Xiaodong Cao
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Xiaodong Cao has a total of 9 co-authors. Here are the top co-authors by number of shared publications:
- Wenguang Zhai
Department of Mathematics, China University of Mining and Technology, Beijing 100083, P. R. China
16 shared publications - Yoshio Tanigawa
Graduate School of Mathematics, Nagoya University, Nagoya, Japan
11 shared publications Wenguang Zhai
2 shared publications- Xu You
Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, People’s Republic of China
2 shared publications - Chen Shuang1 shared publication
- Cristinel Mortici
Valahia University of Targoviste
1 shared publication
Latest Publications
- 17 May 2021
- Published by Springer Nature in The Ramanujan Journal
- Vol. 58 (2) , 597-631
On hybrid moments of $\Delta _2(x)$ and $\Delta _3(x)$- 17 May 2021
- Published by Springer Nature in The Ramanujan Journal
- Vol. 58 (2) , 597-631
On hybrid moments of $\Delta _2(x)$ and $\Delta _3(x)$- 16 April 2021
- Published by Springer Nature in Research in Number Theory
- Vol. 7 (2) , 1-16
Some mean value results related to Hardy’s function
- 16 April 2021
- Published by Springer Nature in Research in Number Theory
- Vol. 7 (2) , 1-16
Some mean value results related to Hardy’s function
- 1 January 2021
- Published by Institute of Mathematics, Polish Academy of Sciences in Acta Arithmetica
- Vol. 199 (3) , 237-251
Average of Hardy’s function at Gram points
Let $Z(t)=\chi ^{-1/2}(1/2+it)\zeta (1/2+it)=e^{i\theta (t)}\zeta (1/2+it)$ be Hardy’s function and $g(n)$ be the $n$th Gram point defined by $\theta (g(n))=\pi n$. Titchmarsh proved that $$ \sum _{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log ^{3/4}N) $$ and $$ \- 1 January 2021
- Published by Institute of Mathematics, Polish Academy of Sciences in Acta Arithmetica
- Vol. 199 (3) , 237-251
Average of Hardy’s function at Gram points
Let $Z(t)=\chi ^{-1/2}(1/2+it)\zeta (1/2+it)=e^{i\theta (t)}\zeta (1/2+it)$ be Hardy’s function and $g(n)$ be the $n$th Gram point defined by $\theta (g(n))=\pi n$. Titchmarsh proved that $$ \sum _{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log ^{3/4}N) $$ and $$ \- Open Access
Average of Hardy's function at Gram points
Let $Z(t)=\chi^{-1/2}(1/2+it)\zeta(1/2+it)=e^{i\theta(t)}\zeta(1/2+it)$ be Hardy's function and $g(n)$ be the $n$-th Gram points defined by $\theta(g(n))=\pi n$. Titchmarsh proved that $\sum_{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log^{3/4}N) $ and $\sum_{n \leq N} Z(g(2n+1)) =-2N+O(N^{3/4}\log^{3/4}N)$. We shall improve the error terms to $O(N^{1/4}\log^{3/4}N \log\log N)$.Open AccessAverage of Hardy's function at Gram points
Let $Z(t)=\chi^{-1/2}(1/2+it)\zeta(1/2+it)=e^{i\theta(t)}\zeta(1/2+it)$ be Hardy's function and $g(n)$ be the $n$-th Gram points defined by $\theta(g(n))=\pi n$. Titchmarsh proved that $\sum_{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log^{3/4}N) $ and $\sum_{n \leq N} Z(g(2n+1)) =-2N+O(N^{3/4}\log^{3/4}N)$. We shall improve the error terms to $O(N^{1/4}\log^{3/4}N \log\log N)$. - Open Access
Some mean value results related to Hardy's function
Let $\zeta(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)\zeta(1/2+it)dt$ and $\int_0^T Z^2(t) \zeta(1/2+it)dt$. Furthermore we derive an upper bound for $\int_0^T Z^3(t)\chi^{\alpha}(1/2+it)dt$ for $-1/2<\alpha<1/2$, where $\chi(s)$ is the function which appears in the functional equation of the Riemann zeta function: $\zeta(s)=\chi(s)\zeta(1-s)$.Open AccessSome mean value results related to Hardy's function
Let $\zeta(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)\zeta(1/2+it)dt$ and $\int_0^T Z^2(t) \zeta(1/2+it)dt$. Furthermore we derive an upper bound for $\int_0^T Z^3(t)\chi^{\alpha}(1/2+it)dt$ for $-1/2<\alpha<1/2$, where $\chi(s)$ is the function which appears in the functional equation of the Riemann zeta function: $\zeta(s)=\chi(s)\zeta(1-s)$. - Open Access
The MC algorithm and its applications
By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such as Catalan's constant, the exponential function, etc.
Open AccessThe MC algorithm and its applications
By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such as Catalan's constant, the exponential function, etc.
- 22 July 2016
- Published by Springer Nature in Science China Mathematics
- Vol. 59 (11) , 2103-2144
Tong-type identity and the mean square of the error term for an extended Selberg class
- 22 July 2016
- Published by Springer Nature in Science China Mathematics
- Vol. 59 (11) , 2103-2144
Tong-type identity and the mean square of the error term for an extended Selberg class
- Open Access
Multiple-correction and summation of the rational series
The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.
Open AccessMultiple-correction and summation of the rational series
The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.
- Open Access
Continued fraction expression of the Mathieu series
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
Open AccessContinued fraction expression of the Mathieu series
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
- Open Access
The fastest possible continued fraction approximations of a class of functions
The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are proposed.
Open AccessThe fastest possible continued fraction approximations of a class of functions
The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are proposed.