In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a se...In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.展开更多
Let q ≥ 3 be an integer, and χ be a Dirichlet character modulo q, L（s, χ） denote the Dirichlet L-function corresponding to χ. In this paper, we show some early information on the mean value of the Dirichlet L-fu...Let q ≥ 3 be an integer, and χ be a Dirichlet character modulo q, L（s, χ） denote the Dirichlet L-function corresponding to χ. In this paper, we show some early information on the mean value of the Dirichlet L-functions and give some new identities forwhere a = 2, 3, or 4. Then we give general identities for the case that the integer a divides q - 1. Keywords Dirichlet L-functions, Dedekind sum, trigonometric formula, MSbius inversion formula展开更多
In this paper, by making use of Abel’s theorem on power series, the reflection formula and the function equation for Hurwitz zeta function, we establish several expressions of Dirichlet Lfunction at positive integers...In this paper, by making use of Abel’s theorem on power series, the reflection formula and the function equation for Hurwitz zeta function, we establish several expressions of Dirichlet Lfunction at positive integers by means of some finite sums of different types. Some special cases as well as immediate consequences of the results presented here are also considered.展开更多
I. INTRODUCTIONFor an integer q】2, 1et x denote a typical Dirichlet character mod q, x<sub>0</sub> the principal character, and L（s, x） the corresponding Dirichlet L-function. The main purpose of this n...I. INTRODUCTIONFor an integer q】2, 1et x denote a typical Dirichlet character mod q, x<sub>0</sub> the principal character, and L（s, x） the corresponding Dirichlet L-function. The main purpose of this note is to give a more accurate asymptotic formula for the fourth power展开更多
A sharper asymptotic formula for the mean value sum from xmodq*L′(σ+it,X)L′(1-σ-it,X)1(where the summation is over all primitive Dirichlet characters mod q and 0<σ<1) is derived by using the analytic method...A sharper asymptotic formula for the mean value sum from xmodq*L′(σ+it,X)L′(1-σ-it,X)1(where the summation is over all primitive Dirichlet characters mod q and 0<σ<1) is derived by using the analytic method and the estimate of character sums.展开更多
For integer q≥3, let X denote a typical Dirichlet character mod q, and L(s, X)be the corresponding Dirichlet L-function. We define the function A(q, k)and B(q, k)as follows:
In this paper,we study multiplication operators on weighted Dirichlet spaces D_(β)(β∈R).Let n be a positive integer and β∈R,we show that the multiplication operator M_(z)^(n) on D_(β) is similar to the operator ...In this paper,we study multiplication operators on weighted Dirichlet spaces D_(β)(β∈R).Let n be a positive integer and β∈R,we show that the multiplication operator M_(z)^(n) on D_(β) is similar to the operator ⊕_(1)^(n)M_(z)on the space⊕_(1)^(n)D_(β).Moreover,we prove that M_(z)^(n)(≥2)on Dβis unitarily equivalent to ⊕_(1)^(n)M_(z) on⊕_(1)^(n)D_(β) if and only if β=0.In addition,we completely characterize the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces z^(k)D_(β)(k≥1),and the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces S_(j)(0≤j<n).Abkar,Cao and Zhu[Complex Anal Oper Theory,2020,14:Art 58]pointed out that it is an important,natural,and difficult question in operator theory to identify the commutant of a bounded linear operator.They characterized the commutant A′( M_(z)^(n))of M_(z)^(n)on a family of analytic function spaces A_(α)^(2)(α∈R)on D(in fact,the family of spaces A_(α)^(2)(α∈R)is the same with the family of spaces D_(β)(β∈R))in terms of the multiplier algebra of the underlying function spaces.In this paper,we give a new characterization of the commutant A′( M_(z)^(n))of M_(z)^(n)on D_(β),and characterize the self-adjoint operators and unitary operators in A'(M_(z)^(n)).We find that the class of self-adjoint operators(unitary operators)in A'(M_(z)^(n))when β≠0 is different from the class of self-adjoint operators(unitary operators)in A′( M_(z)^(n))when β=0.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
We consider the asymptotic behavior of solutions to the porous medium equation with a positive bistable type reaction term and Dirichlet boundary condition.We first prove a convergence result.Furthermore,by investigat...We consider the asymptotic behavior of solutions to the porous medium equation with a positive bistable type reaction term and Dirichlet boundary condition.We first prove a convergence result.Furthermore,by investigating families of initial data of the type{Ф_(σ)}σ>0,where Ф_(σ) belongs to an appropriate class of nonnegative compactly supported functions,we prove small spreading-big spreading dichotomy on the asymptotic behavior of the solutions.展开更多
文摘In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.
