arXiv subjects
Publications and source records attributed to Liangyi Zhao.
We establish sharp upper bounds on shifted moments of quadratic Dirichlet $L$-functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.
We establish sharp upper bounds for shifted moments of quadratic Dirichlet $L$-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.
We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for moments of sums involving with Fourier coefficients of a given modular form twisted by Dirichlet characters.
We evaluate the twisted first moment of central values of the family of primitive quadratic Dirichlet $L$-functions using the method of double Dirichlet series together with a recursive argument. Our main result is an asymptotic formula with an error term of size that is the square root of that of the main term.
We evaluate the first moment of central values of the family of quadratic Dirichlet $L$-functions using the method of double Dirichlet series. Under the generalized Riemann hypothesis, we prove an asymptotic formula with an error term of size that is the fourth root of that of the primary main term.
We evaluate asymptotically the negative first moment at points larger than $1/2$ of the family of quadratic twists of automorphic $L$-functions using multiple Dirichlet series under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture.
We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila.
We establish sharp lower bounds for the $2k$-th moment in the range $k \geq 1/2$ of the family of quadratic twists of modular $L$-functions at the central point.
In this paper, we establish the expected order of magnitude of the $k$th-moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus over function fields for all real $k \geq 0$.
We develop the ratios conjecture with one shift in the numerator and denominator in certain ranges for families of primitive quadratic Hecke $L$-functions of imaginary quadratic number fields with class number one using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain unconditional asymptotic formulas for the first moments of central values of these families of $L$-functions with error terms of size that is the square root of that of the primary main terms.
We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet $L$-functions, using the method of double Dirichlet series. Quantitative non-vanishing result for these $L$-values are also proved.
We develop the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular $L$-functions using multiple Dirichlet series under the generalized Riemann hypothesis.
We establish a subconvexity bound for a double Dirichlet series involving with the quadratic Hecke $L$-functions over the Gaussian field.
We develope the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke $L$-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of $L$-functions, obtaining an error term of size $O(X^{1/2+\varepsilon})$.
We evaluate the twisted first moment of central values of the product of a quadratic Dirichlet $L$-function and a quadratic twist of a modular $L$-function.
In this paper, we evaluate asymptotically the sum \[ \sum_{d \leq X} \left( \sum_{n \leq Y} μ(n)\leg {8d}{n} \right)^2, \] where $\leg {8d}{n}$ is the Kronecker symbol and $d$ runs over positive, odd, square-free integers.
In this paper, we investigate the one-level density of low-lying zeros of quadratic twists of automorphic $L$-functions under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture. We improve upon the known results using only functional equations for quadratic Dirichlet $L$-functions.
We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size $O(X^{1/4+\varepsilon})$ under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all $X$, $Y$ for a smoothed double character sum involving $\sum_{N(m) \leq X, N(n)\leq Y}\left( \frac{m}{n} \right)$, where $\left( \frac{\cdot}{n} \right)$ denotes the quadratic symbol in the Gaussian field.