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 de-rived by using the analytic method an...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 de-rived by using the analytic method and the estimate of character sums.展开更多
We use the analytic methods and the properties of Gauss sums to study one kind mean value problems involving the classical Dedekind sums,and give an interesting identity and asymptotic formula for it.
In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class ...In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class of Dirichlet series L(s) which satisfies a functional equation. Let a(n) be an arithmetical function related to a Dirichlet series L(s), and let E(x) be the error term of ′n xa(n). In this paper, after introducing a class of Diriclet series with a general functional equation(which contains the well-known Selberg class), we establish a Tong-type identity and a Tong-type truncated formula for the error term of the Riesz mean of the coefficients of this Dirichlet series L(s). This kind of Tong-type truncated formula could be used to study the mean square of E(x) under a certain assumption. In other words, we reduce the mean square of E(x) to the problem of finding a suitable constant σ*which is related to the mean square estimate of L(s). We shall represent some results of functions in the Selberg class of degrees 2–4.展开更多
The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point <em>P</em> on the graph such that the tangent at <em>P</em> is paralle...The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point <em>P</em> on the graph such that the tangent at <em>P</em> is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point <em>Q</em> on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points <em>P </em>and <em>Q</em> always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.展开更多
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 de-rived by using the analytic method and the estimate of character sums.
基金supported by National Natural Science Foundation of China(Grant Nos.11001218 and 11071194)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20106101120001)
文摘We use the analytic methods and the properties of Gauss sums to study one kind mean value problems involving the classical Dedekind sums,and give an interesting identity and asymptotic formula for it.
基金supported by National Key Basic Research Program of China (Grant No. 2013CB834201)National Natural Science Foundation of China (Grant No. 11171344)+1 种基金Natural Science Foundation of Beijing (Grant No. 1112010)the Fundamental Research Funds for the Central Universities in China (Grant No. 2012YS01)
文摘In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class of Dirichlet series L(s) which satisfies a functional equation. Let a(n) be an arithmetical function related to a Dirichlet series L(s), and let E(x) be the error term of ′n xa(n). In this paper, after introducing a class of Diriclet series with a general functional equation(which contains the well-known Selberg class), we establish a Tong-type identity and a Tong-type truncated formula for the error term of the Riesz mean of the coefficients of this Dirichlet series L(s). This kind of Tong-type truncated formula could be used to study the mean square of E(x) under a certain assumption. In other words, we reduce the mean square of E(x) to the problem of finding a suitable constant σ*which is related to the mean square estimate of L(s). We shall represent some results of functions in the Selberg class of degrees 2–4.
文摘The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point <em>P</em> on the graph such that the tangent at <em>P</em> is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point <em>Q</em> on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points <em>P </em>and <em>Q</em> always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.
