Let f be a Hecke eigenform of even integral weight k for the full modular group SL_(2)(Z).Denote byλ_(f)(n)the n th normalized coefficient of f.The sum of Fourier coefficients of cusp form over the quadratic polynomi...Let f be a Hecke eigenform of even integral weight k for the full modular group SL_(2)(Z).Denote byλ_(f)(n)the n th normalized coefficient of f.The sum of Fourier coefficients of cusp form over the quadratic polynomial m^(2)+n^(2) is considered,i.e.,∑_( m^( 2)+n^( 2))≤λ^(2)_( f)(m^(2)+n^(2))=CX+O(X ^(337/491+ϵ)),here X large enough and C is a constant.展开更多
In this paper a zero-density estimate of the large sieve type is given for the automorphic L-function L f (s,χ),where f is a holomorphic cusp form and χ a Dirichlet character of mod q.
The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.
We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce new CAP(cuspidal associated to parabolics)representations of metaplectic,symplectic and orthogonal groups.These constructions partially gene...We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce new CAP(cuspidal associated to parabolics)representations of metaplectic,symplectic and orthogonal groups.These constructions partially generalize the theories of Waldspurger on the Shimura correspondence and of Piatetski-Shapiro on the Saito-Kurokawa lifting to higher dimensions.展开更多
Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)at...Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.展开更多
Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both...Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded,when k1^ε <<k^2<<k1^1-ε.These bounds are the mean Lindelof hypothesis in one case and subconvexity bounds on average in other cases.These square moment estimates also imply subconvexity bounds for individual L(1/2+it,f×g) for all g when f is chosen outside a small exceptional set.In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al.(2006) in both the k1 and k2 aspects.展开更多
Abstract Let λf（n） be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f（z） ∈ Sk（F）, In this paper, we established nontrivial estimates for ∑n≤x λf（n^i）λf（n^j）,where 1≤ij≤4.
For the normalized Fourier coefficients of Maass cusp forms λ（n） and the normalized Fourier coefficients of holomorphic cusp forms a（n）, we give the bound of
Let a（n） be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A（x） =∑n≤xa（n）.In this paper, we establish an asymptotic formula of tile fourth power mome...Let a（n） be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A（x） =∑n≤xa（n）.In this paper, we establish an asymptotic formula of tile fourth power moment of A（x） and prove that ∫1^TA^4（x）dx=3/64κπ^4s4;2（a^~）T^2κ＋O（T^2a-δ4＋t） with δ4 = 1/8, which improves the previous result.展开更多
Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x...Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.展开更多
Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag（n） be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag（n）e（an β） and prove that S1 has an asymptotic for...Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag（n） be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag（n）e（an β） and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag（n）e（an β） Ф（n/X） with Ф（x） ∈ C c ∞（0,＋∞） and prove that S2 has better upper bounds than S1 at some special α and β.展开更多
A dimension formula for the cusp form spaces defined on half-spaces of quaternions by Selberg trace formula is deduced and the contributions of several conjugacy classes to dimension formula are calculated.
In this note,we present a simple approach for bounding the shifted convolution sum involving the Fourier coefficients of half-integral weight holomorphic cusp forms and Maass cusp forms.
Letλ_(f×f)(n)be the nth Fourier coeficient of Rankin-Selberg L-function L(f×f,s).In this paper,we are interested in the average behavior of coeficients of Rankin-Selberg L-functions over sparse sequences,an...Letλ_(f×f)(n)be the nth Fourier coeficient of Rankin-Selberg L-function L(f×f,s).In this paper,we are interested in the average behavior of coeficients of Rankin-Selberg L-functions over sparse sequences,and establish the asymptotic formula ofΣ_(n≤xλ_(f×f)n^(m)).展开更多
在解析数论领域,探讨全纯尖形式的傅里叶系数具有重要的理论价值.通过复变函数的积分技巧,结合自守L-函数的凸界以及积分均值估计方法,研究了不同序列上傅里叶系数的一致均值估计,形式化表达为∑n≤xλf nλf n j,j=2,3,其中f是全模群Γ...在解析数论领域,探讨全纯尖形式的傅里叶系数具有重要的理论价值.通过复变函数的积分技巧,结合自守L-函数的凸界以及积分均值估计方法,研究了不同序列上傅里叶系数的一致均值估计,形式化表达为∑n≤xλf nλf n j,j=2,3,其中f是全模群Γ=SL(2,Z)上权为偶数k的Hecke特征型,λf n是其在尖点∞处傅里叶展开的第n个标准化傅里叶系数.展开更多
Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.
Ω results involving the coefficients of automorphic L-functions are important research object in analytic number theory.Let f be a primitive holomorphic cusp form.Denote by λ_(f×f)(n) the nth Fourier coefficien...Ω results involving the coefficients of automorphic L-functions are important research object in analytic number theory.Let f be a primitive holomorphic cusp form.Denote by λ_(f×f)(n) the nth Fourier coefficient of Rankin-Selberg L-function L(f×f,s).This paper combines Kühleitner and Nowak′s Omega theorem and the analytic properties of Rankin-Selberg L-functions to study Omega results for coefficients of Rankin-Selberg L-functions over sparse sequences,and establishes the asymptotic formula for Σ_(n≤x)λf×f(n^(m))(m=2,3).展开更多
基金Supported in part by the Natural Science Foundation of Henan Youth Foundation(Grant No.222300420034)National Natural Science Foundation of China(Grant No.11871193).
