Peng Gao

Large values of quadratic Hecke $L$ -functions with prime-related moduli of the Gaussian field

Peng Gao

In this paper, we apply the resonance method to exhibit large values at the central point of the family of quadratic Hecke $L$ -functions with prime-related moduli of the Gaussian f…

Large values of cubic and quartic Dirichlet $L$ -functions of prime moduli

Peng Gao, 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 gen…

Murmurations of quadratic Hecke $L$ -functions of the Gaussian field

Peng Gao, Liangyi Zhao

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.

Large values of quadratic Dirichlet $L$ -functions of prime-related moduli

Peng Gao

In this paper, we apply the long resonator method to exhibit large values at the central point of the family of quadratic Dirichlet $L$ -functions of prime-related moduli under the…

High moments of random multiplicative functions twisted by Fourier coefficients of modular forms

Peng Gao, Liangyi Zhao

Let $λ (n)$ 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 gener…

Low moments of random multiplicative functions twisted by Fourier coefficients of modular forms

Peng Gao, Liangyi Zhao

Let $λ (n)$ 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 mag…

Shifted moments of modular $L$ -functions to a fixed level

Peng Gao, Liangyi Zhao

We establish upper bounds for shifted moments of modular $L$ -functions to a fixed prime level under the generalized Riemann hypothesis.

First Moment of Quadratic Hecke $L$ -Functions with Lower Order Term

Peng Gao, Liangyi Zhao

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…

Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms

Peng Gao, Liangyi Zhao

For any large prime $q$ , $x \leq 1$ and any real $k \geq 2$ , we prove a lower bound for the following $2 k$ -th moment \begin{equation*} \sum_{\substack{χ\in X_q^*}} \Big| \sum_{n\leq…

The mollified fourth moment of Dirichlet $L$ -functions

Peng Gao, Xiaosheng Wu, Liangyi Zhao

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 l…

Shifted moments of cubic and quartic Dirichlet $L$ -functions

Peng Gao, Liangyi Zhao

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…

Twisted fourth moment of Dirichlet $L$ -functions to a fixed modulus

Peng Gao, Liangyi Zhao

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 savin…

Bounds for Moments of Dirichlet $L$ -functions of fixed modulus on the critical line

Peng Gao, Liangyi Zhao

We study the $2 k$ -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 shar…

Twisted second moment of modular $L$ -functions to a fixed modulus

Peng Gao, Liangyi Zhao

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…

Upper bounds for moments of Dirichlet $L$ -functions to a fixed modulus

Peng Gao, Liangyi Zhao

We study the $2 k$ -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 $k \in [0, 2]$ .

Bounds for Moments of Twisted Fourier coefficients of Modular Forms

Peng Gao, Liangyi Zhao

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 hypothe…

Lower bounds for negative moments of Dirichlet $L$ -functions to a fixed modulus

Peng Gao

We establish lower bounds for the $2 k$ -th moment of central values of the family of primitive Dirichlet $L$ -functions to a fixed prime modulus for all real $k < 0$ , assuming the non-…

Lower bounds for shifted moments of Dirichlet $L$ -functions of fixed modulus

Peng Gao, Liangyi Zhao

We establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet $L$ -function of fixed modulus under the generalized Riemann hypothesis.

Upper bounds for shifted moments of quadratic Dirichlet $L$ -functions over function fields

Peng Gao, 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 Diric…

Upper bounds for shifted moments of Dirichlet $L$ -functions to a fixed modulus over function fields

Stephan Baier, Peng Gao

In this paper, we establish sharp upper bounds on shifted moments of the family of Dirichlet $L$ -functions to a fixed modulus over function fields. We apply the result to obtain up…

Shifted moments of quadratic Dirichlet $L$ -functions

Peng Gao, Liangyi Zhao

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 moment…

Upper bounds for moments of zeta sums

Peng Gao

We establish upper bounds for moments of zeta sums using results on shifted moments of the Riemann zeta function under the Riemann hypothesis.