AI 中文总结
研究狄利克雷L函数非共享零点问题，利用扭曲离散矩短窗口渐近性，结合围道积分与短区间估计，证明无狄利克雷L函数或其非平凡线性组合能在固定L(s,χ₀)每个零点处为零，对正密度x，非平凡组合在高短窗口某些零点处非零。
AI 中文摘要
我们证明了没有狄利克雷L函数（更一般地，没有狄利克雷L函数的非平凡有限线性组合）能在固定的L(s,χ₀)的每个零点处都为零。证明的核心是关于扭曲离散矩∑ρxρL(ρ,χ₁)的短窗口渐近性，其中χ₀≠χ₁是本原狄利克雷特征，ρ = β + iγ遍历L(s,χ₀)满足T - Δ < γ ≤ T的零点，且x∈Z。该渐近性是无条件的，对每个窗口宽度Δ∈[Te^(-C√logT),T/logT]都成立。通过改变x分离线性组合中每个L函数的贡献，我们推断出对于正密度的x∈N，在任何足够高的短窗口中，每个非平凡组合在L(s,χ₀)的某些零点处非零。证明结合了 -L′/L(1 - s,χ₀)L(s,χ₁)的围道积分与狄利克雷卷积(χ₀Λ)*χ₁的短区间估计。
英文摘要
We prove that no Dirichlet $L$-function (and more generally, no nontrivial finite linear combination of Dirichlet $L$-functions) can vanish at every zero of a fixed $L(s,χ_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_ρ x^ρ L(ρ,χ_1)$, where $χ_0\neχ_1$ are primitive Dirichlet characters, $ρ=β+iγ$ runs over zeros of $L(s,χ_0)$ with $T-Δ<γ\le T$, and $x\in\mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $Δ\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. Notably, the main term $\frac{χ_1(x)}{2π}\,Δ\log T$ depends on $x$ only through the single value $χ_1(x)$. Since distinct primitive characters are distinguished by their values, varying $x$ isolates the contribution of each $L$-function within a linear combination, and we deduce that for a positive density of $x\in\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,χ_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L'}{L}(1-s,\overlineχ_0)\,L(s,χ_1)$ with short-interval estimates for the Dirichlet convolution $(χ_0Λ)*χ_1$, which derive from the classical de la Vallée Poussin zero-free region.
Comments18 pages, 1 figure
