arXiv subjects
Publications and source records attributed to Liangyi Zhao.
In this paper, we obtain asymptotic formulas for weighted first moments of central values of families of primitive quadratic Dirichlet $L$-functions whose conductors comprise only primes that split in a given quadratic number field. We then deduce a non-vanishing result of these $L$-functions at the point $s=1/2$.
In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.
In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(ω), ω=\exp \left( \frac {2πi}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$, and establish quantitative non-vanishing result for the $L$-values. We also establish an one level density result for the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$.
In this paper, we study the first moment of central values of Hecke $L$-functions associated with quartic characters.
In this paper, we prove some one level density results for the low-lying zeros of famliies of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that, respectively, at least $94.27 \%$ and $5\%$ of the members of the quadratic family and the quartic family do not vanish at the central point.
Suppose that $d \in \{ 2, 3, 4, 6 \}$ and $a \in \mathbb{Z}$ with $a\neq -1$ and $a$ is not square. Let $P_{(a,d)}$ be the number of primes $p$ not exceeding $x$ such that $p \equiv 1 \pmod{d}$ and $a^{(p-1)/d} \equiv 1 \pmod{p}$. In this paper, we study the mean value of $P_{(a,d)}$.
We prove a lower bound for the large sieve with square moduli.
Let $N$ be any fixed positive integer and define \begin{align*} S_N(x)=\sum_{m, n \leq x}d(n^2+Nm^2), \end{align*} where $d(n)$ is the divisor function. We evaluate asymptotically $S_N(x)$ for several $N$, extending earlier works of Gafurov and Yu on the case $N=1$.
In this paper, we evaluate a smoothed character sum involving $\sum_{m}\sum_{n}\leg {m}{n}$, with quadratic, cubic or quartic Hecke characters $\leg {m}{n}$, and the two sums over $m$ and $n$ are of comparable lengths.
We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace $t$ becomes close to uniform even when $t$ is averaged over very short intervals inside the Hasse-Weil interval.
Let $t \in \mathbb{N}$, $η>0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-η}$, $q$ not a multiple of the conductor of the exceptional character $χ^*$ (if it exists). Suppose further that, \[ \max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^δ, \] where $C$ and $δ$ are suitable positive constants depending on $t$ and $η$. Let $a \in \mathbb{Z}$, $(a,q)=1$ and \[ \mathcal{A} = \{n \in (x/2, x]: n \equiv a \pmod{q} \} . \] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $\mathcal{A}$ with \[ p_t - p_1 \ll qt \exp (\frac{40 t}{9-20 θ}) . \] Here $θ= (\log q) / \log x$.
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.
In this paper, we study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form $[g(n)]$.
In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.
In this paper, we develop a conditional subconvexity bound for Godement-Jacquet $L$-functions associated with Maass forms for $SL(3,Z)$.
In this paper, we prove some one level density results for low-lying zeros of families of $L$-functions. More specifically, the families under consideration are that of $L$-functions of holomorphic Hecke eigenforms of level 1 and weight $k$ twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet $L$-functions.
Let $λ(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $λ(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \leq N$.