The main purpose of this paper is using the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums,and give some interesting mean val...The main purpose of this paper is using the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums,and give some interesting mean value formulae and identities for it.展开更多
The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era,...The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.展开更多
Let q 〉 4 be an integer. The main purpose of this paper is to study the mean value of Cochrane sum C（a, q） in quarter intervals, and obtain a sharp asymptotic formula for it.
Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where ...Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.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
The main purpose of this paper is to use the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean v...The main purpose of this paper is to use the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.展开更多
Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 ＋ a1m0＋a2m0m1＋…＋atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠...Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 ＋ a1m0＋a2m0m1＋…＋atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠0.let each fi be a function defined on {0,1,2…,mi-1} with fi（0）=0.write S（n）=i=0∑tfi（ai）.In this paper, we give the asymptotic formula for x^-1∑n≤xS（n）^k,where k is a positive integer.展开更多
Thek-dimensional Piatetski-Shapiro prime number problem fork?3 is studied. Let π(x 1 c 1,?,c k ) denote the number of primesp withp?x, $p = [n_1^{c_1 } ] = \cdots [n_k^{c_k } ]$ , where 1<c 1<?<c k are fixed...Thek-dimensional Piatetski-Shapiro prime number problem fork?3 is studied. Let π(x 1 c 1,?,c k ) denote the number of primesp withp?x, $p = [n_1^{c_1 } ] = \cdots [n_k^{c_k } ]$ , where 1<c 1<?<c k are fixed constants. It is proved that π(x;c 1,?,c k ) has an asymptotic formula ifc 1 ?1 +?+c k ?1 >k?k/(4k 2+2).展开更多
Let p be an odd prime and let δ be a fixed real number with 0<δ<2.For an integer α with 0<α<p,denote by ā the unique integer between 0 and p satisfying aā≡1(mod p).Further, let{x}denote the fraction...Let p be an odd prime and let δ be a fixed real number with 0<δ<2.For an integer α with 0<α<p,denote by ā the unique integer between 0 and p satisfying aā≡1(mod p).Further, let{x}denote the fractional part of x.We derive an asymptotic formula for the number of pairs of integers(a,b)with 1≤a≤p-1,1≤b≤p-1,|{a^k/p}+{b^k/p}-{(?)~l/p}-{(?)~l/p}|<δ.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10671155)
文摘The main purpose of this paper is using the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums,and give some interesting mean value formulae and identities for it.
文摘The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.
基金supported by China Postdoctoral Science Foundation funded project (20080430202)the N.S.F.(10671155) of P.R.China
文摘Let q 〉 4 be an integer. The main purpose of this paper is to study the mean value of Cochrane sum C（a, q） in quarter intervals, and obtain a sharp asymptotic formula for it.
文摘Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.
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 National Natural Science Foundation of China （Grant No. 10671155） and Northwest University Innovation Fund （Grant No. 08YZZ30） The authors express their gratitude to the referee for his very helpful and detailed comments.
文摘The main purpose of this paper is to use the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.
基金Supported by National Natural Science Foundation of China (Grant No. 10771103)
文摘Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 ＋ a1m0＋a2m0m1＋…＋atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠0.let each fi be a function defined on {0,1,2…,mi-1} with fi（0）=0.write S（n）=i=0∑tfi（ai）.In this paper, we give the asymptotic formula for x^-1∑n≤xS（n）^k,where k is a positive integer.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19801021)the Natural Science Foundation of Shandong Province (Grant No. Q98A02110).
文摘Thek-dimensional Piatetski-Shapiro prime number problem fork?3 is studied. Let π(x 1 c 1,?,c k ) denote the number of primesp withp?x, $p = [n_1^{c_1 } ] = \cdots [n_k^{c_k } ]$ , where 1<c 1<?<c k are fixed constants. It is proved that π(x;c 1,?,c k ) has an asymptotic formula ifc 1 ?1 +?+c k ?1 >k?k/(4k 2+2).
文摘Let p be an odd prime and let δ be a fixed real number with 0<δ<2.For an integer α with 0<α<p,denote by ā the unique integer between 0 and p satisfying aā≡1(mod p).Further, let{x}denote the fractional part of x.We derive an asymptotic formula for the number of pairs of integers(a,b)with 1≤a≤p-1,1≤b≤p-1,|{a^k/p}+{b^k/p}-{(?)~l/p}-{(?)~l/p}|<δ.