文摘Let f be a Hecke eigenform of even integral weight k for the full modular group SL_(2)(Z).Denote byλ_(f)(n)the n th normalized coefficient of f.The sum of Fourier coefficients of cusp form over the quadratic polynomial m^(2)+n^(2) is considered,i.e.,∑_( m^( 2)+n^( 2))≤λ^(2)_( f)(m^(2)+n^(2))=CX+O(X ^(337/491+ϵ)),here X large enough and C is a constant.
基金Supported by the NNSF of China（11071186）Supported by the Science Foundation for the Excellent Youth Scholars of Shanghai（ssc08017）Supported by the Doctoral Research Fund of Shanghai Ocean University
文摘In this paper a zero-density estimate of the large sieve type is given for the automorphic L-function L f (s,χ),where f is a holomorphic cusp form and χ a Dirichlet character of mod q.
文摘The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.
基金supported by JSPS Grant-in-Aid for Scientific Research(Gant Nos.(C)23K03055,(B)19H01778 and(A)22H00096).
文摘We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce new CAP(cuspidal associated to parabolics)representations of metaplectic,symplectic and orthogonal groups.These constructions partially generalize the theories of Waldspurger on the Shimura correspondence and of Piatetski-Shapiro on the Saito-Kurokawa lifting to higher dimensions.
文摘Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.
基金supported by National Natural Science Foundation of China(Grant No.11531008)Ministry of Education of China(Grant No.IRT16R43)+3 种基金Taishan Scholar Project of Shandong Provincesupported by National Natural Science Foundation of China(Grant No.11601271)China Postdoctoral Science Foundation(Grant No.2016M602125)China Scholarship Council(Grant No.201706225004)。
文摘Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded,when k1^ε <<k^2<<k1^1-ε.These bounds are the mean Lindelof hypothesis in one case and subconvexity bounds on average in other cases.These square moment estimates also imply subconvexity bounds for individual L(1/2+it,f×g) for all g when f is chosen outside a small exceptional set.In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al.(2006) in both the k1 and k2 aspects.
基金Supported by National Natural Science Foundation of China(Grant No.11101249)
文摘Abstract Let λf（n） be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f（z） ∈ Sk（F）, In this paper, we established nontrivial estimates for ∑n≤x λf（n^i）λf（n^j）,where 1≤ij≤4.
基金Acknowledgements This work was supported in part by the Natural Science Foundation of Jiangxi Province （Nos. 2012ZBAB211001, 20132BAB2010031）.
文摘For the normalized Fourier coefficients of Maass cusp forms λ（n） and the normalized Fourier coefficients of holomorphic cusp forms a（n）, we give the bound of
文摘Let a（n） be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A（x） =∑n≤xa（n）.In this paper, we establish an asymptotic formula of tile fourth power moment of A（x） and prove that ∫1^TA^4（x）dx=3/64κπ^4s4;2（a^~）T^2κ＋O（T^2a-δ4＋t） with δ4 = 1/8, which improves the previous result.
基金This work is supported by the National Natural Science Foundation of China （Grant No. 10701048）
文摘Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.
基金This work was supported in part by the Natural Science Foundation of Shandong Province （No. ZR2015AM016）.
文摘Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag（n） be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag（n）e（an β） and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag（n）e（an β） Ф（n/X） with Ф（x） ∈ C c ∞（0,＋∞） and prove that S2 has better upper bounds than S1 at some special α and β.
文摘A dimension formula for the cusp form spaces defined on half-spaces of quaternions by Selberg trace formula is deduced and the contributions of several conjugacy classes to dimension formula are calculated.
基金supported by US National Science Foundation (Grant No.DMS-0855600)
文摘In this note,we present a simple approach for bounding the shifted convolution sum involving the Fourier coefficients of half-integral weight holomorphic cusp forms and Maass cusp forms.
基金Supported by Natural Science Foundation of Shandong Province(No.ZR2024MA053)。
文摘Letλ_(f×f)(n)be the nth Fourier coeficient of Rankin-Selberg L-function L(f×f,s).In this paper,we are interested in the average behavior of coeficients of Rankin-Selberg L-functions over sparse sequences,and establish the asymptotic formula ofΣ_(n≤xλ_(f×f)n^(m)).
文摘在解析数论领域,探讨全纯尖形式的傅里叶系数具有重要的理论价值.通过复变函数的积分技巧,结合自守L-函数的凸界以及积分均值估计方法,研究了不同序列上傅里叶系数的一致均值估计,形式化表达为∑n≤xλf nλf n j,j=2,3,其中f是全模群Γ=SL(2,Z)上权为偶数k的Hecke特征型,λf n是其在尖点∞处傅里叶展开的第n个标准化傅里叶系数.
文摘Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.
文摘Ω results involving the coefficients of automorphic L-functions are important research object in analytic number theory.Let f be a primitive holomorphic cusp form.Denote by λ_(f×f)(n) the nth Fourier coefficient of Rankin-Selberg L-function L(f×f,s).This paper combines Kühleitner and Nowak′s Omega theorem and the analytic properties of Rankin-Selberg L-functions to study Omega results for coefficients of Rankin-Selberg L-functions over sparse sequences,and establishes the asymptotic formula for Σ_(n≤x)λf×f(n^(m))(m=2,3).
