arXiv subjects
Publications and source records attributed to Liangxun Li.
Let $H_k$ be an $L^2$-normalized Hecke basis for the space of all holomorphic cusp forms of weight $k$. We show that $\max_{f\in H_k}\Vert F\Vert_6\gg (\log\log k)^{\frac{1}{2}}$ where $F(z)=(\Im z)^{\frac{k}{2}}f(z).$ This confirms that the $L^6$-norm of Hecke eigenforms does not converge uniformly as the weight goes to infinity. We also give some results on the joint mass of degree $6$.
In this note, we generalize a 1950 result of P. Erd\H os on upper bounds of $k$-th moment of Romanoff type representation functions. As an application, we give a conditional proof of a recent conjecture of Y.-G. Chen on Romanoff type sumsets under the assumption of the Hardy-Littlewood conjecture.
Assume that the generalized Ramanujan conjecture holds on the automorphic $L$-function $L(s, π)$ on $\GL_d$ over $\mathbb{Q}$ with $d\geq 3$, we can obtain a small log-saving non-trivial bound on the second integral moment of $L(1/2+it, π)$. Specifically the bound \[ \int_{T}^{2T}\Big|L\big(\frac{1}{2}+it, π\big)\Big |^2 \dd t\ll_π \frac{T^{\frac{d}{2}}}{\log^{η_d}T} \] holds for a small constant $η_d>0$.
In this paper, we give the upper bounds on the variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series respectively. For the cusp form case, the bound comes from a large sieve inequality for symmetric cubes. We also give some nontrivial bounds for higher moments of symmetric cube $L$-functions. For the Eisenstein series case, the upper bound comes from Lindelöf-on-average type bounds for various $L$-functions. In particular, we establish the sharp upper bounds for the fourth moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ $L$-functions and the eighth moment of $\mathrm{GL}(2)$ $L$-functions around special points $1/2+it_j$. Our proof is based on the work of Chandee and Li \cite{C-L20} about bounding the second moment of $\mathrm{GL}(4)\times \mathrm{GL}(2)$ $L$-functions.
Assuming the Ramanujan conjecture, the zero density estimate and some subconvexity type bound, we describe a general method to obtain the log-saving upper bound for the second moment of standard twisted higher degree $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq3$. Under these foundational hypotheses. the bound \[ \sideset{}{^*}{\sum}_{χ\pmod q}\Big|L\big(\frac{1}{2}, F\times χ\big)\Big |^2\ll_{F,η} \frac{q^{\frac{d}{2}}}{\log^ηq} \] holds for some small $η>0$
In this paper, we formulate a conjecture on joint distribution of Hecke--Maass cusp forms. To support our conjecture, we prove two conditional results on joint moments of two Hecke--Maass cusp forms, which confirms statistical independence of orthogonal cusp forms.
In this paper, we prove asymptotic formulas of mixed moments of $\rm GL(2)$ and its symmetric square $L$-functions for both Hecke--Maass cusp forms and holomorphic Hecke eigenforms in short intervals. As an application, we prove quantitative simultaneous non-vanishing of central values of these $L$-functions.