论文标题
Maass-Form $ l $功能的立方时刻的强烈ivić猜想的证明
Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions
论文作者
论文摘要
在本文中，我们证明了以下渐近公式，用于中央$ l $值的光谱立方矩：$$ \ sum_ {t_f \ leqslant t} \ frac {2 l \ big（\ tfrac 1 2，f \ big）^3} {l（1，\ Mathrm {sym}^2 f）} + \ frac {2} + \ frac {2}π\ int_ {0}^0}^{t} ζ\ big（\ tfrac 1 2 + it \ big）\ right |^{6}}} {| ζ（1 + 2 IT）|^2} \ Mathrm {d} t = t^2 p_3（\ log t） + o（t^{1+ \ varepsilon}），$ f $在$ f $范围内以（偶数）hecke-maass cusp form的正常基础和$ p _ $ p _ $ p _ $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3。它在ivić的论文中改进了错误术语$ o（t^{8/7+\ varepsilon}）$，因此证实了他在立方时刻的强烈猜想。这是在立方体情况下（强）猜想第一次得到充分证明。此外，我们建立上述渐近公式的短间变体，长度短至$ t^{\ varepsilon} $。
In this paper, we prove the following asymptotic formula for the spectral cubic moment of central $L$-values: $$ \sum_{t_f \leqslant T} \frac {2 L \big( \tfrac 1 2 , f \big)^3} {L(1, \mathrm{Sym}^2 f)} + \frac {2} π \int_{0}^{T} \frac {\left| ζ\big(\tfrac 1 2 + it \big) \right|^{6} } { | ζ(1 + 2 it ) |^2 } \mathrm{d} t = T^2 P_3 (\log T) + O (T^{1+\varepsilon}) , $$ where $f$ ranges in an orthonormal basis of (even) Hecke--Maass cusp forms, and $P_3$ is a certain polynomial of degree $3$. It improves on the error term $O (T^{8/7+\varepsilon})$ in a paper of Ivić and hence confirms his strong conjecture for the cubic moment. This is the first time that the (strong) moment conjecture is fully proven in a cubic case. Moreover, we establish the short-interval variant of the above asymptotic formula on intervals of length as short as $T^{\varepsilon}$.
