A few remarks on the Baez-Duarte Criterion

Abstract

We study exponentially damped Möbius approximants in $\mathcal{H}=L^2([1,\infty ),dt/t^{-2})$. 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 $⟨1\mid f(u)⟩\to 1$. Vasyunin's formula expresses $F(e^{-u})=\Vert f(u){\Vert }_2^2$ 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 \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] 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.

66 pages