发表机构
Shandong University; The University of Hong Kong（山东大学; 香港大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广Nair--Tenenbaum估计至Kloosterman和的混合矩，证明其渐近公式，并给出Hecke特征值混合矩上界、大Kloosterman值频率及谱计数方差下界。
AI 中文摘要
我们将Nair--Tenenbaum关于非负算术函数在多项式取值处的短和估计推广到涉及Kloosterman和的混合矩。结合这些估计与通过改进Fouvry和Michel方法得到的下界，我们证明：对每个固定的非零整数\\(a\\)及固定的\\(z,\nu>0\\)，有\\[ \sum_{n\leq x}z^{\omega(n)} |\operatorname{Kl}(a;n)|^\nu \asymp_{a,z,\nu} x(\log x)^{z\mathfrak{s}(\nu)-1},\\]其中\\(\omega(n)\\)计数不同素因子的个数，\\(\mathfrak{s}(\nu)\\)是Sato--Tate测度的第\\(\nu\\)阶绝对矩。我们还获得了Hecke特征值与Kloosterman和的混合绝对矩的上界，在对数尺度上确定了大的Kloosterman值的频率，并建立了Li和Sarnak对经典Kloosterman和的对角二阶矩所预测的数量级。我们在对数范围内获得了模曲面上平滑谱计数方差的匹配界，以及下界\\(\int_T^{2T}S(t)^2\\,dt\gg T^2/\log T\\)，其中\\(S(t)\\)是Weyl定律中的余项。
英文摘要
We extend the Nair--Tenenbaum estimates for short sums of nonnegative arithmetic functions at polynomial values to mixed moments involving Kloosterman sums. Combining these estimates with lower bounds obtained by refining the method of Fouvry and Michel, we prove that, for every fixed nonzero integer \(a\) and fixed \(z,ν>0\), \[ \sum_{n\leq x}z^{ω(n)} |\operatorname{Kl}(a;n)|^ν\asymp_{a,z,ν} x(\log x)^{z\mathfrak{s}(ν)-1}, \] where \(ω(n)\) counts distinct prime divisors and \(\mathfrak{s}(ν)\) is the \(ν\)-th absolute moment of the Sato--Tate measure. We also obtain upper bounds for mixed absolute moments of Hecke eigenvalues and Kloosterman sums, determine the frequency of large Kloosterman values on a logarithmic scale, and establish the order of magnitude predicted by Li and Sarnak for the diagonal second moment of classical Kloosterman sums. We obtain matching bounds for the smoothed spectral counting variance on the modular surface in a logarithmic range and the lower bound \(\int_T^{2T}S(t)^2\,dt\gg T^2/\log T\), where \(S(t)\) is the remainder in Weyl's law.
Comments41 pages
