大泽塔和与黎曼ζ函数的零点
Large zeta sums and zeros of the Riemann zeta function
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中文总结 AI 辅助
该研究通过建立S(x,t)的大值与黎曼ζ函数零点的关联，利用Granville和Soundararajan的零点强制机制，推导了相关定量结论并得到S(x,t)的上界估计。
中文摘要 AI 辅助
对于实数t和x≥1，设S(x,t)=∑_{n≤x}n^{it}。我们证明了一个无条件逆定理，它将|t|很大时S(x,t)的大值与高度t附近的黎曼ζ函数（ζ(s)）的零点关联起来。更准确地说，若T≤|t|≤2T，exp(√log T)≤x≤√T，且|S(x,t)|=x/N，其中N≤(log x)^{1/100}，则对于每个满足cN⁶≤L≤(log x)/2的L，以1+iφ为中心且|φ-t|≪N的圆盘至少包含ζ(s)的L/360个零点。由此可得，在直线Re s=1左侧任意窄固定窗口构成的短族中，对零点施加定量限制，可在x的多项式范围内得到S(x,t)≪x/(log x)^{1/100}。该证明采用了Granville与Soundararajan用于大特征和的零点强制机制；在ζ函数的场景中，谱高度被t偏移，且高斯变换中会出现ζ(s)极点带来的额外余项，在本文考虑的范围内，该余项呈指数级小。
英文摘要
For real $t$ and $x\ge 1$, set \[ S(x,t)=\sum_{n\le x} n^{\ii t}. \] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$. More precisely, if $T\le |t|\le 2T$, $\exp(\sqrt{\log T})\le x\le \sqrt T$, and $|S(x,t)|=x/N$ with $N\le (\log x)^{1/100}$, then for every $cN^6\le L\le (\log x)/2$ a disk centered at $1+\iiϕ$, where $|ϕ-t|\ll N$, contains at least $L/360$ zeros of $ζ(s)$. As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line $\Ree s=1$ yields $S(x,t)\ll x/(\log x)^{1/100}$ in polynomial ranges of $x$. The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. In the zeta setting the spectral height is shifted by $t$, and an additional residue from the pole of $ζ(s)$ appears in the Gaussian transform; in the range considered here that residue is exponentially small.
发表机构
- Soochow University（苏州大学）
- Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科技大学）
- Shandong University（山东大学）
- Hunan University（湖南大学）
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