In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,a...In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,and derive an exact computational formula.It can be conveniently programmed by the“Mathematica”software,by which we can get the exact results easily.展开更多
Sequences with nice pseudo-randomness play an important role in not only communication system but also cryptography system. Based on the Legendre-Sidelnikov sequence, a modified Legendre-Sidelnikov sequence was introd...Sequences with nice pseudo-randomness play an important role in not only communication system but also cryptography system. Based on the Legendre-Sidelnikov sequence, a modified Legendre-Sidelnikov sequence was introduced. The exact value of the autocorrelation function was derived by strict computation. According to the values of the autocorrelation functions of the two Legendre-Sidelnikov sequences, it is proven that both of them have perfect pseudo-randomness. Furthermore, a detailed comparison between autocorrelation functions of the two Legendre-Sidelnikov sequences was deduced. It indicates that no matter which parameters are chosen, the modified sequence has pseudo-randomness as good as the primitive sequence, which is of great significance for applications.展开更多
Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, fo...Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2, Φ× uj ×χ) ■Φ,ε(q(1 + |tj |))3/2-θ+ε and L(1/2 + it, Φ×χ) ■Φ,ε(q(1 + |t|))3/4-θ/2+ε for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation, and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.展开更多
文摘In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,and derive an exact computational formula.It can be conveniently programmed by the“Mathematica”software,by which we can get the exact results easily.
基金supported by the National Natural Science Foundation of China (60833008)the Science and Technology on Communication Security Laboratory (9140C110201110C1102)the Fundamental Research Funds for the Central Universities (K5051270003, K50511010007)
文摘Sequences with nice pseudo-randomness play an important role in not only communication system but also cryptography system. Based on the Legendre-Sidelnikov sequence, a modified Legendre-Sidelnikov sequence was introduced. The exact value of the autocorrelation function was derived by strict computation. According to the values of the autocorrelation functions of the two Legendre-Sidelnikov sequences, it is proven that both of them have perfect pseudo-randomness. Furthermore, a detailed comparison between autocorrelation functions of the two Legendre-Sidelnikov sequences was deduced. It indicates that no matter which parameters are chosen, the modified sequence has pseudo-randomness as good as the primitive sequence, which is of great significance for applications.
基金supported by National Natural Science Foundation of China(Grant No.11531008)the Ministry of Education of China(Grant No.IRT 16R43)。
文摘Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2, Φ× uj ×χ) ■Φ,ε(q(1 + |tj |))3/2-θ+ε and L(1/2 + it, Φ×χ) ■Φ,ε(q(1 + |t|))3/4-θ/2+ε for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation, and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.
