AI 中文总结
研究偶数ζ值狄利克雷级数\(D(s)\)的零点集，通过赫克型精确函数方程给出完整无条件描述，包括零点分类、分布规律等，未作黎曼型假设，完整呈现该级数零点特性。
AI 中文摘要
我们给出了狄利克雷级数\(D(s):=\sum_{n\geq1}\zeta(2n)n^{-s}\)零点集的完整无条件描述，该级数亚纯延拓到\(\mathbb{C}\)，在\(s = 1\)处有一个单极点。此级数既没有欧拉乘积也没有自对偶函数方程，如此完整的描述极为罕见。关键是由利普希茨求和公式得到的赫克型精确函数方程，它将左半平面的\(D\)表示为伽马因子乘以关于完全平方数复对数的对偶级数；黎曼函数方程是对偶级数的单列。零点分为四类。半平面\(\sigma\geq\sigma_0 = 1.5001\cdots\)无零点，\(D\)有一个实零点\(\rho_0 = 0.2004\cdots\)，推测是唯一实零点。临界带内的零点是\(\zeta\)在\(a\)接近\(-(\zeta(2)-1)\)时的扰动\(a\)点且计数函数服从黎曼 - 冯·曼戈尔特定律。其余零点形成两个复共轭串，沿明确射线退向左半平面，最终为单零点且满足渐近误差几何衰减。串的几何结构由对偶级数的两个最小频率\(2\log 2\)（由\(D\)的整函数部分贡献）和\(\pm 2\pi i\)（由\(\zeta\)贡献）的干涉决定。整个过程未作任何黎曼型假设。
英文摘要
We give a complete unconditional description of the zero set of the Dirichlet series $\mathbf{D}(s) = \sum_{n \ge 1} ζ(2n)n^{-s}$, which continues meromorphically to $\mathbb{C}$ with a single simple pole at $s = 1$. The series possesses neither an Euler product nor a self-dual functional equation, and descriptions with this level of completeness are exceedingly rare for such series. The key input is an exact functional equation of Hecke type, obtained from the Lipschitz summation formula, which expresses $\mathbf{D}$ in the left half-plane as a gamma factor times a dual series over the complex logarithms of the perfect squares; Riemann's functional equation appears as a single column of the dual series. The zeros fall into four families. The half-plane $σ\ge σ_0 = 1.5001\ldots$ is zero-free, and $\mathbf{D}$ has a unique real zero $ρ_0 = 0.2004\ldots$. The zeros in the critical strip are perturbed $a$-points of $ζ$ for values of $a$ near $-(ζ(2)-1)$, and their counting function obeys a Riemann--von Mangoldt law. The remaining zeros form two complex-conjugate strings that recede into the left half-plane along explicit rays, are eventually simple, and satisfy an asymptotic with geometrically decaying error. The string geometry is governed by interference between the two smallest frequencies of the dual series, $2\log 2$ contributed by the entire part of $\mathbf{D}$ and $\pm 2πi$ contributed by $ζ$. No hypothesis of Riemann type is assumed at any point.
Commentsv2: Conjecture 4.3 of v1 is now Proposition 4.3, proved following an observation of Qiping Zhou; two references added. No other result is affected. 26 pages, 4 figures, 3 tables
