AI 中文总结
该研究建立了高斯角Hecke L函数L(s,λ^k)的光滑加权移位二阶矩的一致渐近公式，通过转化非对角贡献为Weyl和等方法完成证明，所得公式与L函数比率猜想的四交换预测一致，误差为O_{Φ,ε}(K^{1/2+ε})。
AI 中文摘要
我们建立了高斯角Hecke L函数L(s,λ^k)的光滑加权移位二阶矩的一致渐近公式。完整矩被表示为四个显式主项的和，对应四个函数方程交换，误差为O_{Φ,ε}(K^{1/2+ε})。移除阿基米德因子后，所得公式与L函数比率猜想的四交换预测的仅分子特例一致。证明将非对角贡献转化为r²≡-1 mod C根上的Weyl和，通过在i处求值的不完整庞加莱级数将这些和谱实现，并分离艾森斯坦和Maaß谱。两个v型主项分别来自零频率和两个移动艾森斯坦极点的组合留数，而尖点谱被吸收到平方根误差项中。
英文摘要
We establish a uniform asymptotic formula for the smoothly weighted shifted second moment of the Gaussian angular Hecke $L$-functions $L(s,λ^k)$. The completed moment is expressed as the sum of four explicit main terms, corresponding to the four functional-equation swaps, with an error of size $O_{Φ,ε}(K^{1/2+ε})$.After the Archimedean factors are removed, the resulting formula agrees with the numerator-only specialization of the four-swap prediction of the $L$-functions Ratios Conjecture. The proof transforms the off-diagonal contribution into Weyl sums over the roots of $r^2\equiv-1\pmod C$, realizes these sums spectrally through incomplete Poincaré series evaluated at $i$, and separates the Eisenstein and Maaß spectra. The two $v$-type main terms arise respectively from the zero frequency and from the combined residues of two moving Eisenstein poles, while the cuspidal spectrum is absorbed into the square-root error term.
