Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function a...Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.展开更多
In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L（s, f × χ）, where f is a holomorphic cusp form and χ（mod q） is a primitive character.
In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-HoMey contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argumen...In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-HoMey contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established展开更多
Let fk（n） be the characteristic function of n with Ω（n） = k, and T k（x,α）=∑n≤xfk（n）e（nα）.The main purpose of this paper is to establish a Bombieri-type mean-value theorem for Tk（x, α）, via using the ...Let fk（n） be the characteristic function of n with Ω（n） = k, and T k（x,α）=∑n≤xfk（n）e（nα）.The main purpose of this paper is to establish a Bombieri-type mean-value theorem for Tk（x, α）, via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions. The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method展开更多
It is proved that every large integerN≡5 (mod 24) can be written as $N = p_1^2 + ... + p_5^2 $ with each primep j satisfying $|p_j - \sqrt {N/5} | \leqslant N^{\frac{{12}}{{25}} + E} $ , which gives a short interval ...It is proved that every large integerN≡5 (mod 24) can be written as $N = p_1^2 + ... + p_5^2 $ with each primep j satisfying $|p_j - \sqrt {N/5} | \leqslant N^{\frac{{12}}{{25}} + E} $ , which gives a short interval version of a classical theorem of Hua.展开更多
Let f（z） be a Hecke-Maass cusp form for SL2（Z）, and let L（s, f） be the corresponding automorphic L-function associated to f. For sufficiently large T, let N（σ, T） be the number of zeros p =β ＋ iγ of L（s, ...Let f（z） be a Hecke-Maass cusp form for SL2（Z）, and let L（s, f） be the corresponding automorphic L-function associated to f. For sufficiently large T, let N（σ, T） be the number of zeros p =β ＋ iγ of L（s, f） with ｜γ｜ ≤T, β〉 σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4≤ σ≤ 1 - ε, there exists a constant C = C（ε） such that N（σ, T） 〈〈 T2（1-σ）/σ（log T）c, which improves the previous results.展开更多
文摘Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.
基金Mathematical Tianyuan Foundation(No.10826028)National Natural Science Foundation of China(Grant No.10771127,10571107)
文摘In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L（s, f × χ）, where f is a holomorphic cusp form and χ（mod q） is a primitive character.
文摘In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-HoMey contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established
基金Supported by National Natural Science Foundation of China(Grant No.11271249)the First-class Discipline of Universities in Shanghai
文摘Let fk（n） be the characteristic function of n with Ω（n） = k, and T k（x,α）=∑n≤xfk（n）e（nα）.The main purpose of this paper is to establish a Bombieri-type mean-value theorem for Tk（x, α）, via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions. The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method
文摘It is proved that every large integerN≡5 (mod 24) can be written as $N = p_1^2 + ... + p_5^2 $ with each primep j satisfying $|p_j - \sqrt {N/5} | \leqslant N^{\frac{{12}}{{25}} + E} $ , which gives a short interval version of a classical theorem of Hua.
基金The author would like to thank Xu Zhao and the referees for carefully reading the manuscript and detailed comments. This work was supported by the National Natural Science Foundation of China （Grant No. 11126151） and the Scientific Foundation of Henan University （Grant No. 2012YBZR030）.
文摘Let f（z） be a Hecke-Maass cusp form for SL2（Z）, and let L（s, f） be the corresponding automorphic L-function associated to f. For sufficiently large T, let N（σ, T） be the number of zeros p =β ＋ iγ of L（s, f） with ｜γ｜ ≤T, β〉 σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4≤ σ≤ 1 - ε, there exists a constant C = C（ε） such that N（σ, T） 〈〈 T2（1-σ）/σ（log T）c, which improves the previous results.
