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.
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
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
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
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
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
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.
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 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
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.
t Let f（z） be a holomorphic Hecke eigencuspform of weight k for the full mod- ular group. Let Af（n） be the nth normalized Fourier coefficient of f（z）. Suppose that L（sym2f, s） is the symmetric square L-functio...t Let f（z） be a holomorphic Hecke eigencuspform of weight k for the full mod- ular group. Let Af（n） be the nth normalized Fourier coefficient of f（z）. Suppose that L（sym2f, s） is the symmetric square L-function associated with f（z）, and Asym2f（n） de- notes the nth coefficient L（sym2f, s）. In this paper, it is proved that where P2 （t） is a polynomial in t of degree 2. Similarly, it is obtained that where P2（t） is a polynomial in t of degree 2.展开更多
Let L（s, sym2f） be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL（2, Z）, with tf（n, 1） denoting the nthcoefficient of the Dirichlet series for it. It is proved that, for...Let L（s, sym2f） be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL（2, Z）, with tf（n, 1） denoting the nthcoefficient of the Dirichlet series for it. It is proved that, forN≥2and anyα∈ there exists an effective positive constant c such that ∑n≤N∧（n）tf（n,1）e（nα）〈〈N exp （-c√logN,where ∧（n） is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov＇s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.展开更多
Let f be a holomorphic Hecke cusp form with even integral weight k≥2 for the full modular group,and letχbe a primitive Dirichlet character modulo q.Let Lf(s,χ)be the automorphic L-function attached to f andχ-We st...Let f be a holomorphic Hecke cusp form with even integral weight k≥2 for the full modular group,and letχbe a primitive Dirichlet character modulo q.Let Lf(s,χ)be the automorphic L-function attached to f andχ-We study the mean-square estimate of Lf(s,χ)and establish an asymptotic formula.展开更多
Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈Sk(Γ).We establish that, for any ε > 0,1/Xintegral from n=1 to x|sum λ~2f^((n^2)) from n≤x to - c_2x|2dx ?ε X154/1...Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈Sk(Γ).We establish that, for any ε > 0,1/Xintegral from n=1 to x|sum λ~2f^((n^2)) from n≤x to - c_2x|2dx ?ε X154/101+ε,which improves previous results.展开更多
Letλ(n)be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z)∈Sk(Γ)for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivia...Letλ(n)be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z)∈Sk(Γ)for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum ofλ(n)over cubes,i.e.n xλ(n3).In this paper,we are able to use the analytic properties of symmetric power L-functions to solve his problem.More precisely,we prove thatΣn≤zλ(n3)【【x(3/4+ε),Σn≤zλ(n4)【【x(97+ε).展开更多
In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for t...In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for the above L-functions.展开更多
In this paper,we prove the conjectured order lower bound for the k-th moment of central values of quadratic twisted self-dual GL(3)L-functions for all k≥1,based on our recent work on the twisted first moment of centr...In this paper,we prove the conjectured order lower bound for the k-th moment of central values of quadratic twisted self-dual GL(3)L-functions for all k≥1,based on our recent work on the twisted first moment of central values in this family of L-functions.展开更多
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.
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 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.
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).
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.
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）.
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.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Project supported by the National Natural Science Foundation of China(Nos.10971119,11101249)the Shandong Provincial Natural Science Foundation of China(No.ZR2009AQ007)
文摘t Let f（z） be a holomorphic Hecke eigencuspform of weight k for the full mod- ular group. Let Af（n） be the nth normalized Fourier coefficient of f（z）. Suppose that L（sym2f, s） is the symmetric square L-function associated with f（z）, and Asym2f（n） de- notes the nth coefficient L（sym2f, s）. In this paper, it is proved that where P2 （t） is a polynomial in t of degree 2. Similarly, it is obtained that where P2（t） is a polynomial in t of degree 2.
文摘Let L（s, sym2f） be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL（2, Z）, with tf（n, 1） denoting the nthcoefficient of the Dirichlet series for it. It is proved that, forN≥2and anyα∈ there exists an effective positive constant c such that ∑n≤N∧（n）tf（n,1）e（nα）〈〈N exp （-c√logN,where ∧（n） is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov＇s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.
基金the National Natural Science Foundation of China(Grant No.11601309).
文摘Let f be a holomorphic Hecke cusp form with even integral weight k≥2 for the full modular group,and letχbe a primitive Dirichlet character modulo q.Let Lf(s,χ)be the automorphic L-function attached to f andχ-We study the mean-square estimate of Lf(s,χ)and establish an asymptotic formula.
基金supported by National Natural Science Foundation of China(Grant No.11101249)
文摘Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈Sk(Γ).We establish that, for any ε > 0,1/Xintegral from n=1 to x|sum λ~2f^((n^2)) from n≤x to - c_2x|2dx ?ε X154/101+ε,which improves previous results.
基金supported by National Natural Science Foundation of China(Grant No.10701048)Natural Science Foundation of Shandong Province(Grant No.ZR2009AM007)and IIFSDU
文摘Letλ(n)be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z)∈Sk(Γ)for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum ofλ(n)over cubes,i.e.n xλ(n3).In this paper,we are able to use the analytic properties of symmetric power L-functions to solve his problem.More precisely,we prove thatΣn≤zλ(n3)【【x(3/4+ε),Σn≤zλ(n4)【【x(97+ε).
文摘In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for the above L-functions.
基金National Key R&D Program of China(Grant No.2021YFA1000700)NSFC(Grant Nos.12001314 and 12031008)。
文摘In this paper,we prove the conjectured order lower bound for the k-th moment of central values of quadratic twisted self-dual GL(3)L-functions for all k≥1,based on our recent work on the twisted first moment of central values in this family of L-functions.
