正则算术函数，第一卷：理论、应用与实例
Regular Arithmetic Functions, Volume I. Theory, Applications, Examples
浏览论文内容
中文总结 AI 辅助
本文介绍正则算术函数理论，通过单一定义方程导出核的正则指数，并应用于黎曼猜想（英厄姆核指数为1/2）及正交递归展开的衰减界改进，附十七个实例。
中文摘要 AI 辅助
这是关于正则算术函数的两卷著作中的第一卷，也是对其理论的介绍。正则算术函数（RAF）是一个核，其性质源于一个单一的定义方程。设$G(n,k)$为两个整数变量的函数，且$G(n,n)\neq 0$，对每个$\beta$，通过$\sum_{k\le n} a_k G(n,k)=n^{-\beta}$定义序列$(a_k)$，按秩逐项求解，无需任何假设，也无需建立收敛性。当部分和$\sum_{k\le n} a_k$在某个指数处改变行为时，$G$称为正则的：在该指数之下，它们复现强制速率；在该指数之上，它们吸收该速率。这个临界点就是正则指数$\alpha(G)$，一个属于核本身的量。该指数源于类比、实验和观察，本质上具有算术性质。其最显著的应用是黎曼猜想，该猜想成立当且仅当英厄姆核$G(n,k)=(k/n)\lfloor n/k\rfloor$是指数为$1/2$的RAF。这并非唯一应用。在一个性质相当不同的问题上，同一理论改进了单位元正交递归展开的已知衰减界。在画廊中详细展示了十七个核，以便读者掌握这一概念。哪些是猜想性的、条件性的或开放性的，均已标明并收录在登记表中。第二卷专门讨论该核，通过一个规范系统将其与Hasse-Weil zeta函数的世界联系起来。
英文摘要
This is the first of two volumes on regular arithmetic functions, and an introduction to their theory. A regular arithmetic function (RAF) is a kernel whose properties come from a single defining equation. Let $G(n,k)$ be a function of two integer variables with $G(n,n)\neq 0$, and for each $β$ define a sequence $(a_k)$ by $\sum_{k\le n} a_k G(n,k)=n^{-β}$, solved rank by rank with nothing to assume and no convergence to establish. Then $G$ is regular when the partial sums $\sum_{k\le n} a_k$ change behaviour at one exponent. Below it they reproduce the forced rate, above it they absorb it. That tipping point is the regularity index $α(G)$, a quantity belonging to the kernel itself. The index came out of analogies, experiment and observation, and it is arithmetic by nature. Its most visible application is the Riemann hypothesis, which holds if and only if Ingham's kernel $G(n,k)=(k/n)\lfloor n/k\rfloor$ is a RAF of index $1/2$. It is not the only one. On a problem of a quite different kind the same theory improves the known decay bound for the orthorecursive expansion of unity. Seventeen kernels are worked out in a gallery, so that the notion can be handled by the reader. What is conjectural, conditional or open is marked as such and collected in a register. Volume II is given over to that kernel alone, with its connection to the world of Hasse-Weil zeta functions through a gauged system.
