关于Maass型$L$ -函数立方矩的强Ivić猜想的证明
Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions
摘要：
本文证明了中心$L$ -值的谱三阶矩的渐近公式：$$ \sum_{t_f \leqslant T} \frac {2 L \big( \tfrac 1 2 , f \big)^3} {L(1, \mathrm{Sym}^2 f)} + \frac {2} π \int_{0}^{T} \frac {\left| ζ\big(\tfrac 1 2 + it \big) \right|^{6} } { | ζ(1 + 2 it ) |^2 } \mathrm{d} t = T^2 P_3 (\log T) + O (T^{1+\varepsilon}) , $$其中$f$的范围是（偶）Hecke-Maass尖点形式的规范正交基，而$P_3$是$3$的某次多项式.它改善了错误项$O (T^{8/7+\varepsilon})$在一份文件中的Ivić，从而证实了他的强猜想的三次矩。这是第一次，（强）时刻猜想是完全证明了立方的情况下。此外，我们建立了上述渐近公式的短区间变式，其区间长度短至$T^{\varepsilon}$。
英文摘要：
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 $T^{\varepsilon}$.
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