AI 中文总结
本文提出新分析框架，研究欧拉的ℓ-全函数φ_ℓ(n)及其求和函数Φ_ℓ(x)的性质，依据Φ_ℓ(x)的渐近行为建立黎曼假设的充要判据。
AI 中文摘要
本文开发了一种用于研究黎曼假设的新分析框架。对每个固定整数ℓ≥1，定义欧拉的ℓ-全函数φ_ℓ(n)为φ_ℓ(n):=n∏_{p为素数，v_p(n)≥ℓ}(1−1/p)，其求和函数为Φ_ℓ(x):=∑_{n≤x}φ_ℓ(n)。对广义欧拉ℓ-全函数φ_ℓ(n)进行分析研究，包括其欧拉乘积表示、亚纯延拓及极点结构，针对每个ℓ，依据Φ_ℓ(x)的渐近行为建立了黎曼假设的充要判据。
英文摘要
This paper develops a new analytic framework for investigating the Riemann hypothesis. For each fixed integer $\ell \ge 1$, define Euler's $\ell$-totient function $φ_\ell$ by \[ φ_\ell(n):=n\prod_{\substack{p\ \mathrm{prime}\\ v_p(n)\ge \ell}}\left(1-\frac{1}{p}\right), \] and its summatory function by \[ Φ_\ell(x):=\sum_{n\le x}φ_\ell(n). \] An analytic study of the generalized Euler $\ell$-totient function $φ_\ell(n)$ is carried out, including its Euler product representation, meromorphic continuation, and pole structure. For each $\ell$, necessary and sufficient criteria for the Riemann hypothesis are established in terms of the asymptotic behavior of $Φ_\ell(x)$.
Comments16 pages
