AI 中文总结
本文证明有限Dirichlet和的零点垂直密度渐近集中于单条直线，给出一般准则并应用于zeta函数部分和、Dirichlet L-函数及Hecke本征形式，集中线为Re s=1/2或k/2。
AI 中文摘要
我们证明，广泛的有限Dirichlet和族在单条直线上表现出归一化垂直零密度的确定性集中。主要结果给出了一个一般性准则，在该准则下，归一化的Jessen势局部一致收敛到一个具有单个拐点的分段线性凸函数。描述零点实部在垂直平均密度中分布的关联归一化测度随后弱收敛到一个Dirac质量。因此，对于每个固定截断，零点可能占据一个非平凡的实部范围，而渐近地，它们的归一化垂直密度集中在一条直线上。证明将有限Bohr提升与孤立素数坐标的平移一致反集中估计相结合。我们将该准则应用于Riemann zeta函数的部分和、固定的Dirichlet $L$-函数以及固定水平、平凡Dirichlet特征且无复乘的原始全纯Hecke本征形式。在归一化设定中，集中线为$\noperatorname{Re} s=1/2$，而对于权重$k$的形式的经典Fourier系数，它变为$\noperatorname{Re} s=k/2$。
英文摘要
We prove that broad families of finite Dirichlet sums exhibit deterministic concentration of normalized vertical zero density on a single line. The main result gives a general criterion under which the normalized Jessen potentials converge locally uniformly to a piecewise-linear convex function with a single corner. The associated normalized measures describing the distribution of real parts of zeros in vertical mean density then converge weakly to a Dirac mass. Thus, for each fixed truncation, zeros may occupy a nontrivial range of real parts, while asymptotically their normalized vertical density concentrates on one line. The proof combines the finite Bohr lift with a translation-uniform anti-concentration estimate for isolated prime coordinates. We apply the criterion to partial sums of the Riemann zeta function, fixed Dirichlet $L$-functions, and primitive holomorphic Hecke eigenforms of fixed level, trivial Dirichlet character, and without complex multiplication. The concentration line is $\operatorname{Re} s=1/2$ in the normalized setting and becomes $\operatorname{Re} s=k/2$ for the classical Fourier coefficients of a form of weight $k$.
Comments28 pages
