AI 中文总结
该论文围绕贝兹 - 杜阿尔特准则展开，核心方法是推导相关引理，主要贡献是得出有趣引理。
AI 中文摘要
本文旨在推导一些与贝兹 - 杜阿尔特准则相关的非常有趣的引理。
英文摘要
We study exponentially damped Möbius approximants in $\mathscr H=L^2([1,\infty),dt/t^{-2})$. With \[ γ_n(t)=\left\lfloor\frac tn\right\rfloor -\frac{\lfloor t\rfloor}{n},\qquad f(u)(t)=\sum_{n\ge1}μ(n)e^{-nu}γ_n(t),\] we compute the relevant scalar products, characterize the Möbius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $\langle1 \mid f(u)\rangle\to1$. Vasyunin's formula expresses $F(e^{-u})=\|f(u)\|_2^2$ as an arithmetic cotangent sum. To analyze $F(x)$ as $x\uparrow 1$, we define the canonical third-order truncation $\mathcal F_{[3]}$ of $F$ by deleting the sole remainder $ρ_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] For the terms containing $ρ_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.
Comments66 pages
