We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of...We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli＇s numbers play a role similar to that of Euler＇s in the old ones. Our technique differs from that of Byrne-Smith （1997） and Berndt-Yeap （2002）.展开更多
The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power...The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean.展开更多
The analytical calculation of the area moments of inertia used for special mechanical tests in materials science and further generalizations for moments of different orders and broader symmetry properties has led to a...The analytical calculation of the area moments of inertia used for special mechanical tests in materials science and further generalizations for moments of different orders and broader symmetry properties has led to a new type of trigonometric power sums. The corresponding generalized equations are presented, proven, and their characteristics discussed. Although the power sums have a basic form, their results have quite different properties, dependent on the values of the free parameters used. From these equations, a large variety of power reduction formulas can be derived. This is shown by some examples.展开更多
In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences Jintroduced there is unusable for such sums and consequently the results obt...In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences Jintroduced there is unusable for such sums and consequently the results obtained in it are completely incorrect. We here introduce appropriate modified double cosine trigonometric sums making the class Jusable considering a particular double cosine trigonometric series.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
Letk be a positive integer and n a nonnegative integer,0 〈 λ1,...,λk＋1 ≤ 1 be real numbers and w =（λ1,λ2,...,λk＋1）.Let q ≥ max{[1/λi ]：1 ≤ i ≤ k ＋ 1} be a positive integer,and a an integer coprime to ...Letk be a positive integer and n a nonnegative integer,0 〈 λ1,...,λk＋1 ≤ 1 be real numbers and w =（λ1,λ2,...,λk＋1）.Let q ≥ max{[1/λi ]：1 ≤ i ≤ k ＋ 1} be a positive integer,and a an integer coprime to q.Denote by N（a,k,w,q,n） the 2n-th moment of（b1··· bk c） with b1··· bk c ≡ a（mod q）,1 ≤ bi≤λiq（i = 1,...,k）,1 ≤ c ≤λk＋1 q and 2（b1＋ ··· ＋ bk ＋ c）.We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N（a,k,w,q,n）,which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N（a,k,w,q,n） in the case ofw =（1/2,1/2,...,1/2） and n = 0.In order to show our result is close to the best possible,the mean-square value of N（a,k,q） φk（q）/2k＋2and the mean value weighted by the high-dimensional Cochrane sum are studied too.展开更多
Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent...Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent 33/5 obtained by X.D.CAO.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
文摘We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli＇s numbers play a role similar to that of Euler＇s in the old ones. Our technique differs from that of Byrne-Smith （1997） and Berndt-Yeap （2002）.
文摘The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean.
文摘The analytical calculation of the area moments of inertia used for special mechanical tests in materials science and further generalizations for moments of different orders and broader symmetry properties has led to a new type of trigonometric power sums. The corresponding generalized equations are presented, proven, and their characteristics discussed. Although the power sums have a basic form, their results have quite different properties, dependent on the values of the free parameters used. From these equations, a large variety of power reduction formulas can be derived. This is shown by some examples.
文摘In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences Jintroduced there is unusable for such sums and consequently the results obtained in it are completely incorrect. We here introduce appropriate modified double cosine trigonometric sums making the class Jusable considering a particular double cosine trigonometric series.
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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11001218,11201275)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20106101120001)the Natural Science Foundation of Shaanxi Province of China(Grant No.2011JQ1010)
文摘Letk be a positive integer and n a nonnegative integer,0 〈 λ1,...,λk＋1 ≤ 1 be real numbers and w =（λ1,λ2,...,λk＋1）.Let q ≥ max{[1/λi ]：1 ≤ i ≤ k ＋ 1} be a positive integer,and a an integer coprime to q.Denote by N（a,k,w,q,n） the 2n-th moment of（b1··· bk c） with b1··· bk c ≡ a（mod q）,1 ≤ bi≤λiq（i = 1,...,k）,1 ≤ c ≤λk＋1 q and 2（b1＋ ··· ＋ bk ＋ c）.We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N（a,k,w,q,n）,which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N（a,k,w,q,n） in the case ofw =（1/2,1/2,...,1/2） and n = 0.In order to show our result is close to the best possible,the mean-square value of N（a,k,q） φk（q）/2k＋2and the mean value weighted by the high-dimensional Cochrane sum are studied too.
基金Project supported by the National Natural Science Foundation of China
文摘Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent 33/5 obtained by X.D.CAO.
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
