihai Institute of Technology(No.KQ10002)
文摘Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+pαis at least c（g）x,where p is an odd prime and the constant c（g）depends only on g.
基金supported by the 973 Programthe National Natural Science Foundation of China (GrantNo. 10531060)+2 种基金Ministry of Education of China (Grant No. 305009)The second author was supportedby the National Security Agency of USA (Grant No. H98230-06-1-0075)The United States government isauthorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
文摘We compute the n-level correlation of normalized nontrivial zeros of a product of Lfunctions: L(s, π1) … L(s, π k ), where πj, j = 1, …, k, are automorphic cuspidal representations of GL mj (?A). Here the sizes of the groups GL mj (?A) are not necessarily the same. When these L(s, π j ) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s, π j ) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m 1, …, m k ? 4, but are under Hypothesis H in other cases.
