arXiv subjects
Publications and source records attributed to Liangyi Zhao.
In this paper, we apply the resonator method to exhibit large values at the central point of the families of cubic and quartic Dirichlet $L$-functions of prime moduli under the generalized Riemann hypothesis.
In this paper, we evaluate the murmuration density for the family of quadratic Hecke $L$-functions of the Gaussian field under the generalized Riemann hypothesis.
Let $\lambda(n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine, under the generalized Riemann hypothesis, the order of magnitude of $\E|\sum_{n \leq x} h(n)\lambda(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, for all real $x, q$ with $1 \leq q \leq c\log x/\log\log x $ and $c>0$ a small constant.
Let $\lambda(n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine the order of magnitude of \[ \E|\sum_{n \leq x} h(n)\lambda(n)|^{2q} \] for real $x$, $q$ with $0 \leq q \leq 1$.
We establish upper bounds for shifted moments of modular $L$-functions to a fixed prime level under the generalized Riemann hypothesis.
We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindel\"of hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.
For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{\chi \in X_q^*}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k}, \end{equation*} where $X_q^*$ denotes the set of primitive Dirichlet characters modulo $q$ and $\lambda(n)$ the Fourier coefficients of a fixed modular form. The bound we obtain is sharp up to a constant factor under the generalized Riemann Hypothesis.
We prove an asymptotic formula with a power saving error term for the fourth moment of the family of Dirichlet $L$-functions to modulus $q$ mollified by a Dirichlet polynomial of length $q^{\frac1{22}-\ve}$, valid for all moduli $q\not\equiv2 \pmod 4$. This result was previously known only for restricted sets of moduli with smaller power savings. As a special case, when no Dirichlet polynomial is enclosed, this leads to a significant improvement on X. Wu's asymptotic evaluation of the fourth moment of Dirichlet $L$-functions at the cental point.
We establish upper bounds for shifted moments of cubic and quartic Dirichlet $L$-functions under the generalized Riemann hypothesis. As an application, we prove bounds for moments of cubic and quartic Dirichlet character sums.
We evaluate the twisted four moment on the critical line of the family of Dirichlet $L$-functions to a fixed prime power modulus, obtaining an asymptotic formula with a power saving error term.
We study the $2k$-th moment of the family of Dirichlet $L$-functions to a fixed prime modulus on the critical line and establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.
We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus and establish sharp upper bounds for all real $k \in [0,2]$.
We study asymptotically the twisted second moment of the family of modular $L$-functions to a fixed modulus. As an application, we establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$ for the $2k$-th moment of these $L$-functions on the critical line.
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 establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet $L$-function of fixed modulus under the generalized Riemann hypothesis.
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 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.