基金Supported by Basic Research Fund of the Northwestern Polytechnical University of China(Grant Nos.JC2011023 and JC2012252)
文摘Let q ≥ 3 be an integer, and χ be a Dirichlet character modulo q, L（s, χ） denote the Dirichlet L-function corresponding to χ. In this paper, we show some early information on the mean value of the Dirichlet L-functions and give some new identities forwhere a = 2, 3, or 4. Then we give general identities for the case that the integer a divides q - 1. Keywords Dirichlet L-functions, Dedekind sum, trigonometric formula, MSbius inversion formula
基金Supported by the National Natural Science Foundation of China(Grant No.11326050)
文摘In this paper, by making use of Abel’s theorem on power series, the reflection formula and the function equation for Hurwitz zeta function, we establish several expressions of Dirichlet Lfunction at positive integers by means of some finite sums of different types. Some special cases as well as immediate consequences of the results presented here are also considered.
文摘I. INTRODUCTIONFor an integer q】2, 1et x denote a typical Dirichlet character mod q, x<sub>0</sub> the principal character, and L（s, x） the corresponding Dirichlet L-function. The main purpose of this note is to give a more accurate asymptotic formula for the fourth power
基金Project supported by the National Natural Science Foundation of China.
文摘A sharper asymptotic formula for the mean value sum from xmodq*L′(σ+it,X)L′(1-σ-it,X)1(where the summation is over all primitive Dirichlet characters mod q and 0<σ<1) is derived by using the analytic method and the estimate of character sums.
文摘For integer q≥3, let X denote a typical Dirichlet character mod q, and L(s, X)be the corresponding Dirichlet L-function. We define the function A(q, k)and B(q, k)as follows:
基金supported by the National Natural Science Foundation of China(12101179,12171138,12171373)the Natural Science Foundation of Hebei Province of China(A2022207001)。
文摘In this paper,we study multiplication operators on weighted Dirichlet spaces D_(β)(β∈R).Let n be a positive integer and β∈R,we show that the multiplication operator M_(z)^(n) on D_(β) is similar to the operator ⊕_(1)^(n)M_(z)on the space⊕_(1)^(n)D_(β).Moreover,we prove that M_(z)^(n)(≥2)on Dβis unitarily equivalent to ⊕_(1)^(n)M_(z) on⊕_(1)^(n)D_(β) if and only if β=0.In addition,we completely characterize the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces z^(k)D_(β)(k≥1),and the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces S_(j)(0≤j<n).Abkar,Cao and Zhu[Complex Anal Oper Theory,2020,14:Art 58]pointed out that it is an important,natural,and difficult question in operator theory to identify the commutant of a bounded linear operator.They characterized the commutant A′( M_(z)^(n))of M_(z)^(n)on a family of analytic function spaces A_(α)^(2)(α∈R)on D(in fact,the family of spaces A_(α)^(2)(α∈R)is the same with the family of spaces D_(β)(β∈R))in terms of the multiplier algebra of the underlying function spaces.In this paper,we give a new characterization of the commutant A′( M_(z)^(n))of M_(z)^(n)on D_(β),and characterize the self-adjoint operators and unitary operators in A'(M_(z)^(n)).We find that the class of self-adjoint operators(unitary operators)in A'(M_(z)^(n))when β≠0 is different from the class of self-adjoint operators(unitary operators)in A′( M_(z)^(n))when β=0.
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The National Natural Science Foundation of China(12071299,12071300)。
文摘We consider the asymptotic behavior of solutions to the porous medium equation with a positive bistable type reaction term and Dirichlet boundary condition.We first prove a convergence result.Furthermore,by investigating families of initial data of the type{Ф_(σ)}σ>0,where Ф_(σ) belongs to an appropriate class of nonnegative compactly supported functions,we prove small spreading-big spreading dichotomy on the asymptotic behavior of the solutions.
