A few remarks on the Baez-Duarte Criterion

We study exponentially damped Möbius approximants in $\mathcal{H}=L^2([1,\mathrm{\infty }),dt\mathrm{/}{t}^{-2})$. With $${\gamma }_n(t)=\lfloor \frac{t}{n}\rfloor -\frac{\lfloor t\rfloor }{n},f(u)(t)=\sum _{n\ge 1}\mu (n)e^{-nu}{\gamma }_n(t),$$ we compute the relevant scalar products, characterize the Möbius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $⟨1\mid f(u)⟩\to 1$. Vasyunin’s formula expresses $F(e^{-u})=\mathrm{\parallel }f(u){\mathrm{\parallel }}_2^2$ 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 ${\rho }_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and $${\mathcal{F}}_{[3]}(x)\ll \frac{{\log }^2\text{\hspace{0.17em}⁣}(e\mathrm{/}(1-x))}{1-x}\mathrm{.}$$ For the terms containing ${\rho }_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.